{"id":16492,"date":"2022-08-19T00:43:19","date_gmt":"2022-08-18T23:43:19","guid":{"rendered":"https:\/\/complex-systems-ai.com\/?page_id=16492"},"modified":"2024-02-11T15:44:25","modified_gmt":"2024-02-11T14:44:25","slug":"exercices-algorithme-recursif","status":"publish","type":"page","link":"https:\/\/complex-systems-ai.com\/es\/algoritmico\/ejercicios-corregidos-algoritmo-recursivo\/","title":{"rendered":"20 ejercicios de algoritmo recursivo corregidos"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-page\" data-elementor-id=\"16492\" class=\"elementor elementor-16492\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-e68469d elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"e68469d\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-4b215eb\" data-id=\"4b215eb\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-2c6fae4 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"2c6fae4\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/complex-systems-ai.com\/algorithmique\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Algorithmique<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-ff18dc7\" data-id=\"ff18dc7\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-b4f6df9 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"b4f6df9\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/complex-systems-ai.com\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Page d'accueil<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-d9600ed\" data-id=\"d9600ed\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-f63a9f7 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"f63a9f7\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/en.wikipedia.org\/wiki\/Algorithm\" target=\"_blank\" rel=\"noopener\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Wiki<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-67f7df8 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"67f7df8\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-31fc7a0\" data-id=\"31fc7a0\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-d5794ab elementor-widget elementor-widget-heading\" data-id=\"d5794ab\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_82_2 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Contenus<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Alternar tabla de contenidos\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/complex-systems-ai.com\/es\/algoritmico\/ejercicios-corregidos-algoritmo-recursivo\/#Exercices-corriges-algorithme-recursif\" >Exercices corrig\u00e9s : algorithme r\u00e9cursif<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/complex-systems-ai.com\/es\/algoritmico\/ejercicios-corregidos-algoritmo-recursivo\/#Exercice-1\" >Exercice 1<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/complex-systems-ai.com\/es\/algoritmico\/ejercicios-corregidos-algoritmo-recursivo\/#Exercice-2\" >Exercice 2<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/complex-systems-ai.com\/es\/algoritmico\/ejercicios-corregidos-algoritmo-recursivo\/#Exercice-3\" >Exercice 3<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/complex-systems-ai.com\/es\/algoritmico\/ejercicios-corregidos-algoritmo-recursivo\/#Exercice-4\" >Exercice 4<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/complex-systems-ai.com\/es\/algoritmico\/ejercicios-corregidos-algoritmo-recursivo\/#Exercice-5\" >Exercice 5<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/complex-systems-ai.com\/es\/algoritmico\/ejercicios-corregidos-algoritmo-recursivo\/#Exercice-6\" >Exercice 6<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/complex-systems-ai.com\/es\/algoritmico\/ejercicios-corregidos-algoritmo-recursivo\/#Exercice-7\" >Exercice 7<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/complex-systems-ai.com\/es\/algoritmico\/ejercicios-corregidos-algoritmo-recursivo\/#Exercice-8\" >Exercice 8<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/complex-systems-ai.com\/es\/algoritmico\/ejercicios-corregidos-algoritmo-recursivo\/#Exercice-9\" >Exercice 9<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/complex-systems-ai.com\/es\/algoritmico\/ejercicios-corregidos-algoritmo-recursivo\/#Exercice-10\" >Exercice 10<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/complex-systems-ai.com\/es\/algoritmico\/ejercicios-corregidos-algoritmo-recursivo\/#Exercice-11\" >Exercice 11<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/complex-systems-ai.com\/es\/algoritmico\/ejercicios-corregidos-algoritmo-recursivo\/#Exercice-12\" >Exercice 12<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-14\" href=\"https:\/\/complex-systems-ai.com\/es\/algoritmico\/ejercicios-corregidos-algoritmo-recursivo\/#Exercice-13\" >Exercice 13<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-15\" href=\"https:\/\/complex-systems-ai.com\/es\/algoritmico\/ejercicios-corregidos-algoritmo-recursivo\/#Exercice-14\" >Exercice 14<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-16\" href=\"https:\/\/complex-systems-ai.com\/es\/algoritmico\/ejercicios-corregidos-algoritmo-recursivo\/#Exercice-15\" >Exercice 15<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-17\" href=\"https:\/\/complex-systems-ai.com\/es\/algoritmico\/ejercicios-corregidos-algoritmo-recursivo\/#Exercice-16\" >Exercice 16<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-18\" href=\"https:\/\/complex-systems-ai.com\/es\/algoritmico\/ejercicios-corregidos-algoritmo-recursivo\/#Exercice-17\" >Exercice 17<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-19\" href=\"https:\/\/complex-systems-ai.com\/es\/algoritmico\/ejercicios-corregidos-algoritmo-recursivo\/#Exercice-18\" >Exercice 18<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-20\" href=\"https:\/\/complex-systems-ai.com\/es\/algoritmico\/ejercicios-corregidos-algoritmo-recursivo\/#Exercice-19\" >Exercice 19<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-21\" href=\"https:\/\/complex-systems-ai.com\/es\/algoritmico\/ejercicios-corregidos-algoritmo-recursivo\/#Exercice-20\" >Exercice 20<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Exercices-corriges-algorithme-recursif\"><\/span>Exercices corrig\u00e9s : algorithme r\u00e9cursif<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-42d92ed elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"42d92ed\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-2db48f2\" data-id=\"2db48f2\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-463e36c elementor-widget elementor-widget-text-editor\" data-id=\"463e36c\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Les exercices corrig\u00e9s suivants concernent le principe d&rsquo;algorithme r\u00e9cursif, par exemple Fibonacci, les tours de Hano\u00ef et bien d&rsquo;autres cas <a href=\"https:\/\/complex-systems-ai.com\/es\/logica-matematica-27\/\">math\u00e9matiques<\/a>. Les exercices comprennent la r\u00e9cursion simple, la r\u00e9cursion terminale, la r\u00e9cursion crois\u00e9e ou r\u00e9cursion mutuelle, et le principe de m\u00e9mo\u00efsation (pour plus de d\u00e9tail sur ce dernier point merci d&rsquo;aller voir le principe de <a href=\"https:\/\/complex-systems-ai.com\/es\/algoritmico\/programacion-dinamica-2\/\">programmation dynamique<\/a>).<\/p><p><img decoding=\"async\" class=\"aligncenter wp-image-11096 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/09\/cropped-Capture.png\" alt=\"algorithme r\u00e9cursif\" width=\"97\" height=\"97\" title=\"\"><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-99b8970 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"99b8970\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-155fc23\" data-id=\"155fc23\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-3f02098 elementor-widget elementor-widget-heading\" data-id=\"3f02098\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Exercice-1\"><\/span>Exercice 1<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-cec9667 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"cec9667\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-642aeac\" data-id=\"642aeac\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-3b52c76 elementor-widget elementor-widget-text-editor\" data-id=\"3b52c76\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">R\u00e9\u00e9crire les algorithmes suivants sous forme r\u00e9cursive sous forme terminal quand c\u2019est possible.<\/span><\/p><p><img decoding=\"async\" class=\"alignnone wp-image-16517 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo31.png\" alt=\"algorithme r\u00e9cursif exercices corrig\u00e9s r\u00e9cursif terminal r\u00e9cursit\u00e9 multiple arbre d&#039;appels\" width=\"194\" height=\"201\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo31.png 194w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo31-12x12.png 12w\" sizes=\"(max-width: 194px) 100vw, 194px\" \/><\/p><p><img decoding=\"async\" class=\"alignnone wp-image-16518 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo32.png\" alt=\"algorithme r\u00e9cursif\" width=\"193\" height=\"251\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo32.png 193w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo32-9x12.png 9w\" sizes=\"(max-width: 193px) 100vw, 193px\" \/><\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-16519 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo33.png\" alt=\"algorithme r\u00e9cursif\" width=\"213\" height=\"190\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo33.png 213w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo33-13x12.png 13w\" sizes=\"(max-width: 213px) 100vw, 213px\" \/><\/p><p><span style=\"font-weight: 400;\">Un dernier pour la route. Cette fois il faut comprendre ce qu\u2019il fait avant de le rendre r\u00e9cursif terminal !<\/span><\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-16520 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo34.png\" alt=\"algorithme pgcd\" width=\"514\" height=\"299\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo34.png 514w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo34-300x175.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo34-18x10.png 18w\" sizes=\"(max-width: 514px) 100vw, 514px\" \/><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-1537c70 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"1537c70\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-c2703cf\" data-id=\"c2703cf\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-e9364ac elementor-widget elementor-widget-toggle\" data-id=\"e9364ac\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-2441\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-2441\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Solution<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-2441\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-2441\"><p>On consid\u00e8re que n est un entier positif ou nul.<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-16528 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo41.png\" alt=\"algorithme r\u00e9cursif terminal\" width=\"277\" height=\"181\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo41.png 277w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo41-18x12.png 18w\" sizes=\"(max-width: 277px) 100vw, 277px\" \/><\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-16529 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo42.png\" alt=\"algorithme r\u00e9cursif terminal\" width=\"493\" height=\"179\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo42.png 493w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo42-300x109.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo42-18x7.png 18w\" sizes=\"(max-width: 493px) 100vw, 493px\" \/><\/p><p><span style=\"font-weight: 400;\">algo3 est d\u00e9j\u00e0 r\u00e9cursif mais non terminal. Puisque l\u2019algorithme renvoit 2^<\/span><span style=\"font-weight: 400;\">n<\/span><span style=\"font-weight: 400;\">, on va changer sa structure pour le rendre terminal.<\/span><\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-16530 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo43.png\" alt=\"algorithme r\u00e9cursif terminal\" width=\"288\" height=\"180\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo43.png 288w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo43-18x12.png 18w\" sizes=\"(max-width: 288px) 100vw, 288px\" \/><\/p><p><span style=\"font-weight: 400;\">Pour le dernier <a href=\"https:\/\/complex-systems-ai.com\/es\/algoritmico\/\">algorithme<\/a>, il faut d\u2019abord comprendre qu\u2019il calcule le pgcd entre a et b. Faites un test \u00e0 la main, vous comprendrez que le premier embranchement ne sert \u00e0 rien, on va donc l\u2019enlever pour la r\u00e9cursion. La variable r n\u2019est pas pr\u00e9sente pour remplacer la valeur de a et b, donc elle ne rentrera pas dans la r\u00e9cursion.<\/span><\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-16531 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo44.png\" alt=\"algorithme pgcd\" width=\"261\" height=\"203\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo44.png 261w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo44-15x12.png 15w\" sizes=\"(max-width: 261px) 100vw, 261px\" \/><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-548c36c elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"548c36c\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-c7ce3c3\" data-id=\"c7ce3c3\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-acd3488 elementor-widget-divider--view-line elementor-widget elementor-widget-divider\" data-id=\"acd3488\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"divider.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-divider\">\n\t\t\t<span class=\"elementor-divider-separator\">\n\t\t\t\t\t\t<\/span>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-1533556 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"1533556\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-63b7965\" data-id=\"63b7965\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-1e4fd5e elementor-widget elementor-widget-heading\" data-id=\"1e4fd5e\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Exercice-2\"><\/span>Exercice 2<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-a07b8dc elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"a07b8dc\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-8b48f8e\" data-id=\"8b48f8e\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-e9c0ad6 elementor-widget elementor-widget-text-editor\" data-id=\"e9c0ad6\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Le probl\u00e8me de Fibonacci (1170 &#8211; 1250) :<\/span><\/p><p><span style=\"font-weight: 400;\">\u00ab\u00a0Poss\u00e9dant initialement un couple de lapins, combien de couples obtient-on en douze mois si chaque couple engendre tous les mois un nouveau couple \u00e0 compter du second mois de son existence ? Les lapins ne meurent jamais.\u201d<\/span><\/p><p><span style=\"font-weight: 400;\">Donc si on prend un exemple les 7 premiers mois.<\/span><\/p><table><tbody><tr><td><p><span style=\"font-weight: 400;\">Mois<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">1<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">2<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">3<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">4<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">5<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">6<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">7<\/span><\/p><\/td><\/tr><tr><td><p><span style=\"font-weight: 400;\">Couple<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">1<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">1<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">2<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">3<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">5<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">8<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">13<\/span><\/p><\/td><\/tr><\/tbody><\/table><p><span style=\"font-weight: 400;\">a &#8211; Exprimez la population de couples du mois n en fonction des mois pr\u00e9c\u00e9dents.<\/span><\/p><p><span style=\"font-weight: 400;\">b &#8211; Ecrire un algorithme it\u00e9ratif qui calcule le nombre de couples au bout de 12 mois.<\/span><\/p><p><span style=\"font-weight: 400;\">c &#8211; Modifier l\u2019algorithme it\u00e9ratif pour calculer le nombre de mois au bout duquel la population de couples de lapins d\u00e9passe 300.<\/span><\/p><p><span style=\"font-weight: 400;\">d &#8211; Ecrire l\u2019algorithme r\u00e9cursif pour calculer le nombre de couples de lapins apr\u00e8s n mois. <\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-8ae198f elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"8ae198f\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-51b6e78\" data-id=\"51b6e78\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-f94e4a4 elementor-widget elementor-widget-toggle\" data-id=\"f94e4a4\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-2611\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-2611\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Solution<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-2611\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-2611\"><p><span style=\"font-weight: 400;\">a &#8211;<\/span><\/p><p><span style=\"font-weight: 400;\">Le nombre de couples de lapins au mois n est la somme des couples existants le mois d\u2019avant (au mois n \u2212 1) et des couples naissants au mois n.<\/span><\/p><p><span style=\"font-weight: 400;\">Or, le texte pr\u00e9cise que tous les couples existants deux mois auparavant (au mois n \u2212 2) se reproduisent.<\/span><\/p><p><span style=\"font-weight: 400;\">Par cons\u00e9quent, au mois n \u2265 2 le nombre de lapins est \u00e9gale \u00e0 la somme du mois n-1 et n-2. Au mois 0 et au mois 1, il n\u2019existe qu\u2019un seul couple.<\/span><\/p><p><span style=\"font-weight: 400;\">b &#8211;<\/span><\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-16532\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo45.png\" alt=\"\" width=\"169\" height=\"277\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo45.png 169w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo45-7x12.png 7w\" sizes=\"(max-width: 169px) 100vw, 169px\" \/><\/p><p>c &#8211;<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-16533\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo46-160x300.png\" alt=\"\" width=\"160\" height=\"300\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo46-160x300.png 160w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo46-6x12.png 6w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo46.png 171w\" sizes=\"(max-width: 160px) 100vw, 160px\" \/><\/p><p>d &#8211;<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-16534 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo47.png\" alt=\"algorithme r\u00e9cursif fibonacci\" width=\"313\" height=\"172\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo47.png 313w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo47-300x165.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo47-18x10.png 18w\" sizes=\"(max-width: 313px) 100vw, 313px\" \/><\/p><p>Il est possible de faire Fibonacci sous forme terminale :<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-19808 size-medium\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/fibo-300x208.png\" alt=\"Fibonacci terminal r\u00e9cursive\" width=\"300\" height=\"208\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/fibo-300x208.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/fibo-18x12.png 18w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/fibo.png 313w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p><p>Ou alors en utilisant la m\u00e9mo\u00efsation :<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-19809 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/fibo2.png\" alt=\"fibonacci m\u00e9mo\u00efsation\" width=\"331\" height=\"203\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/fibo2.png 331w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/fibo2-300x184.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/fibo2-18x12.png 18w\" sizes=\"(max-width: 331px) 100vw, 331px\" \/><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-4cf7b2e elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"4cf7b2e\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-4408bb7\" data-id=\"4408bb7\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-ce6b6a1 elementor-widget-divider--view-line elementor-widget elementor-widget-divider\" data-id=\"ce6b6a1\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"divider.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-divider\">\n\t\t\t<span class=\"elementor-divider-separator\">\n\t\t\t\t\t\t<\/span>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-fa8a266 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"fa8a266\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-c532b07\" data-id=\"c532b07\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-45242e4 elementor-widget elementor-widget-heading\" data-id=\"45242e4\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Exercice-3\"><\/span>Exercice 3<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-73b1af6 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"73b1af6\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-b8e7ca9\" data-id=\"b8e7ca9\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-294ecb9 elementor-widget elementor-widget-text-editor\" data-id=\"294ecb9\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">a &#8211; Ecrire un algorithme qui calcule le maximum de 2 nombres r\u00e9els\u00a0<\/span><\/p><p><span style=\"font-weight: 400;\">b &#8211; Ecrire un algorithme r\u00e9cursif qui calcule le maximum de 3 nombres r\u00e9els en utilisant l&rsquo;algorithme du a. Montrer l\u2019arbre d\u2019appels.<\/span><\/p><p><span style=\"font-weight: 400;\">c &#8211; Ecrire un algorithme r\u00e9cursif\u00a0 qui calcule le maximum de 4 nombres r\u00e9els en utilisant l&rsquo;algorithme du a. Montrer l\u2019arbre d\u2019appels.<\/span><\/p><p>d &#8211; L&rsquo;algorithme est-il r\u00e9cursif ?<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-e79c953 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"e79c953\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-b647d67\" data-id=\"b647d67\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-cb629f2 elementor-widget elementor-widget-toggle\" data-id=\"cb629f2\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-2131\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-2131\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Solution<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-2131\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-2131\"><p>a &#8211;<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-16535 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo48.png\" alt=\"algorithme r\u00e9cursif maximum\" width=\"307\" height=\"200\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo48.png 307w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo48-300x195.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo48-18x12.png 18w\" sizes=\"(max-width: 307px) 100vw, 307px\" \/><\/p><p>b &#8211;<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-16536 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo49.png\" alt=\"algorithme r\u00e9cursif maximum\" width=\"374\" height=\"109\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo49.png 374w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo49-300x87.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo49-18x5.png 18w\" sizes=\"(max-width: 374px) 100vw, 374px\" \/><\/p><p><span style=\"font-weight: 400;\">Ici, Maximum2 fait office de condition terminal de l\u2019algorithme r\u00e9cursif. La pile d\u2019appels est la suivante :<\/span><\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-16537 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo50.png\" alt=\"algorithme r\u00e9cursif maximum\" width=\"529\" height=\"150\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo50.png 529w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo50-300x85.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo50-18x5.png 18w\" sizes=\"(max-width: 529px) 100vw, 529px\" \/><\/p><p>c &#8211;<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-16538 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo51.png\" alt=\"algorithme r\u00e9cursif maximum\" width=\"466\" height=\"106\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo51.png 466w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo51-300x68.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo51-18x4.png 18w\" sizes=\"(max-width: 466px) 100vw, 466px\" \/><\/p><p><span style=\"font-weight: 400;\">La pile d\u2019appels est la suivante :<\/span><\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-16539 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo52.png\" alt=\"algorithme r\u00e9cursif maximum\" width=\"514\" height=\"213\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo52.png 514w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo52-300x124.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo52-18x7.png 18w\" sizes=\"(max-width: 514px) 100vw, 514px\" \/><\/p><p>d &#8211; Bien qu&rsquo;on ait l&rsquo;impression de faire une r\u00e9cursion, il s&rsquo;agit plut\u00f4t d&rsquo;une surcharge de fonction ou d&rsquo;appel de fonction pour r\u00e9duire la taille du probl\u00e8me. Ce n&rsquo;est pas un algorithme r\u00e9cursif \u00e0 proprement parler.<\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-8fecfec elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"8fecfec\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-c1be450\" data-id=\"c1be450\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-f88e273 elementor-widget-divider--view-line elementor-widget elementor-widget-divider\" data-id=\"f88e273\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"divider.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-divider\">\n\t\t\t<span class=\"elementor-divider-separator\">\n\t\t\t\t\t\t<\/span>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-d09c54e elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"d09c54e\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-fe6c62e\" data-id=\"fe6c62e\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-7a66d92 elementor-widget elementor-widget-heading\" data-id=\"7a66d92\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Exercice-4\"><\/span>Exercice 4<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-1dcfc65 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"1dcfc65\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-3708eac\" data-id=\"3708eac\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-af4a52c elementor-widget elementor-widget-text-editor\" data-id=\"af4a52c\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">a &#8211; Est-ce que les algorithmes ci-dessous sont des algorithmes r\u00e9cursifs ?\u00a0<\/span><\/p><p><span style=\"font-weight: 400;\">b &#8211; Est-ce qu\u2019ils sont terminaux ?<\/span><\/p><p><span style=\"font-weight: 400;\">c &#8211; Est-ce qu&rsquo;ils se terminent ?<\/span><\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-16521 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo35.png\" alt=\"algorithme r\u00e9cursif log\" width=\"363\" height=\"241\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo35.png 363w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo35-300x199.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo35-18x12.png 18w\" sizes=\"(max-width: 363px) 100vw, 363px\" \/><\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-16522 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo36.png\" alt=\"algorithme r\u00e9cursif puissance\" width=\"418\" height=\"253\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo36.png 418w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo36-300x182.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo36-18x12.png 18w\" sizes=\"(max-width: 418px) 100vw, 418px\" \/><\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-16523 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo37.png\" alt=\"algorithme r\u00e9cursif somme\" width=\"350\" height=\"234\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo37.png 350w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo37-300x201.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo37-18x12.png 18w\" sizes=\"(max-width: 350px) 100vw, 350px\" \/><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-2a886a2 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"2a886a2\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-10c7eb5\" data-id=\"10c7eb5\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-c3bf688 elementor-widget elementor-widget-toggle\" data-id=\"c3bf688\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-2051\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-2051\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Solution<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-2051\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-2051\"><p><span style=\"font-weight: 400;\">a &#8211;<\/span><\/p><p><span style=\"font-weight: 400;\">Les algorithmes log et somme sont r\u00e9cursifs : chacun contient au moins un appel \u00e0 lui m\u00eame, par contre, puissance ne l\u2019est pas : il fait appel \u00e0 l\u2019algorithme puis. Il faut remplacer puis par puissance.<\/span><\/p><p><span style=\"font-weight: 400;\">b &#8211;\u00a0<\/span><\/p><p><span style=\"font-weight: 400;\">log est terminal, puissance et somme ne le sont pas car ils demandent un calcul lors de la r\u00e9cursion.<\/span><\/p><p><span style=\"font-weight: 400;\">c &#8211;<\/span><\/p><p><span style=\"font-weight: 400;\">log se termine pour tout entier x. L\u2019it\u00e9ration de la division enti\u00e8re par 2 m\u00e8ne \u00e0 0, et le cas de base 0 se termine par l\u2019ex\u00e9cution de retourner.\u00a0<\/span><\/p><p><span style=\"font-weight: 400;\">Pour l\u2019algorithme puissance (corrig\u00e9), le cas de base 0 se termine par l\u2019exc\u00e9ution de retourner. Si l\u2019algorithme se termine pour la valeur n\u22121 alors il se termine aussi pour la valeur n est ex\u00e9cutant retourner. N\u2019oubliez pas qu\u2019un utilisateur peut rentrer n\u2019importe quoi comme puissance(2, &#8211; 5) donc il faut bien v\u00e9rifier sa <a href=\"https:\/\/complex-systems-ai.com\/es\/algoritmico\/terminacion-y-correccion-2\/\">terminaison<\/a>.<\/span><\/p><p><span style=\"font-weight: 400;\">somme ne se termine pas lorsque n est strictement positif. En effet, l\u2019algorithme pour n se termine seulement si l\u2019algorithme se termine pour n+1. Or, il n\u2019existe pas d\u2019entier strictement positif pour lesquels l\u2019agorithme s\u2019arr\u00eate.<\/span><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-d9157eb elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"d9157eb\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-bb2792d\" data-id=\"bb2792d\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-6682648 elementor-widget-divider--view-line elementor-widget elementor-widget-divider\" data-id=\"6682648\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"divider.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-divider\">\n\t\t\t<span class=\"elementor-divider-separator\">\n\t\t\t\t\t\t<\/span>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-02bd916 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"02bd916\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-aa2024f\" data-id=\"aa2024f\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-1dad5e4 elementor-widget elementor-widget-heading\" data-id=\"1dad5e4\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Exercice-5\"><\/span>Exercice 5<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-4afaede elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"4afaede\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-99abf4e\" data-id=\"99abf4e\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-86de030 elementor-widget elementor-widget-text-editor\" data-id=\"86de030\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">a &#8211; On souhaite \u00e9crire une fonction r\u00e9cursive qui calcule le carr\u00e9 d&rsquo;un entier. Pour cela il va falloir se servir d\u2019une relation entre carre(n) et carre(n-1) sachant que carre(1)=1. Repr\u00e9senter la pile d\u2019appels pour carre(5).<\/span><\/p><p><span style=\"font-weight: 400;\">b &#8211; Les combinaisons sont un concept de math\u00e9matiques d\u00e9crivant les diff\u00e9rentes fa\u00e7ons de choisir un nombre donn\u00e9 d&rsquo;objets dans un ensemble de taille donn\u00e9e. Par exemple combien il est possible de tirer 6 boules de Loto sur 60 dans le moulinet. Ou de piocher 3 cartes dans un jeu de tarot. La combinaison de choisir p \u00e9l\u00e9ments parmi n \u00e9l\u00e9ments se notent C(p,n)=(n\/p)*C(p-1, n-1). On a aurri C(0,n)=1 et C(n,n)=1.\u00a0<\/span><\/p><p><span style=\"font-weight: 400;\">Ecrire la fonction r\u00e9cursive non terminale puis la fonction r\u00e9cursive terminale, et montrer la pile d\u2019appels pour C(3,7). Via la fonction terminale, en d\u00e9duire la fonction it\u00e9rative.<\/span><\/p><p><span style=\"font-weight: 400;\">c &#8211; Pour les combinaisons, on pr\u00e9f\u00e8re g\u00e9n\u00e9ralement \u00e9viter la m\u00e9thode du b car les divisions peuvent cr\u00e9er des probl\u00e8mes d\u2019arrondis. On va plut\u00f4t se servir de la relation suivante : C(p,n)=C(p,n-1)+C(p-1, n-1). Ecrire la fonction r\u00e9cursive et la pile ou <a href=\"https:\/\/complex-systems-ai.com\/es\/teoria-de-grafos\/arboles-y-arboles\/\">arbre<\/a> d\u2019appels pour C(2,4).<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-7b0e67d elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"7b0e67d\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-5a8cc86\" data-id=\"5a8cc86\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-fe66652 elementor-widget elementor-widget-toggle\" data-id=\"fe66652\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-2661\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-2661\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Solution<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-2661\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-2661\"><p><span style=\"font-weight: 400;\">a &#8211; Pour trouver un lien entre carre(n) et carre(n-1), on utilise la formule : (n+ 1)\u00b2 = n\u00b2 + 2n+ 1. On l\u2019\u00e9crit en fonction de n vers n-1 : n\u00b2 = (n\u22121)\u00b2+2(n\u22121)+1 d\u2019o\u00f9 carre(n)=carre(n-1)+2*n-1. Le tour est jou\u00e9 !<\/span><\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-16540 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo53.png\" alt=\"algorithme r\u00e9cursif carr\u00e9\" width=\"233\" height=\"180\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo53.png 233w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo53-16x12.png 16w\" sizes=\"(max-width: 233px) 100vw, 233px\" \/><\/p><p><span style=\"font-weight: 400;\">La pile d\u2019appels est :<\/span><\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-16541 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo54.png\" alt=\"algorithme r\u00e9cursif carr\u00e9\" width=\"392\" height=\"252\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo54.png 392w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo54-300x193.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo54-18x12.png 18w\" sizes=\"(max-width: 392px) 100vw, 392px\" \/><\/p><p><span style=\"font-weight: 400;\">b &#8211; Tout est dit dans l\u2019\u00e9nonc\u00e9, il ne reste qu\u2019\u00e0 l\u2019\u00e9crire.<\/span><\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-16542 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo55.png\" alt=\"algorithme r\u00e9cursif combinaison\" width=\"234\" height=\"181\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo55.png 234w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo55-16x12.png 16w\" sizes=\"(max-width: 234px) 100vw, 234px\" \/><\/p><p><span style=\"font-weight: 400;\">Voici la liste d\u2019appels pour C(3,7) :<\/span><\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-16543 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo56.png\" alt=\"algorithme r\u00e9cursif combinaison\" width=\"316\" height=\"205\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo56.png 316w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo56-300x195.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo56-18x12.png 18w\" sizes=\"(max-width: 316px) 100vw, 316px\" \/><\/p><p><span style=\"font-weight: 400;\">Et maintenant pour la version terminale, c\u2019est un peu plus compliqu\u00e9 car il faut se questionner sur ce qu\u2019on doit garder en m\u00e9moire pour la r\u00e9cursion. A chaque it\u00e9ration on a fait n\/p fois le pr\u00e9c\u00e9dent, donc n\/p * (n-1)\/(p-1). On aurait pu l\u2019\u00e9crire n(n-1)\/p(p-1). Donc on pourrait stocker d\u2019un c\u00f4t\u00e9 la multiplication de n d\u00e9croissant et de l\u2019autre de p d\u00e9croissant.<\/span><\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-16544 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo57.png\" alt=\"algorithme r\u00e9cursif combinaison\" width=\"402\" height=\"179\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo57.png 402w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo57-300x134.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo57-18x8.png 18w\" sizes=\"(max-width: 402px) 100vw, 402px\" \/><\/p><p><span style=\"font-weight: 400;\">Voici la pile d\u2019appels pour C(3,7,1,1) :<\/span><\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-16545 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo58.png\" alt=\"algorithme r\u00e9cursif combinaison\" width=\"365\" height=\"122\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo58.png 365w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo58-300x100.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo58-18x6.png 18w\" sizes=\"(max-width: 365px) 100vw, 365px\" \/><\/p><p><span style=\"font-weight: 400;\">On d\u00e9duit de la fonction terminale la fonction it\u00e9rative. Il suffit de faire un Tant Que la condition d\u2019arr\u00eat n\u2019est pas atteinte.<\/span><\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-16546 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo59.png\" alt=\"algorithme r\u00e9cursif combinaison\" width=\"234\" height=\"303\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo59.png 234w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo59-232x300.png 232w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo59-9x12.png 9w\" sizes=\"(max-width: 234px) 100vw, 234px\" \/><\/p><p><span style=\"font-weight: 400;\">c &#8211; On a d\u00e9j\u00e0 la relation r\u00e9cursive dans l\u2019\u00e9nonc\u00e9.<\/span><\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-16547 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo60.png\" alt=\"algorithme r\u00e9cursif combinaison\" width=\"276\" height=\"187\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo60.png 276w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo60-18x12.png 18w\" sizes=\"(max-width: 276px) 100vw, 276px\" \/><\/p><p><span style=\"font-weight: 400;\">Voici l\u2019arbre d\u2019appels pour C(2,4) :<\/span><\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-16548 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo61.png\" alt=\"algorithme r\u00e9cursif combinaison\" width=\"566\" height=\"318\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo61.png 566w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo61-300x169.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo61-18x10.png 18w\" sizes=\"(max-width: 566px) 100vw, 566px\" \/><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-78ad094 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"78ad094\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-31ac958\" data-id=\"31ac958\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-0f5589c elementor-widget-divider--view-line elementor-widget elementor-widget-divider\" data-id=\"0f5589c\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"divider.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-divider\">\n\t\t\t<span class=\"elementor-divider-separator\">\n\t\t\t\t\t\t<\/span>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-3702014 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"3702014\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-11d7197\" data-id=\"11d7197\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-78490ca elementor-widget elementor-widget-heading\" data-id=\"78490ca\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Exercice-6\"><\/span>Exercice 6<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-95504b1 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"95504b1\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-bdc64c9\" data-id=\"bdc64c9\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-58171eb elementor-widget elementor-widget-text-editor\" data-id=\"58171eb\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">On d\u00e9finit la fonction de McCarthy comme suivant :<\/span><\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-16524 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo38.png\" alt=\"algorithme de mccarthy\" width=\"265\" height=\"177\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo38.png 265w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo38-18x12.png 18w\" sizes=\"(max-width: 265px) 100vw, 265px\" \/><\/p><p><span style=\"font-weight: 400;\">Que fait la fonction pour n&gt;100 ? pour n=98, 99, 100 ? En g\u00e9n\u00e9ral pour n&lt;=100 ?<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-48eedc8 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"48eedc8\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-cb3cbda\" data-id=\"cb3cbda\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-3081e7b elementor-widget elementor-widget-toggle\" data-id=\"3081e7b\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-5081\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-5081\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Solution<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-5081\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-5081\"><p><span style=\"font-weight: 400;\">On va faire un peu de math\u00e9matique\u2026 Si n &gt; 100 on aura n-10. Le probl\u00e8me se pose quand on commence avec n&lt;=100. Calculer pour 98, 99 et 100. On va faire une d\u00e9monstration directe.<\/span><\/p><p><span style=\"font-weight: 400;\">Pour n entre 90 et 100 compris \u2013 soit 11 valeur cons\u00e9cutive, \u00e7a nous servira plus tard. On fera Carthy(Carthy(n+11)) donc on d\u00e9passera la valeur de 100. Donc on aura Carthy(n+1) au final. Si n+1 n\u2019est toujours pas sup\u00e9rieur \u00e0 100, on remet le couvert. Donc on s\u2019arr\u00eate quand on passera de 100 \u00e0 101, or Carthy(101)=101-10=91.<\/span><\/p><p><span style=\"font-weight: 400;\">En reprend ce qu\u2019on a laiss\u00e9 au d\u00e9but du pr\u00e9c\u00e9dent paragraphe. Sur les 11 derni\u00e8res valeurs cons\u00e9cutives, on aura un r\u00e9sultat de 91. On note que tout nombre entre 0 et 89, auquel on rajoute 11 tant que c\u2019est inf\u00e9rieur \u00e0 90 aura une valeur finale entre 90 et 100 compris.<\/span><\/p><p><span style=\"font-weight: 400;\">On en d\u00e9duit que Carthy sur une valeur de 100 ou moins donnera la valeur de 91.<\/span><\/p><p><span style=\"font-weight: 400;\">Il fallait bien un peu de math\u00e9matique. L\u2019informatique n\u2019est qu\u2019une autre fa\u00e7on syntaxique de faire des maths !<\/span><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-c2d9f28 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"c2d9f28\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-1756d51\" data-id=\"1756d51\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-b58b6b2 elementor-widget-divider--view-line elementor-widget elementor-widget-divider\" data-id=\"b58b6b2\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"divider.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-divider\">\n\t\t\t<span class=\"elementor-divider-separator\">\n\t\t\t\t\t\t<\/span>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-93a392d elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"93a392d\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-308df3d\" data-id=\"308df3d\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-193aac6 elementor-widget elementor-widget-heading\" data-id=\"193aac6\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Exercice-7\"><\/span>Exercice 7<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-54e7d26 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"54e7d26\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-7d6832a\" data-id=\"7d6832a\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-aa23df9 elementor-widget elementor-widget-text-editor\" data-id=\"aa23df9\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">On parle de <a href=\"https:\/\/complex-systems-ai.com\/es\/algoritmico\/programacion-recursiva\/\">r\u00e9cursivit\u00e9<\/a> crois\u00e9e lorsque deux fonctions s\u2019appellent l\u2019une l\u2019autre r\u00e9cursivement.<\/span><\/p><p><span style=\"font-weight: 400;\">On va tester sur une r\u00e9cursivit\u00e9 crois\u00e9e pour savoir si un nombre est pair (vrai) ou impair (faux). Voici la proposition :<\/span><\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-16525 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo39.png\" alt=\"algorithme r\u00e9cursif crois\u00e9\" width=\"228\" height=\"380\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo39.png 228w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo39-180x300.png 180w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo39-7x12.png 7w\" sizes=\"(max-width: 228px) 100vw, 228px\" \/><\/p><p><span style=\"font-weight: 400;\">Testez pour pair(2), impair(3), pair(3) et impair(2). L\u2019algorithme est-il correct ? Sinon le changer pour v\u00e9rifier la correction \u2190 P.S: si le prof demande \u00e7a c\u2019est que l\u2019algo a un probl\u00e8me CQFD<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-caebf67 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"caebf67\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-cbb542b\" data-id=\"cbb542b\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-5ad65fe elementor-widget elementor-widget-toggle\" data-id=\"5ad65fe\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-9521\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-9521\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Solution<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-9521\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-9521\"><p><span style=\"font-weight: 400;\">Evidemment l\u2019algo n\u2019est pas correct. Prenons pair(3) \u2192 impair(2) \u2192 pair(1) \u2192 impair(0) \u2192 pair(-1) \u2192 impair(-2) \u2192 etc. C\u2019est que les conditions d\u2019arr\u00eats ne marchent que si le nombre de la m\u00e9thode pair est pair et la m\u00e9thode impair est impair. On va seulement changer la m\u00e9thode impair.<\/span><\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-16549 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo62.png\" alt=\"r\u00e9cursivit\u00e9 crois\u00e9e pair impair\" width=\"220\" height=\"185\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo62.png 220w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo62-14x12.png 14w\" sizes=\"(max-width: 220px) 100vw, 220px\" \/><\/p><p><span style=\"font-weight: 400;\">A vous de retester si c\u2019est correct maintenant !<\/span><\/p><p>Pour avoir un algorithme plus solide, il faudrait v\u00e9rifier que l&rsquo;entier en entr\u00e9e soit positif (ou alors faire la valeur absolue).<\/p><p>Dans le cas d&rsquo;une entr\u00e9e avec un double ou flottante, il est possible de le rendre entier en l&rsquo;arrondissant. Pour cela, il faut connaitre le chiffre apr\u00e8s la virgule avec le calcul suivant virgule=n*10[10]. Si virgule&lt;5 alors on cast (entier)n, sinon on cast (entier)n+1.<\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-6c7a689 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"6c7a689\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-a5cee89\" data-id=\"a5cee89\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-619bbca elementor-widget-divider--view-line elementor-widget elementor-widget-divider\" data-id=\"619bbca\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"divider.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-divider\">\n\t\t\t<span class=\"elementor-divider-separator\">\n\t\t\t\t\t\t<\/span>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-2811967 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"2811967\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-8adb19a\" data-id=\"8adb19a\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-cb45e7e elementor-widget elementor-widget-heading\" data-id=\"cb45e7e\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Exercice-8\"><\/span>Exercice 8<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-c9db541 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"c9db541\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-9e65fab\" data-id=\"9e65fab\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-0bb47b0 elementor-widget elementor-widget-text-editor\" data-id=\"0bb47b0\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Il est impossible de conclure sur le r\u00e9cursif sans parler des fameuses Tours de Hano\u00ef (la hantise des \u00e9tudiants et parfois aussi des charg\u00e9s de TD et prof !). On va donc s\u2019y prendre doucement.<\/span><\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-16526 size-medium\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo40-300x171.png\" alt=\"tours de hanoi\" width=\"300\" height=\"171\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo40-300x171.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo40-18x10.png 18w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo40.png 373w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p><p><span style=\"font-weight: 400;\">On dispose de n plateaux de taille diff\u00e9rentes, et de 3 tiges, num\u00e9rot\u00e9es de 0 \u00e0 2.<\/span><\/p><p><span style=\"font-weight: 400;\">Initialement, tous les plateaux sont situ\u00e9s sur la tige i. Le but est de les transf\u00e9rer sur la\u00a0<\/span><span style=\"font-weight: 400;\">tige j, en respectant les r\u00e8gles suivantes :<\/span><\/p><p><span style=\"font-weight: 400;\">\u2013 On ne peux d\u00e9placer qu\u2019un plateau \u00e0 la fois.<\/span><\/p><p><span style=\"font-weight: 400;\">\u2013 Les plateaux doivent toujours rang\u00e9s par taille d\u00e9croissante sur une tige (le plus grand en bas puis par ordre d\u00e9croissant en remontant sur la tige).<\/span><\/p><p><span style=\"font-weight: 400;\">1 &#8211; R\u00e9solvez le probl\u00e8me \u00e0 la main si n = 2, i = 0 et j = 2 (donc 2 plateaux, ils sont tous de base sur la tige 0 et on veut les mettre sur la tige 2).<\/span><\/p><p><span style=\"font-weight: 400;\">2 &#8211; R\u00e9solvez le probl\u00e8me \u00e0 la main si n = 3, i = 0 et j = 2 (donc 3 plateaux, ils sont tous de base sur la tige 0 et on veut les mettre sur la tige 2).<\/span><\/p><p><span style=\"font-weight: 400;\">3 &#8211; Supposez qu\u2019un de vos amis sache r\u00e9soudre le probl\u00e8me pour un certain n, et n\u2019importe <\/span><span style=\"font-weight: 400;\">quels i et j. On vous demande de r\u00e9soudre le probl\u00e8me pour n + 1. Vous avez le droit\u00a0<\/span><span style=\"font-weight: 400;\">d\u2019utiliser l\u2019aide de votre ami. Comment vous y prenez-vous ?<\/span><\/p><p><span style=\"font-weight: 400;\">4 &#8211; Ecrire la fonction r\u00e9cursive qui permet de r\u00e9soudre ce probl\u00e8me pour tous n, i, j (Si vous y arrivez sans tricher f\u00e9licitation ! Sinon c\u2019est normal on est presque tous pass\u00e9 par ce grand moment d\u2019incompr\u00e9hension et de remise en cause de sa carri\u00e8re\u2026)<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-f2263e8 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"f2263e8\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-6a709a4\" data-id=\"6a709a4\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-fe02bea elementor-widget elementor-widget-toggle\" data-id=\"fe02bea\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-2661\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-2661\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Solution<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-2661\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-2661\"><p><span style=\"font-weight: 400;\">1 &#8211;\u00a0<\/span><\/p><p><span style=\"font-weight: 400;\">On transf\u00e8re le petit plateau de 0 \u00e0 1.\u00a0<\/span><\/p><p><span style=\"font-weight: 400;\">On transf\u00e8re le grand plateau de 0 \u00e0 2.\u00a0<\/span><\/p><p><span style=\"font-weight: 400;\">On transf\u00e8re le petit plateau de 1 \u00e0 2.<\/span><\/p><p><span style=\"font-weight: 400;\">2 &#8211;<\/span><\/p><p><span style=\"font-weight: 400;\">On transf\u00e8re le petit plateau de 0 \u00e0 2.\u00a0<\/span><\/p><p><span style=\"font-weight: 400;\">On transf\u00e8re le plateau moyen de 0 \u00e0 1.<\/span><\/p><p><span style=\"font-weight: 400;\">On transf\u00e8re le petit plateau de 2 \u00e0 1. (on a d\u00e9plac\u00e9 2 plateaux de 0 vers 1).\u00a0<\/span><\/p><p><span style=\"font-weight: 400;\">On transf\u00e8re le grand plateau de 0 \u00e0 2.\u00a0<\/span><\/p><p><span style=\"font-weight: 400;\">On transf\u00e8re le petit plateau de 1 \u00e0 0.\u00a0<\/span><\/p><p><span style=\"font-weight: 400;\">On transf\u00e8re le plateau moyen de 1 \u00e0 2.\u00a0<\/span><\/p><p><span style=\"font-weight: 400;\">On transf\u00e8re le petit plateau de 0 \u00e0 2. (on a d\u00e9plac\u00e9 2 plateaux de 1 vers 2).<\/span><\/p><p><span style=\"font-weight: 400;\">3 &#8211;<\/span><\/p><p><span style=\"font-weight: 400;\">On d\u00e9placer n plateaux de i \u00e0 l\u2019emplacement qui n\u2019est ni i ni j (par les s\u00e9ries de trois manipulations du point 2). On d\u00e9place ensuite le dernier plateau (le plus gros) de i \u00e0 j. On remet \u00e0 contribution l\u2019ami pour d\u00e9placer les n plateaux vers j.\u00a0<\/span><\/p><p><span style=\"font-weight: 400;\">4 &#8211;<\/span><\/p><p><span style=\"font-weight: 400;\">Si vous trouvez \u00e7a\u2026 champagne ! On va renomm\u00e9 pour plus de facilit\u00e9 les tiges en d pour celle de d\u00e9part, a pour celle d\u2019arriv\u00e9 et r pour celle qui reste<\/span><\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-16550 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo63.png\" alt=\"tours de hanoi\" width=\"338\" height=\"212\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo63.png 338w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo63-300x188.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo63-18x12.png 18w\" sizes=\"(max-width: 338px) 100vw, 338px\" \/><\/p><p>Regardons son fonctionnement, voici la structure g\u00e9n\u00e9rale de l&rsquo;algorithme :<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-19810\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/hanoi1-300x146.png\" alt=\"hanoi r\u00e9cursif\" width=\"300\" height=\"146\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/hanoi1-300x146.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/hanoi1-18x9.png 18w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/hanoi1.png 385w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p><p>Prenons par exemple Hanoi(2,d,a,r), notons l&rsquo;ensemble des r\u00e9cursions avec l&rsquo;ordre de base (d,a,r) m\u00eame si cela n&rsquo;a pas de sens d&rsquo;un poids de vue r\u00e9cursif :<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-19811\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/hanoi2-300x131.png\" alt=\"hanoi r\u00e9cursif\" width=\"300\" height=\"131\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/hanoi2-300x131.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/hanoi2-18x8.png 18w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/hanoi2.png 461w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p><p>Notons que les noeuds ayant une valeur n de 0 ne servent \u00e0 rien car la fonction sera vide.<\/p><p>Prenons maintenant l&rsquo;arbre avec l&rsquo;ordre des actions. Pour parcourir l&rsquo;arbre ne faisons un parcours d&rsquo;arbre<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-19813\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/parcours-300x162.png\" alt=\"parcours arbre\" width=\"300\" height=\"162\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/parcours-300x162.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/parcours-18x10.png 18w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/parcours.png 723w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p><p>Un noeud r\u00e9alise son action lorsque nous passons en dessous :<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-19814 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/hanoi3.png\" alt=\"hanoi r\u00e9cursif\" width=\"480\" height=\"201\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/hanoi3.png 480w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/hanoi3-300x126.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/hanoi3-18x8.png 18w\" sizes=\"(max-width: 480px) 100vw, 480px\" \/><\/p><p>Ce qui donnerait les d\u00e9placements suivant :<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-19815 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/hanoi4.png\" alt=\"hanoi r\u00e9cursif exemple\" width=\"422\" height=\"302\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/hanoi4.png 422w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/hanoi4-300x215.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/hanoi4-18x12.png 18w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/hanoi4-120x85.png 120w\" sizes=\"(max-width: 422px) 100vw, 422px\" \/><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-9ec6132 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"9ec6132\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-dee40e3\" data-id=\"dee40e3\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-682c7af elementor-widget elementor-widget-heading\" data-id=\"682c7af\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Exercice-9\"><\/span>Exercice 9<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-79a87b0 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"79a87b0\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-1c02efe\" data-id=\"1c02efe\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-97a57f7 elementor-widget elementor-widget-text-editor\" data-id=\"97a57f7\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>\u00c9crivez un programme pour trouver le PGCD de deux nombres en utilisant la r\u00e9cursivit\u00e9.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-3d1a0b2 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"3d1a0b2\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-9ebb093\" data-id=\"9ebb093\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-559c8a4 elementor-widget elementor-widget-toggle\" data-id=\"559c8a4\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-8971\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-8971\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Solution<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-8971\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-8971\"><p>Voici l&rsquo;id\u00e9e derri\u00e8re l&rsquo;algorithme :<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-19287 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/pgcd1.png\" alt=\"pgcd algorithme\" width=\"363\" height=\"352\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/pgcd1.png 363w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/pgcd1-300x291.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/pgcd1-12x12.png 12w\" sizes=\"(max-width: 363px) 100vw, 363px\" \/><\/p><p>Ce qui donne l&rsquo;algorithme suivant :<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-19816 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/pgcd.png\" alt=\"pgcd\" width=\"460\" height=\"240\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/pgcd.png 460w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/pgcd-300x157.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/pgcd-18x9.png 18w\" sizes=\"(max-width: 460px) 100vw, 460px\" \/><\/p><p>Il est aussi possible de faire directement le modulo (plut\u00f4t que a-b plusieurs fois) pour faire une r\u00e9cursion avec le reste :<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-16531\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo44.png\" alt=\"algorithme pgcd\" width=\"261\" height=\"203\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo44.png 261w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/08\/algo44-15x12.png 15w\" sizes=\"(max-width: 261px) 100vw, 261px\" \/><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-732fe05 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"732fe05\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-771c53d\" data-id=\"771c53d\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-a1624bc elementor-widget elementor-widget-heading\" data-id=\"a1624bc\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Exercice-10\"><\/span>Exercice 10<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-9136802 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"9136802\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-c97b86f\" data-id=\"c97b86f\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-4d72cda elementor-widget elementor-widget-text-editor\" data-id=\"4d72cda\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>\u00c9crivez un programme pour convertir un nombre d\u00e9cimal en binaire en utilisant la r\u00e9cursivit\u00e9.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-0bec766 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"0bec766\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-5c82724\" data-id=\"5c82724\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-0478e99 elementor-widget elementor-widget-toggle\" data-id=\"0478e99\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-4681\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-4681\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Solution<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-4681\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-4681\"><p>Voici l&rsquo;id\u00e9e derri\u00e8re l&rsquo;algorithme :<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-19288\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/binary.png\" alt=\"algorithme binaire\" width=\"298\" height=\"268\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/binary.png 298w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/binary-13x12.png 13w\" sizes=\"(max-width: 298px) 100vw, 298px\" \/><\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-19289 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/binary1.png\" alt=\"algorithme r\u00e9cursif binaire\" width=\"831\" height=\"759\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/binary1.png 831w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/binary1-300x274.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/binary1-768x701.png 768w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/binary1-13x12.png 13w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/binary1-600x548.png 600w\" sizes=\"(max-width: 831px) 100vw, 831px\" \/><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-33e58d1 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"33e58d1\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-d6ff461\" data-id=\"d6ff461\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-646bedc elementor-widget elementor-widget-heading\" data-id=\"646bedc\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Exercice-11\"><\/span>Exercice 11<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-cfd0167 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"cfd0167\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-6c759c6\" data-id=\"6c759c6\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-b4a860b elementor-widget elementor-widget-text-editor\" data-id=\"b4a860b\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>\u00c9crivez un programme pour trouver le LCM de deux nombres en utilisant la r\u00e9cursivit\u00e9.<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-19291\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/LCM-300x243.png\" alt=\"LCM\" width=\"300\" height=\"243\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/LCM-300x243.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/LCM-15x12.png 15w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/LCM.png 383w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-47ed49f elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"47ed49f\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-b6baa9f\" data-id=\"b6baa9f\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-972e747 elementor-widget elementor-widget-toggle\" data-id=\"972e747\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-1581\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-1581\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Solution<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-1581\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-1581\"><p>En voici l&rsquo;algorithme :<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-19292 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/LCM1.png\" alt=\"LCM\" width=\"834\" height=\"836\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/LCM1.png 834w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/LCM1-300x300.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/LCM1-150x150.png 150w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/LCM1-768x770.png 768w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/LCM1-12x12.png 12w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/LCM1-600x601.png 600w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/LCM1-100x100.png 100w\" sizes=\"(max-width: 834px) 100vw, 834px\" \/><\/p><p>La fonction r\u00e9cursive lcm() prend deux entiers comme argument. Prenez une variable statique m et initialisez-la avec z\u00e9ro, puis mettez \u00e0 jour avec l&rsquo;un des nombres. Ici, nous mettons \u00e0 jour la variable m avec b, donc la variable m sera toujours divisible par la variable b. Par cons\u00e9quent, nous devons v\u00e9rifier uniquement avec la variable a, c&rsquo;est-\u00e0-dire (m%a == 0).<\/p><p>Puisque nous ajoutons le plus grand nombre \u00e0 la variable m, la variable b doit donc contenir le plus grand nombre. Lors de l&rsquo;appel de lcm(), nous devons transmettre le plus petit nombre comme a et le plus grand nombre comme b.<\/p><p>Si nous ajoutons le plus grand nombre, le nombre d&rsquo;appels r\u00e9cursifs sera le minimum, mais si nous ajoutons le plus petit nombre, le nombre d&rsquo;appels r\u00e9cursifs sera sup\u00e9rieur \u00e0 celui du cas pr\u00e9c\u00e9dent. Nous savons que moins d\u2019appels r\u00e9cursifs donnent des performances \u00e9lev\u00e9es.<\/p><p>Prenons l&rsquo;exemple des nombres 3 et 7. Si nous ajoutons le plus grand nombre 7 \u00e0 la variable somme alors,<br \/>0+7 = 7 (appel depuis la fonction principale), 7%3 != 0<br \/>7+7 = 14 (premier appel r\u00e9cursif), 14%3 != 0<br \/>14+7 = 21 (deuxi\u00e8me appel r\u00e9cursif)<br \/>21 est divisible par 3 et par 7<br \/>Il n\u2019y aura donc que 2 appels r\u00e9cursifs.<\/p><p>De m\u00eame, si l\u2019on prend le plus petit nombre (3) alors,<br \/>0+3 = 3 , 3%7 != 0<br \/>3+3 = 6 , 6%7 != 0<br \/>6+3 = 9 , 9%7 != 0<br \/>9+3 = 12 , 12%7 != 0<br \/>12+3 = 15 , 15%7 != 0<br \/>15+3 = 18 , 18%7 != 0<br \/>18+3 = 21<br \/>21 est divisible par 3 et par 7<br \/>Il y aura donc 6 appels r\u00e9cursifs.<\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-3bcfd8c elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"3bcfd8c\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-2a80ac3\" data-id=\"2a80ac3\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-97d4cb6 elementor-widget elementor-widget-heading\" data-id=\"97d4cb6\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Exercice-12\"><\/span>Exercice 12<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-e193935 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"e193935\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-9a05c94\" data-id=\"9a05c94\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-48d8970 elementor-widget elementor-widget-text-editor\" data-id=\"48d8970\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>\u00c9crivez un programme pour v\u00e9rifier si une cha\u00eene donn\u00e9e est Palindrome ou non.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-5800bbe elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"5800bbe\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-e772a19\" data-id=\"e772a19\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-15b4e08 elementor-widget elementor-widget-toggle\" data-id=\"15b4e08\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-2271\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-2271\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Solution<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-2271\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-2271\"><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-19293 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/palindrome.png\" alt=\"palindrome\" width=\"727\" height=\"755\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/palindrome.png 727w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/palindrome-289x300.png 289w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/palindrome-12x12.png 12w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/palindrome-600x623.png 600w\" sizes=\"(max-width: 727px) 100vw, 727px\" \/><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-fcac776 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"fcac776\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-4bd91f1\" data-id=\"4bd91f1\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-cd4cfcf elementor-widget elementor-widget-heading\" data-id=\"cd4cfcf\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Exercice-13\"><\/span>Exercice 13<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-374a11c elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"374a11c\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-ab77700\" data-id=\"ab77700\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-e39c15d elementor-widget elementor-widget-text-editor\" data-id=\"e39c15d\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Le nombre Pi peut \u00eatre calcul\u00e9 par la formule suivante :<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-19297 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/pi_recursif.png\" alt=\"pi r\u00e9cursif\" width=\"443\" height=\"74\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/pi_recursif.png 443w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/pi_recursif-300x50.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/pi_recursif-18x3.png 18w\" sizes=\"(max-width: 443px) 100vw, 443px\" \/><\/p><p>Ecrire un algorithme r\u00e9cursif permettant de calculer la valeur de Pi en consid\u00e9rant la s\u00e9quence jusqu&rsquo;au rang n.<\/p><p>Faire de m\u00eame avec les approximations de Pi suivants.<\/p><p>S\u00e9rie de Gregory-Leibniz :<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-19300 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/pi2.png\" alt=\"\" width=\"550\" height=\"145\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/pi2.png 550w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/pi2-300x79.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/pi2-18x5.png 18w\" sizes=\"(max-width: 550px) 100vw, 550px\" \/><\/p><p>Et les s\u00e9ries plus anciennes de Madhava, Ramanujan, David et Chudnovsky (la factorielle sera un appel r\u00e9cursif \u00e0 part enti\u00e8re) :<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-19301 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/pi3.png\" alt=\"pi r\u00e9cursif\" width=\"682\" height=\"274\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/pi3.png 682w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/pi3-300x121.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/pi3-18x7.png 18w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/pi3-600x241.png 600w\" sizes=\"(max-width: 682px) 100vw, 682px\" \/><\/p><p>\u00a0<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-90b1b72 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"90b1b72\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-898d184\" data-id=\"898d184\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-3ea89ce elementor-widget elementor-widget-toggle\" data-id=\"3ea89ce\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-6571\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-6571\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Solution<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-6571\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-6571\"><p>Voici l&rsquo;algorithme en r\u00e9cursif :<\/p><pre>gregory(n)<br \/>{<br \/>si(n==0) retourner 4<br \/>retourner 4*pow(-1,n)\/(2*n+1)+gregory(n-1)<br \/>}<\/pre><p>Les autres formules suivent le m\u00eame principe, inutile de tout d\u00e9tailler. Attention tout de m\u00eame \u00e0 utiliser le bon calcul dans la r\u00e9cursion ! (les algorithmes seront plus facile \u00e0 \u00e9crire en r\u00e9cursif non terminal.<\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-d9aa26e elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"d9aa26e\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-ab24a09\" data-id=\"ab24a09\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-b6b4857 elementor-widget elementor-widget-heading\" data-id=\"b6b4857\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Exercice-14\"><\/span>Exercice 14<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-0252485 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"0252485\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-2f6e202\" data-id=\"2f6e202\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-654b56a elementor-widget elementor-widget-text-editor\" data-id=\"654b56a\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Dans les ann\u00e9es 1920, Wilhelm Ackermann et Gabriel Sudan, alors \u00e9tudiants sous la direction de David Hilbert, \u00e9tudient les fondements de la calculabilit\u00e9. Sudan est le premier \u00e0 donner un exemple de fonction r\u00e9cursive mais non r\u00e9cursive primitive, appel\u00e9e alors fonction de Sudan. Peu apr\u00e8s et ind\u00e9pendamment, en 1928, Ackermann publie son propre exemple de fonction r\u00e9cursive mais non r\u00e9cursive primitive. \u00c0 l&rsquo;origine, Ackermann consid\u00e8re une fonction \u03d5(m, n, p) d\u00e9pendante de trois variables.<\/p><p>Une fonction de seulement deux variables est donn\u00e9e plus tard par R\u00f3zsa P\u00e9ter et Raphael Robinson ; c&rsquo;est cette derni\u00e8re qui est connue aujourd&rsquo;hui sous le nom de fonction d&rsquo;Ackermann.<\/p><p>La fonction d&rsquo;Ackermann-P\u00e9ter est d\u00e9finie r\u00e9cursivement comme suit :<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-19308 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/ackermann.png\" alt=\"ackermann\" width=\"431\" height=\"68\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/ackermann.png 431w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/ackermann-300x47.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/ackermann-18x3.png 18w\" sizes=\"(max-width: 431px) 100vw, 431px\" \/><\/p><p>La fonction grandit tellement vite qu&rsquo;il est rapidement impossible de l&rsquo;\u00e9crire avec un nombre classique et qu&rsquo;il faut utiliser la notation de Knuth !<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-19310 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/ackermann2.png\" alt=\"ackermann\" width=\"630\" height=\"242\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/ackermann2.png 630w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/ackermann2-300x115.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/ackermann2-18x7.png 18w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/ackermann2-600x230.png 600w\" sizes=\"(max-width: 630px) 100vw, 630px\" \/><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-42c1dee elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"42c1dee\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-98fa7ae\" data-id=\"98fa7ae\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-040be5f elementor-widget elementor-widget-toggle\" data-id=\"040be5f\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-4241\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-4241\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Solution<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-4241\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-4241\"><p>La solution est tr\u00e8s simple, mais je vous d\u00e9conseille de lancer l&rsquo;algorithme sur un PC.<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-19311 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/ackermann3.jpeg\" alt=\"ackermann r\u00e9cursif\" width=\"398\" height=\"340\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/ackermann3.jpeg 398w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/ackermann3-300x256.jpeg 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/ackermann3-14x12.jpeg 14w\" sizes=\"(max-width: 398px) 100vw, 398px\" \/><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-db2610f elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"db2610f\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-cb6a752\" data-id=\"cb6a752\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-945c789 elementor-widget elementor-widget-heading\" data-id=\"945c789\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Exercice-15\"><\/span>Exercice 15<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-c0cd5b6 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"c0cd5b6\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-f86f65b\" data-id=\"f86f65b\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-9265241 elementor-widget elementor-widget-text-editor\" data-id=\"9265241\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Voici la fonction de Sudan, autre \u00e9l\u00e8ve de David Hilbert. Cette fonction a \u00e9t\u00e9 con\u00e7u en 1927 donc une ann\u00e9e plus t\u00f4t que la fonction d&rsquo;Ackermann.<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-19312 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/sudan.png\" alt=\"sudan r\u00e9cursif\" width=\"469\" height=\"84\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/sudan.png 469w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/sudan-300x54.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/sudan-18x3.png 18w\" sizes=\"(max-width: 469px) 100vw, 469px\" \/><\/p><p>Ici, plus la valeur de n grandit, plus le r\u00e9sultat de la fonction de la fonction explose. En voici l&rsquo;example pour F_1(14,14).<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-19313 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/sudan2.png\" alt=\"sudan r\u00e9cursif\" width=\"878\" height=\"470\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/sudan2.png 878w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/sudan2-300x161.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/sudan2-768x411.png 768w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/sudan2-18x10.png 18w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/sudan2-600x321.png 600w\" sizes=\"(max-width: 878px) 100vw, 878px\" \/><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-0a32359 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"0a32359\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-665cc0d\" data-id=\"665cc0d\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-e7013df elementor-widget elementor-widget-toggle\" data-id=\"e7013df\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-2421\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-2421\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Solution<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-2421\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-2421\"><p>Rien de bien compliqu\u00e9, il suffit de recopier la formule math\u00e9matique sous forme de r\u00e9cursion.<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-19314 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/sudan3.png\" alt=\"sudan r\u00e9cursif\" width=\"505\" height=\"225\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/sudan3.png 505w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/sudan3-300x134.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/sudan3-18x8.png 18w\" sizes=\"(max-width: 505px) 100vw, 505px\" \/><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-6e76db7 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"6e76db7\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-b0ee6c5\" data-id=\"b0ee6c5\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-3accca1 elementor-widget elementor-widget-heading\" data-id=\"3accca1\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Exercice-16\"><\/span>Exercice 16<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-5e69f71 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"5e69f71\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-7faf1a2\" data-id=\"7faf1a2\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-080b621 elementor-widget elementor-widget-text-editor\" data-id=\"080b621\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>La fonction de Takeuchi, abr\u00e9g\u00e9e tak ou parfois tarai, est la pr\u00e9sentation r\u00e9cursive d&rsquo;une fonction qui doit son nom \u00e0 Ikuo Takeuchi (\u7af9\u5185\u90c1\u96c4). La pr\u00e9sentation de la fonction, qui, par ailleurs, admet une d\u00e9finition non r\u00e9cursive assez simple, peut requ\u00e9rir des calculs tr\u00e8s longs si le compilateur qui l&rsquo;implante n&rsquo;est pas performant. Pour cette raison, elle est souvent utilis\u00e9e pour tester les performances de l&rsquo;implantation des fonctions r\u00e9cursives par le compilateur d&rsquo;un langage de programmation.<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-19315 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/takeuchi.png\" alt=\"takeuchi\" width=\"524\" height=\"51\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/takeuchi.png 524w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/takeuchi-300x29.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/takeuchi-18x2.png 18w\" sizes=\"(max-width: 524px) 100vw, 524px\" \/><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-537bbb1 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"537bbb1\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-055c182\" data-id=\"055c182\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-2b3be31 elementor-widget elementor-widget-toggle\" data-id=\"2b3be31\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-4531\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-4531\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Solution<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-4531\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-4531\"><p>La d\u00e9finition originale de Takeuchi \u00e9tait la suivante :<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-19316\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/takeuchi2.png\" alt=\"takeuchi\" width=\"215\" height=\"209\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/takeuchi2.png 215w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/takeuchi2-12x12.png 12w\" sizes=\"(max-width: 215px) 100vw, 215px\" \/><\/p><p>tarai est l&rsquo;abr\u00e9viation de \u00ab\u00a0passer autour\u00a0\u00bb en japonais. John McCarthy a nomm\u00e9 cette fonction tak() d&rsquo;apr\u00e8s Takeuchi.<\/p><p>La r\u00e9cursivit\u00e9 \u00e0 \u00e9t\u00e9 am\u00e9lior\u00e9 gr\u00e2ce \u00e0 une r\u00e9cursivit\u00e9 crois\u00e9e :<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-19317\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/takeuchi3-300x300.png\" alt=\"takeuchi\" width=\"300\" height=\"300\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/takeuchi3-300x300.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/takeuchi3-150x150.png 150w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/takeuchi3-12x12.png 12w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/takeuchi3-100x100.png 100w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/takeuchi3.png 353w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-9177b60 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"9177b60\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-3034e4f\" data-id=\"3034e4f\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-d377bd7 elementor-widget elementor-widget-heading\" data-id=\"d377bd7\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Exercice-17\"><\/span>Exercice 17<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-5d2f954 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"5d2f954\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-bb02353\" data-id=\"bb02353\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-def36f0 elementor-widget elementor-widget-text-editor\" data-id=\"def36f0\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>La s\u00e9quence Male-Female Hofstadter est d\u00e9finie par :<\/p><p>F(0)=1, M(0)=0 et<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-19324 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/hofstadter.png\" alt=\"hofstadter\" width=\"331\" height=\"81\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/hofstadter.png 331w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/hofstadter-300x73.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/hofstadter-18x4.png 18w\" sizes=\"(max-width: 331px) 100vw, 331px\" \/><\/p><p>Ecrire un algorithme r\u00e9cursif permettant de calculer la s\u00e9quence de Hofstadter.<\/p><p>De m\u00eame pour la s\u00e9quence G :<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-19829\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/Hofstadter1.png\" alt=\"Hofstadter G\" width=\"255\" height=\"98\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/Hofstadter1.png 255w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/Hofstadter1-18x7.png 18w\" sizes=\"(max-width: 255px) 100vw, 255px\" \/><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-01d7c12 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"01d7c12\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-6ae6088\" data-id=\"6ae6088\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-f206909 elementor-widget elementor-widget-toggle\" data-id=\"f206909\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-2531\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-2531\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Solution<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-2531\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-2531\"><p>Il s&rsquo;agit d&rsquo;un algorithme r\u00e9cursif crois\u00e9 (aussi dit parfait).<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-19827\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/Hofstadter.png\" alt=\"Hofstadter male female\" width=\"267\" height=\"248\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/Hofstadter.png 267w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/Hofstadter-13x12.png 13w\" sizes=\"(max-width: 267px) 100vw, 267px\" \/><\/p><p>Pour la s\u00e9quence G, le principe est le m\u00eame :<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-19830\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/Hofstadter2.png\" alt=\"Hofstadter G\" width=\"254\" height=\"109\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/Hofstadter2.png 254w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/Hofstadter2-18x8.png 18w\" sizes=\"(max-width: 254px) 100vw, 254px\" \/><\/p><p>Vous trouverez ci-dessous un exemple d&rsquo;arbre de n\u0153uds form\u00e9 lorsque le programme est ex\u00e9cut\u00e9 avec des valeurs de n comprises entre 1 et 21. Par exemple, G(15) produira la valeur 9, et G(9) produira la valeur 6, etc. Il est int\u00e9ressant de noter que la s\u00e9quence de Fibonacci est visible sur le c\u00f4t\u00e9 droit de l&rsquo;arbre.<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-19831 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/hofstadter3.webp\" alt=\"hofstadter G\" width=\"538\" height=\"404\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/hofstadter3.webp 538w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/hofstadter3-300x225.webp 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/hofstadter3-16x12.webp 16w\" sizes=\"(max-width: 538px) 100vw, 538px\" \/><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-09e7786 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"09e7786\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-f58ae6a\" data-id=\"f58ae6a\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-cf5de6f elementor-widget elementor-widget-heading\" data-id=\"cf5de6f\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Exercice-18\"><\/span>Exercice 18<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-6691b00 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"6691b00\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-7ef1d76\" data-id=\"7ef1d76\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-8dc0ec0 elementor-widget elementor-widget-text-editor\" data-id=\"8dc0ec0\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Certains langages de programmation permettent la m\u00e9mo\u00efsation. Il s&rsquo;agit d&rsquo;une mise en cache des valeurs de retour d&rsquo;une fonction selon ses valeurs d&rsquo;entr\u00e9e. Le but de cette technique d&rsquo;optimisation de code est de diminuer le temps d&rsquo;ex\u00e9cution d&rsquo;un programme informatique en m\u00e9morisant les valeurs retourn\u00e9es par une fonction.<\/p><p>Ainsi, en r\u00e9alisant la suite de Fibonacci, il est possible de se m\u00e9moriser les valeurs d\u00e9j\u00e0 calcul\u00e9es afin d&rsquo;\u00e9viter de saturer la m\u00e9moire avec des valeurs d\u00e9j\u00e0 obtenu dans une autre branche de l&rsquo;arbre d&rsquo;appel.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-5cdb911 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"5cdb911\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-9adc4ae\" data-id=\"9adc4ae\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-7e65ab9 elementor-widget elementor-widget-toggle\" data-id=\"7e65ab9\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-1321\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-1321\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Solution<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-1321\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-1321\"><p>En fait, chaque fois que nous calculons fib (i), nous pourrions simplement mettre en cache ce r\u00e9sultat et l&rsquo;utiliser plus tard. Ainsi, lorsque nous appelons fib (n), nous ne devrions pas avoir \u00e0 faire beaucoup plus que 0 (n) appels, car il n&rsquo;y a que O(n) valeurs possibles que nous pouvons lancer sur fib(n).<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-19809\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/fibo2-300x184.png\" alt=\"fibonacci m\u00e9mo\u00efsation\" width=\"300\" height=\"184\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/fibo2-300x184.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/fibo2-18x12.png 18w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/fibo2.png 331w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p><p>Ici l&rsquo;arbre de r\u00e9cursion sera bien diff\u00e9rent de celui de la variante non terminale :<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-19836 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/fibo3.png\" alt=\"fibonacci m\u00e9moisation\" width=\"689\" height=\"242\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/fibo3.png 689w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/fibo3-300x105.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/fibo3-18x6.png 18w\" sizes=\"(max-width: 689px) 100vw, 689px\" \/><\/p><p>Notez que l&rsquo;arbre ci-avant ne prend pas compte du sens d&rsquo;appel gauche puis d&rsquo;appel droit. Pour plus de confort, il est coutume de mettre le premier appel en fils gauche et le deuxi\u00e8me en fils droit, donc le contraire de l&rsquo;arbre pr\u00e9sent\u00e9 !<\/p><p>Les temps de calcul s&rsquo;en retrouvent grandement diminu\u00e9 comme le montre le sch\u00e9ma suivant sur Fibo(100) sous ses variantes :<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-19837 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/comparison.png\" alt=\"comparison fibonacci\" width=\"547\" height=\"550\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/comparison.png 547w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/comparison-298x300.png 298w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/comparison-150x150.png 150w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/comparison-12x12.png 12w\" sizes=\"(max-width: 547px) 100vw, 547px\" \/><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-ee0d46e elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"ee0d46e\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-ba54d96\" data-id=\"ba54d96\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-6e011c1 elementor-widget elementor-widget-heading\" data-id=\"6e011c1\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Exercice-19\"><\/span>Exercice 19<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-4683b19 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"4683b19\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-e503fa8\" data-id=\"e503fa8\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-24f4b25 elementor-widget elementor-widget-text-editor\" data-id=\"24f4b25\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Certains langages de programmation permettent la m\u00e9mo\u00efsation. Il s&rsquo;agit d&rsquo;une mise en cache des valeurs de retour d&rsquo;une fonction selon ses valeurs d&rsquo;entr\u00e9e. Le but de cette technique d&rsquo;optimisation de code est de diminuer le temps d&rsquo;ex\u00e9cution d&rsquo;un programme informatique en m\u00e9morisant les valeurs retourn\u00e9es par une fonction.<\/p><p>Ecrire un algorithme calculant factoriel n (n!) utilisant le principe de m\u00e9mo\u00efsation.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-4a2ac1a elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"4a2ac1a\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-28ba65b\" data-id=\"28ba65b\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-7ff84d8 elementor-widget elementor-widget-toggle\" data-id=\"7ff84d8\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-1341\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-1341\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Solution<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-1341\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-1341\"><p>Voici la solution classique pour faire le calcul factoriel en r\u00e9cursif :<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-19841 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/factoriel.png\" alt=\"factoriel r\u00e9cursif\" width=\"573\" height=\"157\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/factoriel.png 573w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/factoriel-300x82.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2023\/11\/factoriel-18x5.png 18w\" sizes=\"(max-width: 573px) 100vw, 573px\" \/><\/p><p>L&rsquo;impl\u00e9mentation non m\u00e9moris\u00e9e ci-dessus, \u00e9tant donn\u00e9 la nature de l&rsquo;algorithme r\u00e9cursif impliqu\u00e9, n\u00e9cessiterait n + 1 invocations de factorielles pour arriver \u00e0 un r\u00e9sultat, et chacune de ces invocations, \u00e0 son tour, a un co\u00fbt associ\u00e9 en termes de temps n\u00e9cessaire \u00e0 la fonction. pour renvoyer la valeur calcul\u00e9e. Selon la machine, ce co\u00fbt peut \u00eatre la somme de :<\/p><p>Le co\u00fbt de configuration du cadre de pile d\u2019appels fonctionnel.<br \/>Le co\u00fbt pour comparer n \u00e0 0.<br \/>Le co\u00fbt pour soustraire 1 de n.<br \/>Le co\u00fbt de configuration du cadre de pile d\u2019appels r\u00e9cursifs. (Comme ci-dessus.)<br \/>Le co\u00fbt pour multiplier le r\u00e9sultat de l\u2019appel r\u00e9cursif \u00e0 factoriel par n.<br \/>Le co\u00fbt de stockage du r\u00e9sultat renvoy\u00e9 afin qu\u2019il puisse \u00eatre utilis\u00e9 par le contexte appelant.<br \/>Dans une impl\u00e9mentation non m\u00e9moris\u00e9e, chaque appel de niveau sup\u00e9rieur \u00e0 factorial inclut le co\u00fbt cumul\u00e9 des \u00e9tapes 2 \u00e0 6 proportionnel \u00e0 la valeur initiale de n.<\/p><p>Une version m\u00e9moris\u00e9e de la fonction factorielle suit\u00a0:<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-19327 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/factmemo.png\" alt=\"factoriel m\u00e9mo\u00efsation\" width=\"358\" height=\"153\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/factmemo.png 358w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/factmemo-300x128.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/factmemo-18x8.png 18w\" sizes=\"(max-width: 358px) 100vw, 358px\" \/><\/p><p>Dans cet exemple particulier, si factorial est d&rsquo;abord invoqu\u00e9 avec 5, puis invoqu\u00e9 ult\u00e9rieurement avec une valeur inf\u00e9rieure ou \u00e9gale \u00e0 cinq, ces valeurs de retour auront \u00e9galement \u00e9t\u00e9 m\u00e9moris\u00e9es, puisque factorial aura \u00e9t\u00e9 appel\u00e9 de mani\u00e8re r\u00e9cursive avec les valeurs 5, 4, 3, 2, 1 et 0, et les valeurs de retour pour chacune d&rsquo;entre elles auront \u00e9t\u00e9 stock\u00e9es. S&rsquo;il est ensuite appel\u00e9 avec un nombre sup\u00e9rieur \u00e0 5, comme 7, seuls 2 appels r\u00e9cursifs seront effectu\u00e9s (7 et 6), et la valeur pour 5 ! aura \u00e9t\u00e9 stock\u00e9 \u00e0 partir de l\u2019appel pr\u00e9c\u00e9dent.<\/p><p>De cette mani\u00e8re, la m\u00e9morisation permet \u00e0 une fonction de devenir plus efficace en termes de temps \u00e0 mesure qu&rsquo;elle est appel\u00e9e plus souvent, ce qui entra\u00eene une \u00e9ventuelle acc\u00e9l\u00e9ration globale.<\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-68a4cef elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"68a4cef\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-0c3c046\" data-id=\"0c3c046\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-ff20f4a elementor-widget elementor-widget-heading\" data-id=\"ff20f4a\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Exercice-20\"><\/span>Exercice 20<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-6719970 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"6719970\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-52a5da5\" data-id=\"52a5da5\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-c4d7b90 elementor-widget elementor-widget-text-editor\" data-id=\"c4d7b90\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Et voici un petit exercice m\u00ealant tout ce que vous avez pu voir r\u00e9cursivit\u00e9, crois\u00e9 et m\u00e9mo\u00efsation.<\/p><p>Il y a n oranges dans la cuisine et vous avez d\u00e9cid\u00e9 de manger quelques-unes de ces oranges tous les jours comme suit :<\/p><ul><li>Mangez une orange.<\/li><li>Si le nombre d&rsquo;oranges restantes n est divisible par 2 alors vous pouvez manger n\/2 oranges.<\/li><li>Si le nombre d&rsquo;oranges restantes n est divisible par 3 alors vous pouvez manger 2*(n\/3) oranges.<\/li><\/ul><p>Vous ne pouvez choisir qu&rsquo;une seule des actions par jour.<\/p><p>\u00c9tant donn\u00e9 l&rsquo;entier n, renvoie le nombre minimum de jours pour manger n oranges.<\/p><p>Entr\u00e9e : n = 10<br \/>Sortie\u00a0: 4<\/p><p>Explication : Vous avez 10 oranges.<\/p><p>Jour 1 : Mangez 1 orange, 10 &#8211; 1 = 9.<br \/>Jour 2 : Mange 6 oranges, 9 &#8211; 2*(9\/3) = 9 &#8211; 6 = 3. (Puisque 9 est divisible par 3)<br \/>Jour 3 : Mange 2 oranges, 3 &#8211; 2*(3\/3) = 3 &#8211; 2 = 1.<br \/>Jour 4 : Mangez la derni\u00e8re orange 1 &#8211; 1 = 0.<\/p><p>Il faut au moins 4 jours pour manger les 10 oranges.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-8ac3969 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"8ac3969\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-ef86eac\" data-id=\"ef86eac\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-12e281b elementor-widget elementor-widget-toggle\" data-id=\"12e281b\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-1981\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-1981\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Solution<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-1981\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-1981\"><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-19332 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/orange.png\" alt=\"orange m\u00e9mo\u00efsation\" width=\"503\" height=\"839\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/orange.png 503w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/orange-180x300.png 180w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2022\/12\/orange-7x12.png 7w\" sizes=\"(max-width: 503px) 100vw, 503px\" \/><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>Algorithmics Home page Wiki Ejercicios corregidos: algoritmo recursivo Los siguientes ejercicios corregidos se relacionan con el principio del algoritmo recursivo, por ejemplo Fibonacci, las torres de Hanoi\u2026 <\/p>","protected":false},"author":1,"featured_media":0,"parent":1062,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-16492","page","type-page","status-publish","hentry"],"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/complex-systems-ai.com\/es\/wp-json\/wp\/v2\/pages\/16492","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/complex-systems-ai.com\/es\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/complex-systems-ai.com\/es\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/complex-systems-ai.com\/es\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/complex-systems-ai.com\/es\/wp-json\/wp\/v2\/comments?post=16492"}],"version-history":[{"count":55,"href":"https:\/\/complex-systems-ai.com\/es\/wp-json\/wp\/v2\/pages\/16492\/revisions"}],"predecessor-version":[{"id":20229,"href":"https:\/\/complex-systems-ai.com\/es\/wp-json\/wp\/v2\/pages\/16492\/revisions\/20229"}],"up":[{"embeddable":true,"href":"https:\/\/complex-systems-ai.com\/es\/wp-json\/wp\/v2\/pages\/1062"}],"wp:attachment":[{"href":"https:\/\/complex-systems-ai.com\/es\/wp-json\/wp\/v2\/media?parent=16492"}],"curies":[{"name":"gracias","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}