{"id":1892,"date":"2019-12-28T18:08:33","date_gmt":"2019-12-28T17:08:33","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=1892"},"modified":"2019-12-29T10:59:55","modified_gmt":"2019-12-29T09:59:55","slug":"le-jeu-du-2020","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2019\/12\/28\/le-jeu-du-2020\/","title":{"rendered":"Le jeu du 2020"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">\u00c0 l&rsquo;occasion de la nouvelle ann\u00e9e qui approche, je vous propose un jeu qui m&rsquo;a \u00e9t\u00e9 inspir\u00e9 directement du nombre 2020&#8230;<\/p>\n\n\n\n<!--more-->\n\n\n\n<p class=\"wp-block-paragraph\">\u00ab\u00a0Mais St\u00e9phane, dis-moi vite les r\u00e8gles de ce jeu ! Je n&rsquo;en peux plus d&rsquo;attendre !\u00a0\u00bb<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Je vous entends de chez moi dire \u00e7a, alors sans plus attendre, les voici:<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\"><p>On dispose d&rsquo;une infinit\u00e9 de jetons sur lesquels sont indiqu\u00e9s \u00ab\u00a00\u00a0\u00bb, \u00ab\u00a02\u00a0\u00bb ainsi que les op\u00e9rations \u00ab\u00a0+\u00a0\u00bb, \u00ab\u00a0-\u00ab\u00a0, \u00ab\u00a0*\u00a0\u00bb, \u00ab\u00a0\/\u00a0\u00bb, \u00ab\u00a0^\u00a0\u00bb et des parenth\u00e8ses \u00ab\u00a0(\u00a0\u00bb et \u00ab\u00a0)\u00a0\u00bb.<\/p><p>Le but du jeu est de cr\u00e9er une op\u00e9ration qui comporte alternativement des \u00ab\u00a02\u00a0\u00bb et des\u00a0\u00bb0&Prime; s\u00e9par\u00e9s par des parenth\u00e8ses et des op\u00e9rations permises, l&rsquo;op\u00e9ration commen\u00e7ant par \u00ab\u00a02\u00a0\u00bb.<\/p><p>Par exemple, \u00ab\u00a0(2+(0+2)\/(0+2))*(0+2)\u00a0\u00bb est une op\u00e9ration permise car elle comporte alternativement des \u00ab\u00a02\u00a0\u00bb et des \u00ab\u00a00\u00a0\u00bb s\u00e9par\u00e9s par des op\u00e9rations et des parenth\u00e8ses, tout en commen\u00e7ant par \u00ab\u00a02\u00a0\u00bb, mais son r\u00e9sultat n&rsquo;est pas \u00e9gal \u00e0 2020.<\/p><p>Trouver une op\u00e9ration qui donne pour r\u00e9sultat 2020.<\/p><cite>R\u00e8gles du jeu du 2020<\/cite><\/blockquote>\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_88 counter-hierarchy ez-toc-counter ez-toc-white ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Au menu sur cette page...<\/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=\"Toggle Table of Content\"><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:\/\/www.mathweb.fr\/euclide\/2019\/12\/28\/le-jeu-du-2020\/#Une_solution\" >Une solution<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.mathweb.fr\/euclide\/2019\/12\/28\/le-jeu-du-2020\/#Une_autre_solution\" >Une autre solution<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Une_solution\"><\/span>Une solution<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Comme je suis quelqu&rsquo;un de tr\u00e8s gentil, je vous propose une solution comment\u00e9e. <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Le premier r\u00e9flexe que j&rsquo;ai eu est de d\u00e9composer en produit de facteurs premiers le nombre 2020; je trouve : \\(2020 = 2\\times2\\times5\\times101\\). Je cherche donc \u00e0 \u00e9crire les nombres \u00ab\u00a02\u00a0\u00bb (facile celui-l\u00e0 !), \u00ab\u00a05\u00a0\u00bb et \u00ab\u00a0101\u00a0\u00bb en utilisant que des \u00ab\u00a02\u00a0\u00bb et des \u00ab\u00a00\u00a0\u00bb. Autant \u00ab\u00a02\u00a0\u00bb et \u00ab\u00a05\u00a0\u00bb sont assez faciles, autant \u00ab\u00a0101\u00a0\u00bb est un peu plus long&#8230; En effet, $$101 = 1+100=1+50\\times2=1+5^2\\times2\\times2$$. Ouf ! J&rsquo;y suis ! Je n&rsquo;utilise que des \u00ab\u00a02\u00a0\u00bb et des \u00ab\u00a05\u00a0\u00bb, avec un petit \u00ab\u00a01\u00a0\u00bb qui tra\u00eene mais qui n&rsquo;est pas bien m\u00e9chant car je peux le consid\u00e9rer comme \u00e9tant le quotient de \u00ab\u00a02\u00a0\u00bb par \u00ab\u00a02\u00a0\u00bb.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Et tout ceci me donne alors :<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>(2+0)*2*(0+(2+0)\/2+0+(2+0)*2)*(0+(2+0)\/2+0+2*(0+2*(0+((2+0)*2+(0+2)\/(0+2))^(0+2))))<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Vous pouvez v\u00e9rifier avec Xcas par exemple : cela donne bien 2020 ! Et les \u00ab\u00a02\u00a0\u00bb et \u00ab\u00a00\u00a0\u00bb sont bien altern\u00e9s&#8230;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Maintenant, est-ce la meilleure fa\u00e7on d&rsquo;obtenir 2020 avec ces r\u00e8gles ? Je ne demande qu&rsquo;\u00e0 voir alors proposez vos solutions si elles sont diff\u00e9rentes de la mienne !<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Une_autre_solution\"><\/span>Une autre solution<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Cette solution est bas\u00e9e sur les puissances de 2. Comme 2020 est tr\u00e8s proche de 2048, qui est la puissance 11 de 2, je cherche d&rsquo;abord \u00e0 calculer 2048, puis \u00e0 soustraire le n\u00e9cessaire. Cela donne:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>(2+0)^((2+0)*((2+0)\/(2+0)+(2+0)*(2+0)))*(2+0)-(2+0)^((2+0)^2+(0+2)\/(0+2))+0+2+0+2<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Vous constaterez alors que cette deuxi\u00e8me expression admet l\u00e9g\u00e8rement moins de symboles que la premi\u00e8re, mais le gain n&rsquo;est pas ouf non plus&#8230; 81 contre 83 ! Alors, sur cette m\u00eame logique, proposons une autre expression:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>2^(0+2*(0+2*(0+2)+(0+2)))\/(0+2)-(0+2)^((0+2)^(0+2)+(0+2)\/(0+2))+0+2+0+2<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">C&rsquo;est bien mieux non? Et seulement 71 symboles !<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Qui pourra trouver une expression plus courte ? Je suis s\u00fbr que c&rsquo;est possible&#8230;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u00c0 l&rsquo;occasion de la nouvelle ann\u00e9e qui approche, je vous propose un jeu qui m&rsquo;a \u00e9t\u00e9 inspir\u00e9 directement du nombre 2020&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6],"tags":[154,68],"class_list":["post-1892","post","type-post","status-publish","format-standard","hentry","category-mathematiques","tag-arithmetique","tag-jeu"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.5 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Le jeu du 2020 - Mathweb.fr<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.mathweb.fr\/euclide\/2019\/12\/28\/le-jeu-du-2020\/\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Le jeu du 2020 - 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