{"id":2139,"date":"2020-03-09T17:09:09","date_gmt":"2020-03-09T16:09:09","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=2139"},"modified":"2020-03-09T17:39:04","modified_gmt":"2020-03-09T16:39:04","slug":"tour-de-magie","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2020\/03\/09\/tour-de-magie\/","title":{"rendered":"Tour de magie"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Cet article est inspir\u00e9 d&rsquo;un probl\u00e8me tir\u00e9 du 7\u00e8me Championnat International des Jeux Math\u00e9matiques et Logiques ( demi-finale, 20 mars 1993).<\/p>\n\n\n\n<!--more-->\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_87 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\/2020\/03\/09\/tour-de-magie\/#Le_tour\" >Le tour<\/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\/2020\/03\/09\/tour-de-magie\/#Explications\" >Explications<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Le_tour\"><\/span>Le tour<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">On dispose de six cartes A, B, C, D, E et F sur lesquelles sont mis des nombres de -19 \u00e0 +19. Une personne choisi un nombre au hasard compris entre -19 et +19 et dit sur quelles cartes ce nombre se trouve. Gr\u00e2ce \u00e0 cela, on est capable de dire le nombre choisi.<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"350\" height=\"250\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/03\/tour-magie-1.png\" alt=\"\" class=\"wp-image-2142\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/03\/tour-magie-1.png 350w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/03\/tour-magie-1-300x214.png 300w\" sizes=\"auto, (max-width: 350px) 100vw, 350px\" \/><figcaption>Les 6 cartes<\/figcaption><\/figure><\/div>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Explications\"><\/span>Explications<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">\u00c0 chaque carte, nous avons au pr\u00e9alable associ\u00e9 un nombre : \\(x_1\\) pour la carte A, \\(x_2\\) pour la carte B, etc.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">En ajoutant uniquement les \\(x_i\\) des cartes o\u00f9 figurent le nombre choisi au hasard, on obtient justement ce nombre. Il ne reste plus qu&rsquo;\u00e0 trouver les \\(x_i\\)&#8230;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Pour cela, il suffit de prendre 6 nombres au hasard: je vais prendre 0, 1, -1, 2, -2 et 3.<\/p>\n\n\n\n<ol class=\"wp-block-list\"><li>\u00ab\u00a00\u00a0\u00bb appara\u00eet sur toutes les cartes, donc:$$x_1+x_2+x_3+x_4+x_5+x_6=0.$$<\/li><li>\u00ab\u00a01\u00a0\u00bb appara\u00eet sur les cartes A, E et F donc:$$x_1+x_5+x_6=1.$$<\/li><li>\u00ab\u00a0-1\u00a0\u00bb appara\u00eet sur les cartes A, B, E et F donc:$$x_1+x_2+x_5+x_6=-1.$$<\/li><li>\u00ab\u00a02\u00a0\u00bb appara\u00eet sur les cartes A, C, D, E et F donc:$$x_1+x_3+x_4+x_5+x_6=2.$$<\/li><li>\u00ab\u00a0-2\u00a0\u00bb appara\u00eet sur les cartes A, B, C, E et F donc:$$x_1+x_2+x_3+x_5+x_6=-2.$$<\/li><li>\u00ab\u00a03\u00a0\u00bb appara\u00eet sur les cartes A, D, E et F donc:$$x_1+x_4+x_5+x_6=3.$$<\/li><\/ol>\n\n\n\n<p class=\"wp-block-paragraph\">Nous avons 6 \u00e9quations \u00e0 6 inconnues, mais si on cherche \u00e0 r\u00e9soudre le syst\u00e8me form\u00e9 par ces 6 \u00e9quations, on n&rsquo;arrivera pas \u00e0 trouver toutes les inconnues. On peut seulement obtenir:$$x_2=-2,\\ x_3=-1,\\ x_4=2.$$ En effet, on arrive \u00e0 montrer que \\(x_1=1-x_5-x_6\\) sans obtenir \\(x_5\\) et \\(x_6\\). Il nous faut donc prendre d&rsquo;autres nombres. <\/p>\n\n\n\n<ol class=\"wp-block-list\"><li>\u00ab\u00a019\u00a0\u00bb appara\u00eet sur les cartes D, E et F donc:$$x_4+x_5+x_6=19\\text{ soit }x_5+x_6=17.$$<\/li><li>\u00ab\u00a04\u00a0\u00bb appara\u00eet sur les cartes B et E donc:$$x_2+x_5=4\\text{ soit }x_5=6.$$<\/li><\/ol>\n\n\n\n<p class=\"wp-block-paragraph\">On en d\u00e9duit alors que:$$x_6=11$$et donc que:$$x_1=1-6-11=-16.$$<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Cet article est inspir\u00e9 d&rsquo;un probl\u00e8me tir\u00e9 du 7\u00e8me Championnat International des Jeux Math\u00e9matiques et Logiques ( demi-finale, 20 mars 1993).<\/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":[189],"class_list":["post-2139","post","type-post","status-publish","format-standard","hentry","category-mathematiques","tag-magie"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.4 - 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