{"id":3389,"date":"2023-12-01T18:47:07","date_gmt":"2023-12-01T17:47:07","guid":{"rendered":"https:\/\/math.univ-lyon1.fr\/dream\/?page_id=3389"},"modified":"2026-04-27T12:45:38","modified_gmt":"2026-04-27T10:45:38","slug":"le-probleme-qui-dechire-2","status":"publish","type":"post","link":"https:\/\/math.univ-lyon1.fr\/dream\/?p=3389","title":{"rendered":"Le probl\u00e8me qui d\u00e9chire"},"content":{"rendered":"\r\n<h2>\u00c9nonc\u00e9 du probl\u00e8me<\/h2>\r\n<h2><iframe loading=\"lazy\" title=\"Le Probl\u00e8me qui D\u00e9chire\" src=\"https:\/\/tube-sciences-technologies.apps.education.fr\/videos\/embed\/ce2f6954-c61d-4d43-a15a-3141a320ffce\" width=\"560\" height=\"315\" frameborder=\"0\" sandbox=\"allow-same-origin allow-scripts allow-popups allow-forms\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/h2>\r\n<p>Tout part d&rsquo;une feuille de papier que l&rsquo;on va couper en plusieurs morceaux.<br \/>Imaginons : je la coupe en deux, puis je prends un des deux morceaux et je le recoupe en deux, puis je prends un des morceaux et je le recoupe en deux et ainsi de suite. Combien de fois je devrais faire cette op\u00e9ration pour avoir 2016 morceaux de papier ?<\/p>\r\n<p>Maintenant : je la coupe en trois, puis je prends un des trois morceaux et je le recoupe en trois, puis je prends un des morceaux et je le recoupe en trois et ainsi de suite. Est-ce que je pourrais avoir un jour 2016 morceaux ? Et si je faisais la m\u00eame op\u00e9ration mais en coupant chaque fois en quatre ? En cinq ?. . .<\/p>\r\n<p>Plus g\u00e9n\u00e9ralement, quelles sont les d\u00e9coupes qui me permettraient d&rsquo;obtenir 2016 morceaux ? Et si je voulais atteindre 2017 ? 2018 ?<\/p>\r\n<p>Maintenant, je choisis de couper ma feuille en deux ou en trois parties. Est-ce que je peux atteindre 2016 ? De combien de fa\u00e7ons diff\u00e9rentes ? Et si je coupe en trois ou quatre parties ? En six ou huit parties ?. . . Est-ce que je peux toujours obtenir 2016 (2017,2018,. . .) morceaux de papier ? Si oui, pourquoi et si non, quand est-ce que je ne peux pas ?<\/p>\r\n<h1>Les math\u00e9matiques en jeu<\/h1>\r\n<ul>\r\n<li>Multiples et diviseurs des nombres d&rsquo;usage courant. Crit\u00e8res de divisibilit\u00e9 (2,3, 4, 5, 9, 10)<\/li>\r\n<li>Division euclidienne<\/li>\r\n<li>R\u00e9soudre des probl\u00e8mes mettant en jeu les quatre op\u00e9rations. Sens des op\u00e9rations.<\/li>\r\n<li>Calculs<\/li>\r\n<li>Utiliser des outils pour repr\u00e9senter un probl\u00e8me<\/li>\r\n<li>Calcul litt\u00e9ral<\/li>\r\n<li>Suites<\/li>\r\n<li>\u00c9quations diophantiennes<\/li>\r\n<\/ul>\r\n<h4>Analyse didactique<\/h4>\r\n<p><a href=\"https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2022\/03\/lePbQuDechire_Analyse_didactique.pdf\" target=\"_blank\" rel=\"noopener\"> T\u00e9l\u00e9charger le document <\/a><\/p>\r\n<h4>Analyse math\u00e9matique<\/h4>\r\n<p><a href=\"https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2022\/03\/lePbQuiDechire_Analyse_Maths.pdf\" target=\"_blank\" rel=\"noopener\"> T\u00e9l\u00e9charger le document <\/a><\/p>\r\n<h4>Exemples de mise en \u0153uvre<\/h4>\r\n<p><a href=\"https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2022\/03\/lePbQuiDechire_Mise_en_oeuvre.pdf\" target=\"_blank\" rel=\"noopener\"> T\u00e9l\u00e9charger le document <\/a><\/p>\r\n<h4>Pour aller plus loin<\/h4>\r\n<p><a href=\"https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2022\/03\/lePbQuDechire_Pour_Aller_Plus_Loin.pdf\" target=\"_blank\" rel=\"noopener\"> T\u00e9l\u00e9charger le document <\/a><\/p>\r\n<h2>Retours d&rsquo;exp\u00e9riences<\/h2>\r\n<p>Des retours d&rsquo;exp\u00e9riences partag\u00e9s par les enseignant.e.s du groupe d&rsquo;exp\u00e9rimentation qui ont mis en place le probl\u00e8me qui d\u00e9chire dans le cadre d&rsquo;un enseignement fond\u00e9 sur les probl\u00e8mes<\/p>\r\n<p><a href=\"https:\/\/math.univ-lyon1.fr\/dream\/?page_id=77\" target=\"_self\"> Retour aux Exemples de SDRP <\/a><\/p>\r\n","protected":false},"excerpt":{"rendered":"<p>Tout part d\u2019une feuille de papier que l\u2019on va couper en plusieurs morceaux. Est-il possible d&rsquo;atteindre 2160 morceaux en suivant une d\u00e9coupe impos\u00e9e ?<\/p>\n","protected":false},"author":3,"featured_media":2916,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[106],"tags":[],"niveau":[112,113,114],"type_situation":[115],"theme_maths":[118],"probleme_associe":[123],"type-expe":[],"class_list":{"0":"post-3389","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-sdrp","8":"niveau-cycle_3","9":"niveau-cycle-4","10":"niveau-lycee","11":"type_situation-sdrp","12":"theme_maths-nombres-et-calculs","13":"probleme_associe-le-probleme-qui-dechire","14":"czr-hentry"},"rttpg_featured_image_url":{"full":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/12\/PBDechire.png",340,350,false],"landscape":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/12\/PBDechire.png",340,350,false],"portraits":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/12\/PBDechire.png",340,350,false],"thumbnail":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/12\/PBDechire-150x150.png",150,150,true],"medium":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/12\/PBDechire-291x300.png",291,300,true],"large":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/12\/PBDechire.png",340,350,false],"1536x1536":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/12\/PBDechire.png",340,350,false],"2048x2048":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/12\/PBDechire.png",340,350,false],"tc-grid-full":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/12\/PBDechire.png",340,350,false],"tc-grid":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/12\/PBDechire.png",340,350,false],"tc-thumb":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/12\/PBDechire-270x250.png",270,250,true],"slider-full":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/12\/PBDechire.png",340,350,false],"slider":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/12\/PBDechire.png",340,350,false],"tc-sq-thumb":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/12\/PBDechire.png",340,350,false],"tc-ws-thumb":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/12\/PBDechire.png",340,350,false],"tc-ws-small-thumb":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/12\/PBDechire-340x297.png",340,297,true],"tc-slider-small":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/12\/PBDechire-340x235.png",340,235,true]},"rttpg_author":{"display_name":"St\u00e9phanie CROQUELOIS","author_link":"https:\/\/math.univ-lyon1.fr\/dream\/?author=3"},"rttpg_comment":0,"rttpg_category":"<a href=\"https:\/\/math.univ-lyon1.fr\/dream\/?cat=106\" rel=\"category\">SDRP<\/a>","rttpg_excerpt":"Tout part d\u2019une feuille de papier que l\u2019on va couper en plusieurs morceaux. Est-il possible d'atteindre 2160 morceaux en suivant une d\u00e9coupe impos\u00e9e ?","_links":{"self":[{"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=\/wp\/v2\/posts\/3389","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=3389"}],"version-history":[{"count":11,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=\/wp\/v2\/posts\/3389\/revisions"}],"predecessor-version":[{"id":3772,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=\/wp\/v2\/posts\/3389\/revisions\/3772"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=\/wp\/v2\/media\/2916"}],"wp:attachment":[{"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=3389"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=3389"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=3389"},{"taxonomy":"niveau","embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=%2Fwp%2Fv2%2Fniveau&post=3389"},{"taxonomy":"type_situation","embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=%2Fwp%2Fv2%2Ftype_situation&post=3389"},{"taxonomy":"theme_maths","embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=%2Fwp%2Fv2%2Ftheme_maths&post=3389"},{"taxonomy":"probleme_associe","embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=%2Fwp%2Fv2%2Fprobleme_associe&post=3389"},{"taxonomy":"type-expe","embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=%2Fwp%2Fv2%2Ftype-expe&post=3389"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}