{"id":407,"date":"2018-07-06T10:29:45","date_gmt":"2018-07-06T08:29:45","guid":{"rendered":"https:\/\/blogs.futura-sciences.com\/lehning\/?p=407"},"modified":"2018-07-28T17:00:16","modified_gmt":"2018-07-28T15:00:16","slug":"les-suites-qui-se-racontent-et-le-jeu-de-robinson","status":"publish","type":"post","link":"https:\/\/blogs.futura-sciences.com\/lehning\/2018\/07\/06\/les-suites-qui-se-racontent-et-le-jeu-de-robinson\/","title":{"rendered":"Les suites qui se racontent et le jeu de Robinson"},"content":{"rendered":"

Les suites qui se racontent sont entr\u00e9es un jour dans ma vie pour ne plus en sortir \u00e0 travers une \u00e9nigme math\u00e9matique que me posa un ami : <\/span><\/span><\/p>\n

Si une suite commence par 0, 10, 1110, 3110, 132110, 13123110, quel est le nombre suivant\u00a0?<\/span><\/span><\/em><\/p>\n

Ma r\u00e9ponse f\u00fbt 23124110 (dans 0, je compte un 0 ce que j’\u00e9cris 10\u00a0; dans 10, je compte un 1, un 0 ce que j’\u00e9cris 1110 ; dans 1110, je compte trois 1, un 0 ce qui donne 3110, etc.). En quelque sorte, chaque terme “raconte” le pr\u00e9c\u00e9dent.<\/span><\/span><\/p>\n

L’\u00e9nigme se transforme en suite \u2026 et en conjecture<\/h2>\n

J’aurais pu m’arr\u00eater l\u00e0 mais, allez savoir pourquoi, je continuais\u00a0: 1413223110, 1423224110, 2413323110, 1433223110, 1433223110. La suite est donc constante \u00e0 partir de ce terme. Le r\u00e9sultat me sembla surprenant, c’est pourquoi j’essayais d’autres valeurs initiales. J’avais beau examiner un grand nombre de valeurs, je trouvais toujours une suite p\u00e9riodique, la p\u00e9riode \u00e9tant 1, 2 ou 3. Tr\u00e8s vite convaincu de ce r\u00e9sultat, je tentais de prouver ce qui \u00e9tait devenu une conjecture. Apr\u00e8s plusieurs mois de recherche, je trouvais 109 points fixes, 31 cycles de p\u00e9riode deux et 10 cycles de p\u00e9riode trois (voir plus loin leur liste exhaustive).\u00a0<\/span><\/span><\/p>\n

Puis la conjecture en th\u00e9or\u00e8me<\/h2>\n

Pour trouver tous les cycles des suites qui se racontent, on peut utiliser un ordinateur \u00e0 condition de r\u00e9duire d’abord le nombre de cas \u00e0 essayer. Il est \u00e9galement possible de faire un raisonnement analytique classique. Pour ceux que le sujet int\u00e9resse, j’ai racont\u00e9 cette qu\u00eate de la preuve dans\u00a0:<\/p>\n

Herv\u00e9 Lehning, \u00ab\u00a0Computer-aided or analytic proof?\u00a0\u00bb, College Mathematics Journal<\/em>, vol.\u00a021, no<\/sup>\u00a03,\u200e 1990, p.\u00a0228-239<\/p>\n

Herv\u00e9 Lehning,\u00a0\u00ab Quelle est la meilleure preuve ?\u00a0\u00bb,\u00a0Quadrature<\/em>, n\u00b0 11, 1992 (Ce dernier article est illustr\u00e9 Par Charb.)<\/p>\n

\"\"
Page de l’article avec l’illustration de Charb<\/figcaption><\/figure>\n

Le jeu de Robinson<\/h2>\n

Quand j’ai propos\u00e9 mon article, on me fit remarquer que je r\u00e9solvais sans le savoir une conjecture de Douglas Hofstadter (parue dans Ma Th\u00e9magie<\/em>) li\u00e9e \u00e0 un jeu invent\u00e9 dans les ann\u00e9es soixante-dix par Rapha\u00ebl Robinson (1911 \u2013 1995), un math\u00e9maticien am\u00e9ricain. Le but est de remplir les blancs de la phrase suivante afin qu’elle devienne vraie\u00a0:<\/span><\/span><\/p>\n

Dans cette phrase, il y a __ 0, __ 1, __ 2, __ 3, __ 4, __ 5, __ 6, __ 7, __ 8, et __ 9<\/span><\/span><\/p>\n

On remarque imm\u00e9diatement que tout point fixe utilisant les dix chiffres d\u2019une suite qui se raconte est solution, et r\u00e9ciproquement. En utilisant la liste des 109 points fixes, on trouve deux solutions :<\/span><\/span><\/p>\n

Dans cette phrase, il y a 1 0, 11 1, 2 2, 1 3, 1 4, 1 5, 1 6, 1 7, 1 8, et 1 9.<\/span><\/span><\/p>\n

Dans cette phrase, il y a 1 0, 7 1, 3 2, 2 3, 1 4, 1 5, 1 6, 2 7, 1 8, et 1 9.<\/span><\/span><\/p>\n

Les autres points fixes fournissent des solutions \u00e0 un jeu de Robinson l\u00e9g\u00e8rement modifi\u00e9 o\u00f9 certains chiffres peuvent \u00eatre supprim\u00e9s. Par exemple\u00a0:<\/span><\/span><\/p>\n

Dans cette phrase, il y a 3 1, 2 2, 3 3, 1 4 et 1 5.<\/span><\/span><\/p>\n

Suppl\u00e9ments : cycles des suites qui se racontent<\/h2>\n

Pour en simplifier la lecture, je noterai <n<\/em>> le fait que n<\/em> chiffres, choisis arbitrairement parmi les chiffres possibles, soient pr\u00e9c\u00e9d\u00e9s d’un 1.<\/p>\n

Liste des points fixes<\/h2>\n

1 9 \/ 1 8 \/ 1 7 \/ 1 6 \/ 1 5 \/ 1 4 \/ 1 3 \/ 2 2 \/ 11 1 \/ 1 0 et<\/p>\n

1 9 \/ 1 8 \/ 2 7 \/ 1 6 \/ 1 5 \/ 2 3 \/ 3 2 \/ 7 1 \/ 1 0 ce qui fait 2 points fixes,<\/p>\n

11\u00a01\u00a0\/\u00a0<8> ce qui fait 9 points fixes,\u00a02\u00a06\u00a0\/\u00a02\u00a03\u00a0\/\u00a03\u00a02\u00a0\/\u00a06\u00a01\u00a0\/\u00a0<5> ce qui fait 6 points fixes,\u00a02\u00a05\u00a0\/\u00a02\u00a03\u00a0\/\u00a03\u00a02\u00a0\/\u00a05\u00a01\u00a0\/\u00a0<4> ce qui fait 15 points fixes,\u00a02\u00a04\u00a0\/\u00a02\u00a03\u00a0\/\u00a03\u00a02\u00a0\/\u00a04\u00a01\u00a0\/\u00a0<3> ce qui fait 20 points fixes,\u00a03\u00a03\u00a0\/\u00a02\u00a02\u00a0\/\u00a03\u00a01\u00a0\/\u00a0<2> ce qui fait 21 points fixes,\u00a02\u00a03\u00a0\/\u00a03\u00a02\u00a0\/\u00a02\u00a01\u00a0\/\u00a0<1> ce qui fait 7 points fixes,\u00a03\u00a03\u00a0\/\u00a03\u00a01\u00a0\/\u00a0<2> ce qui fait 28 points fixes,\u00a02\u00a02 ce qui fait 1 point fixe,\u00a0d’o\u00f9 un total de 109 points fixes.<\/p>\n

Liste des cycles de p\u00e9riode deux<\/h2>\n

2\u00a08\u00a0\/\u00a01\u00a07\u00a0\/\u00a0\u00a01\u00a04\u00a0\/\u00a04\u00a02\u00a0\/\u00a07\u00a01\u00a0\/\u00a0<5>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 1\u00a08\u00a0\/\u00a02\u00a07\u00a0\/\u00a02\u00a04\u00a0\/\u00a02\u00a02\u00a0\/\u00a08\u00a01\u00a0\/\u00a0<5> ce qui fait 1 cycle,\u00a0<\/span><\/span><\/p>\n

2\u00a07\u00a0\/\u00a01\u00a06\u00a0\/\u00a01\u00a04\u00a0\/\u00a04\u00a02\u00a0\/\u00a06\u00a01\u00a0\/\u00a0<4>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 1\u00a07\u00a0\/\u00a02\u00a06\u00a0\/\u00a02\u00a04\u00a0\/\u00a02\u00a02\u00a0\/\u00a07\u00a01\u00a0\/\u00a0<4> ce qui fait 5 cycles,<\/span><\/span><\/p>\n

2\u00a06\u00a0\/\u00a01\u00a05\u00a0\/\u00a01\u00a04\u00a0\/\u00a04\u00a02\u00a0\/\u00a05\u00a01\u00a0\/\u00a0<3>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 1\u00a06\u00a0\/\u00a02\u00a05\u00a0\/\u00a02\u00a04\u00a0\/\u00a02\u00a02\u00a0\/\u00a06\u00a01\u00a0\/\u00a0<3> ce qui fait 10 cycles,<\/span><\/span><\/p>\n

2\u00a04\u00a0\/\u00a01\u00a03\u00a0\/\u00a04\u00a02\u00a0\/\u00a03\u00a01\u00a0\/\u00a0<2>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 2\u00a04\u00a0\/\u00a02\u00a03\u00a0\/\u00a02\u00a02\u00a0\/\u00a04\u00a01\u00a0\/\u00a0<2> ce qui fait 15 cycles,<\/span><\/span><\/p>\n

d’o\u00f9 un total de 31 cycles de p\u00e9riode 2.<\/span><\/span><\/p>\n

Liste des cycles de p\u00e9riode trois<\/h2>\n

2\u00a05\u00a0\/\u00a01\u00a04\u00a0\/\u00a01\u00a03\u00a0\/\u00a04\u00a02\u00a0\/\u00a04\u00a01\u00a0\/\u00a0<2>\u00a0\u00a0\u00a0\u00a0\u00a01\u00a05\u00a0\/\u00a03\u00a04\u00a0\/\u00a01\u00a03\u00a0\/\u00a02\u00a02\u00a0\/\u00a05\u00a01\u00a0\/\u00a0<2>\u00a0\u00a0\u00a0\u00a0\u00a0 2\u00a05\u00a0\/\u00a01\u00a04\u00a0\/\u00a02\u00a03\u00a0\/\u00a02\u00a02\u00a0\/\u00a05\u00a01\/\u00a0<2><\/span><\/span><\/p>\n

ce qui fait 10 cycles de p\u00e9riode 3.<\/span><\/span><\/p>\n

\u00a0<\/span><\/span><\/p>\n

 <\/p>\n","protected":false},"excerpt":{"rendered":"

Les suites qui se racontent sont entr\u00e9es un jour dans ma vie pour ne plus en sortir \u00e0 travers une \u00e9nigme math\u00e9matique que me posa un ami : Si une suite commence par 0, 10, 1110, 3110, 132110, 13123110, quel est le nombre suivant\u00a0? Ma r\u00e9ponse f\u00fbt 23124110 (dans 0, je compte un 0 ce … Continuer la lecture de Les suites qui se racontent et le jeu de Robinson<\/span> →<\/span><\/a><\/p>\n","protected":false},"author":12,"featured_media":410,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":"","_links_to":"","_links_to_target":""},"categories":[8,29],"tags":[],"class_list":["post-407","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-enigmes","category-maths-2"],"yoast_head":"\nLes suites qui se racontent et le jeu de Robinson, par Herv\u00e9 Lehning<\/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:\/\/blogs.futura-sciences.com\/lehning\/2018\/07\/06\/les-suites-qui-se-racontent-et-le-jeu-de-robinson\/\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Les suites qui se racontent et le jeu de Robinson, par Herv\u00e9 Lehning\" \/>\n<meta property=\"og:description\" content=\"Les suites qui se racontent sont entr\u00e9es un jour dans ma vie pour ne plus en sortir \u00e0 travers une \u00e9nigme math\u00e9matique que me posa un ami : Si une suite commence par 0, 10, 1110, 3110, 132110, 13123110, quel est le nombre suivant\u00a0? 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