From 466b84168367ecf351e726019f7f7c0a07449e77 Mon Sep 17 00:00:00 2001
From: Jean-Michel Sarlat Composition d’un jeu avec \(p+1\) symboles par carte (\(p=7\) pour le cas qui nous intéresse), \(p\) étant un nombre premier. Ingrédients : \(1+p+p^2\) symboles distincts deux à deux, répartis de la façon suivante: 1 symbole \(a\), \(p\) symboles \(b_i\) avec \(0\leqslant i\leqslant p-1\), \(p^2\) symboles \(c_{i,j}\) avec \(0\leqslant i,j\leqslant p-1\). Ce qui précède n’est qu’une façon de ranger les symboles pour mieux les redistribuer ensuite, ils ne sont pas de natures différentes. Les cartes : on peut en obtenir jusqu’à \(1+p+p^2\), autant qu’il y a de symboles... 1 carte \(A=\{a, b_0, \dots b_{p-1}\}\), \(p\) cartes \(B_i\) (\(0\leqslant i\leqslant p-1\)) telles que
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\(p^2\) cartes \(C_{i,j}\) (\(0\leqslant i,j\leqslant p-1\)) telles que:
Questions en suspens:
+Si \(p=7\) alors \(p^2+p+1=57\), pourquoi le jeu se limite-t-il à \(55\) cartes ?
Si \(p\) n’est pas premier (utilisé pour démontrer que les cartes \(C_{i,j}\) ont un symbole et un seul en commun), le problème a-t-il une solution ? Si oui, laquelle ?
Le Dobble est jeu de 55 cartes comportant 8 symboles chacune; deux cartes distinctes ont un, et un seul, symbole en commun.
+Les fichiers présentés ici proposent une construction du jeu au format PDF à partir d'un source luatex. L'algorithme utilisé est décrit ci-dessous.
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