All the Suckers's Bets Decks with, 4, decks, with deck sizes, 6, 6, 6, 6, using , 7, denominations labelled from 1 to, 7 Where No denomination appears in more than one deck (hence there is never a \ tie) By Shalosh B. Ekhad Up to obvious cyclic symmetry, there are, 38, distinct such collection of decks where Deck , 1, beats deck, 2 Deck , 2, beats deck, 3 Deck , 3, beats deck, 4 but surprise, surprise Deck, 4, beats deck 1 Here there all are ------------------------------------------------------- Set Number, 1, is as follows Deck , 1, has denominations , 1, 1, 5, 5, 5, 5 Deck , 2, has denominations , 4, 4, 4, 4, 4, 4 Deck , 3, has denominations , 3, 3, 3, 3, 3, 3 Deck , 4, has denominations , 2, 2, 2, 2, 6, 7 It is readily seen that Deck , 1, beats deck , 2, with probability , 2/3 Deck , 2, beats deck , 3, with probability , 1 Deck , 3, beats deck , 4, with probability , 2/3 yet Deck, 4, beats deck 1, with probability , 5/9 ------------------------------------------------------- Set Number, 2, is as follows Deck , 1, has denominations , 1, 1, 5, 5, 5, 5 Deck , 2, has denominations , 4, 4, 4, 4, 4, 4 Deck , 3, has denominations , 3, 3, 3, 3, 3, 6 Deck , 4, has denominations , 2, 2, 2, 2, 7, 7 It is readily seen that Deck , 1, beats deck , 2, with probability , 2/3 Deck , 2, beats deck , 3, with probability , 5/6 Deck , 3, beats deck , 4, with probability , 2/3 yet Deck, 4, beats deck 1, with probability , 5/9 ------------------------------------------------------- Set Number, 3, is as follows Deck , 1, has denominations , 1, 1, 5, 5, 5, 5 Deck , 2, has denominations , 4, 4, 4, 4, 4, 4 Deck , 3, has denominations , 3, 3, 3, 3, 3, 7 Deck , 4, has denominations , 2, 2, 2, 2, 6, 6 It is readily seen that Deck , 1, beats deck , 2, with probability , 2/3 Deck , 2, beats deck , 3, with probability , 5/6 13 Deck , 3, beats deck , 4, with probability , -- 18 yet Deck, 4, beats deck 1, with probability , 5/9 ------------------------------------------------------- Set Number, 4, is as follows Deck , 1, has denominations , 1, 1, 5, 5, 5, 5 Deck , 2, has denominations , 4, 4, 4, 4, 4, 4 Deck , 3, has denominations , 3, 3, 3, 3, 3, 7 Deck , 4, has denominations , 2, 2, 2, 6, 6, 6 It is readily seen that Deck , 1, beats deck , 2, with probability , 2/3 Deck , 2, beats deck , 3, with probability , 5/6 Deck , 3, beats deck , 4, with probability , 7/12 yet Deck, 4, beats deck 1, with probability , 2/3 ------------------------------------------------------- Set Number, 5, is as follows Deck , 1, has denominations , 1, 1, 5, 5, 5, 5 Deck , 2, has denominations , 4, 4, 4, 4, 4, 4 Deck , 3, has denominations , 3, 3, 3, 3, 6, 6 Deck , 4, has denominations , 2, 2, 2, 2, 7, 7 It is readily seen that Deck , 1, beats deck , 2, with probability , 2/3 Deck , 2, beats deck , 3, with probability , 2/3 Deck , 3, beats deck , 4, with probability , 2/3 yet Deck, 4, beats deck 1, with probability , 5/9 ------------------------------------------------------- Set Number, 6, is as follows Deck , 1, has denominations , 1, 1, 5, 5, 5, 5 Deck , 2, has denominations , 4, 4, 4, 4, 4, 4 Deck , 3, has denominations , 3, 3, 3, 3, 7, 7 Deck , 4, has denominations , 2, 2, 2, 2, 6, 6 It is readily seen that Deck , 1, beats deck , 2, with probability , 2/3 Deck , 2, beats deck , 3, with probability , 2/3 Deck , 3, beats deck , 4, with probability , 7/9 yet Deck, 4, beats deck 1, with probability , 5/9 ------------------------------------------------------- Set Number, 7, is as follows Deck , 1, has denominations , 1, 1, 5, 5, 5, 5 Deck , 2, has denominations , 4, 4, 4, 4, 4, 4 Deck , 3, has denominations , 3, 3, 3, 3, 7, 7 Deck , 4, has denominations , 2, 2, 2, 6, 6, 6 It is readily seen that Deck , 1, beats deck , 2, with probability , 2/3 Deck , 2, beats deck , 3, with probability , 2/3 Deck , 3, beats deck , 4, with probability , 2/3 yet Deck, 4, beats deck 1, with probability , 2/3 ------------------------------------------------------- Set Number, 8, is as follows Deck , 1, has denominations , 1, 1, 5, 5, 5, 5 Deck , 2, has denominations , 4, 4, 4, 4, 4, 4 Deck , 3, has denominations , 3, 3, 3, 3, 7, 7 Deck , 4, has denominations , 2, 2, 6, 6, 6, 6 It is readily seen that Deck , 1, beats deck , 2, with probability , 2/3 Deck , 2, beats deck , 3, with probability , 2/3 Deck , 3, beats deck , 4, with probability , 5/9 yet Deck, 4, beats deck 1, with probability , 7/9 ------------------------------------------------------- Set Number, 9, is as follows Deck , 1, has denominations , 1, 1, 5, 5, 5, 5 Deck , 2, has denominations , 4, 4, 4, 4, 4, 6 Deck , 3, has denominations , 3, 3, 3, 3, 3, 3 Deck , 4, has denominations , 2, 2, 2, 2, 7, 7 It is readily seen that Deck , 1, beats deck , 2, with probability , 5/9 Deck , 2, beats deck , 3, with probability , 1 Deck , 3, beats deck , 4, with probability , 2/3 yet Deck, 4, beats deck 1, with probability , 5/9 ------------------------------------------------------- Set Number, 10, is as follows Deck , 1, has denominations , 1, 1, 5, 5, 5, 5 Deck , 2, has denominations , 4, 4, 4, 4, 4, 7 Deck , 3, has denominations , 3, 3, 3, 3, 3, 3 Deck , 4, has denominations , 2, 2, 2, 2, 6, 6 It is readily seen that Deck , 1, beats deck , 2, with probability , 5/9 Deck , 2, beats deck , 3, with probability , 1 Deck , 3, beats deck , 4, with probability , 2/3 yet Deck, 4, beats deck 1, with probability , 5/9 ------------------------------------------------------- Set Number, 11, is as follows Deck , 1, has denominations , 1, 1, 5, 5, 5, 6 Deck , 2, has denominations , 4, 4, 4, 4, 4, 4 Deck , 3, has denominations , 3, 3, 3, 3, 3, 3 Deck , 4, has denominations , 2, 2, 2, 2, 7, 7 It is readily seen that Deck , 1, beats deck , 2, with probability , 2/3 Deck , 2, beats deck , 3, with probability , 1 Deck , 3, beats deck , 4, with probability , 2/3 yet Deck, 4, beats deck 1, with probability , 5/9 ------------------------------------------------------- Set Number, 12, is as follows Deck , 1, has denominations , 1, 1, 5, 5, 6, 6 Deck , 2, has denominations , 4, 4, 4, 4, 4, 4 Deck , 3, has denominations , 3, 3, 3, 3, 3, 3 Deck , 4, has denominations , 2, 2, 2, 2, 7, 7 It is readily seen that Deck , 1, beats deck , 2, with probability , 2/3 Deck , 2, beats deck , 3, with probability , 1 Deck , 3, beats deck , 4, with probability , 2/3 yet Deck, 4, beats deck 1, with probability , 5/9 ------------------------------------------------------- Set Number, 13, is as follows Deck , 1, has denominations , 1, 1, 5, 6, 6, 6 Deck , 2, has denominations , 4, 4, 4, 4, 4, 4 Deck , 3, has denominations , 3, 3, 3, 3, 3, 3 Deck , 4, has denominations , 2, 2, 2, 2, 7, 7 It is readily seen that Deck , 1, beats deck , 2, with probability , 2/3 Deck , 2, beats deck , 3, with probability , 1 Deck , 3, beats deck , 4, with probability , 2/3 yet Deck, 4, beats deck 1, with probability , 5/9 ------------------------------------------------------- Set Number, 14, is as follows Deck , 1, has denominations , 1, 1, 6, 6, 6, 6 Deck , 2, has denominations , 2, 2, 5, 5, 5, 5 Deck , 3, has denominations , 4, 4, 4, 4, 4, 4 Deck , 4, has denominations , 3, 3, 3, 3, 7, 7 It is readily seen that Deck , 1, beats deck , 2, with probability , 2/3 Deck , 2, beats deck , 3, with probability , 2/3 Deck , 3, beats deck , 4, with probability , 2/3 yet Deck, 4, beats deck 1, with probability , 5/9 ------------------------------------------------------- Set Number, 15, is as follows Deck , 1, has denominations , 1, 1, 6, 6, 6, 6 Deck , 2, has denominations , 2, 5, 5, 5, 5, 5 Deck , 3, has denominations , 4, 4, 4, 4, 4, 4 Deck , 4, has denominations , 3, 3, 3, 3, 7, 7 It is readily seen that Deck , 1, beats deck , 2, with probability , 2/3 Deck , 2, beats deck , 3, with probability , 5/6 Deck , 3, beats deck , 4, with probability , 2/3 yet Deck, 4, beats deck 1, with probability , 5/9 ------------------------------------------------------- Set Number, 16, is as follows Deck , 1, has denominations , 1, 1, 6, 6, 6, 6 Deck , 2, has denominations , 3, 3, 5, 5, 5, 5 Deck , 3, has denominations , 4, 4, 4, 4, 4, 4 Deck , 4, has denominations , 2, 2, 2, 2, 7, 7 It is readily seen that Deck , 1, beats deck , 2, with probability , 2/3 Deck , 2, beats deck , 3, with probability , 2/3 Deck , 3, beats deck , 4, with probability , 2/3 yet Deck, 4, beats deck 1, with probability , 5/9 ------------------------------------------------------- Set Number, 17, is as follows Deck , 1, has denominations , 1, 1, 6, 6, 6, 6 Deck , 2, has denominations , 3, 5, 5, 5, 5, 5 Deck , 3, has denominations , 4, 4, 4, 4, 4, 4 Deck , 4, has denominations , 2, 2, 2, 2, 7, 7 It is readily seen that Deck , 1, beats deck , 2, with probability , 2/3 Deck , 2, beats deck , 3, with probability , 5/6 Deck , 3, beats deck , 4, with probability , 2/3 yet Deck, 4, beats deck 1, with probability , 5/9 ------------------------------------------------------- Set Number, 18, is as follows Deck , 1, has denominations , 1, 1, 6, 6, 6, 6 Deck , 2, has denominations , 4, 4, 4, 4, 4, 4 Deck , 3, has denominations , 3, 3, 3, 3, 3, 5 Deck , 4, has denominations , 2, 2, 2, 2, 7, 7 It is readily seen that Deck , 1, beats deck , 2, with probability , 2/3 Deck , 2, beats deck , 3, with probability , 5/6 Deck , 3, beats deck , 4, with probability , 2/3 yet Deck, 4, beats deck 1, with probability , 5/9 ------------------------------------------------------- Set Number, 19, is as follows Deck , 1, has denominations , 1, 1, 6, 6, 6, 6 Deck , 2, has denominations , 4, 4, 4, 4, 4, 4 Deck , 3, has denominations , 3, 3, 3, 3, 5, 5 Deck , 4, has denominations , 2, 2, 2, 2, 7, 7 It is readily seen that Deck , 1, beats deck , 2, with probability , 2/3 Deck , 2, beats deck , 3, with probability , 2/3 Deck , 3, beats deck , 4, with probability , 2/3 yet Deck, 4, beats deck 1, with probability , 5/9 ------------------------------------------------------- Set Number, 20, is as follows Deck , 1, has denominations , 1, 1, 6, 6, 6, 6 Deck , 2, has denominations , 4, 4, 4, 4, 4, 5 Deck , 3, has denominations , 3, 3, 3, 3, 3, 3 Deck , 4, has denominations , 2, 2, 2, 2, 7, 7 It is readily seen that Deck , 1, beats deck , 2, with probability , 2/3 Deck , 2, beats deck , 3, with probability , 1 Deck , 3, beats deck , 4, with probability , 2/3 yet Deck, 4, beats deck 1, with probability , 5/9 ------------------------------------------------------- Set Number, 21, is as follows Deck , 1, has denominations , 1, 1, 6, 6, 6, 6 Deck , 2, has denominations , 4, 4, 4, 4, 5, 5 Deck , 3, has denominations , 3, 3, 3, 3, 3, 3 Deck , 4, has denominations , 2, 2, 2, 2, 7, 7 It is readily seen that Deck , 1, beats deck , 2, with probability , 2/3 Deck , 2, beats deck , 3, with probability , 1 Deck , 3, beats deck , 4, with probability , 2/3 yet Deck, 4, beats deck 1, with probability , 5/9 ------------------------------------------------------- Set Number, 22, is as follows Deck , 1, has denominations , 1, 1, 6, 6, 6, 6 Deck , 2, has denominations , 4, 4, 4, 5, 5, 5 Deck , 3, has denominations , 3, 3, 3, 3, 3, 3 Deck , 4, has denominations , 2, 2, 2, 2, 7, 7 It is readily seen that Deck , 1, beats deck , 2, with probability , 2/3 Deck , 2, beats deck , 3, with probability , 1 Deck , 3, beats deck , 4, with probability , 2/3 yet Deck, 4, beats deck 1, with probability , 5/9 ------------------------------------------------------- Set Number, 23, is as follows Deck , 1, has denominations , 1, 1, 6, 6, 6, 6 Deck , 2, has denominations , 4, 4, 5, 5, 5, 5 Deck , 3, has denominations , 3, 3, 3, 3, 3, 3 Deck , 4, has denominations , 2, 2, 2, 2, 7, 7 It is readily seen that Deck , 1, beats deck , 2, with probability , 2/3 Deck , 2, beats deck , 3, with probability , 1 Deck , 3, beats deck , 4, with probability , 2/3 yet Deck, 4, beats deck 1, with probability , 5/9 ------------------------------------------------------- Set Number, 24, is as follows Deck , 1, has denominations , 1, 1, 6, 6, 6, 6 Deck , 2, has denominations , 4, 5, 5, 5, 5, 5 Deck , 3, has denominations , 3, 3, 3, 3, 3, 3 Deck , 4, has denominations , 2, 2, 2, 2, 7, 7 It is readily seen that Deck , 1, beats deck , 2, with probability , 2/3 Deck , 2, beats deck , 3, with probability , 1 Deck , 3, beats deck , 4, with probability , 2/3 yet Deck, 4, beats deck 1, with probability , 5/9 ------------------------------------------------------- Set Number, 25, is as follows Deck , 1, has denominations , 1, 1, 6, 6, 6, 6 Deck , 2, has denominations , 5, 5, 5, 5, 5, 5 Deck , 3, has denominations , 2, 4, 4, 4, 4, 4 Deck , 4, has denominations , 3, 3, 3, 3, 7, 7 It is readily seen that Deck , 1, beats deck , 2, with probability , 2/3 Deck , 2, beats deck , 3, with probability , 1 Deck , 3, beats deck , 4, with probability , 5/9 yet Deck, 4, beats deck 1, with probability , 5/9 ------------------------------------------------------- Set Number, 26, is as follows Deck , 1, has denominations , 1, 1, 6, 6, 6, 6 Deck , 2, has denominations , 5, 5, 5, 5, 5, 5 Deck , 3, has denominations , 3, 3, 3, 3, 3, 4 Deck , 4, has denominations , 2, 2, 2, 2, 7, 7 It is readily seen that Deck , 1, beats deck , 2, with probability , 2/3 Deck , 2, beats deck , 3, with probability , 1 Deck , 3, beats deck , 4, with probability , 2/3 yet Deck, 4, beats deck 1, with probability , 5/9 ------------------------------------------------------- Set Number, 27, is as follows Deck , 1, has denominations , 1, 1, 6, 6, 6, 6 Deck , 2, has denominations , 5, 5, 5, 5, 5, 5 Deck , 3, has denominations , 3, 3, 3, 3, 4, 4 Deck , 4, has denominations , 2, 2, 2, 2, 7, 7 It is readily seen that Deck , 1, beats deck , 2, with probability , 2/3 Deck , 2, beats deck , 3, with probability , 1 Deck , 3, beats deck , 4, with probability , 2/3 yet Deck, 4, beats deck 1, with probability , 5/9 ------------------------------------------------------- Set Number, 28, is as follows Deck , 1, has denominations , 1, 1, 6, 6, 6, 6 Deck , 2, has denominations , 5, 5, 5, 5, 5, 5 Deck , 3, has denominations , 3, 3, 3, 4, 4, 4 Deck , 4, has denominations , 2, 2, 2, 2, 7, 7 It is readily seen that Deck , 1, beats deck , 2, with probability , 2/3 Deck , 2, beats deck , 3, with probability , 1 Deck , 3, beats deck , 4, with probability , 2/3 yet Deck, 4, beats deck 1, with probability , 5/9 ------------------------------------------------------- Set Number, 29, is as follows Deck , 1, has denominations , 1, 1, 6, 6, 6, 6 Deck , 2, has denominations , 5, 5, 5, 5, 5, 5 Deck , 3, has denominations , 3, 3, 4, 4, 4, 4 Deck , 4, has denominations , 2, 2, 2, 2, 7, 7 It is readily seen that Deck , 1, beats deck , 2, with probability , 2/3 Deck , 2, beats deck , 3, with probability , 1 Deck , 3, beats deck , 4, with probability , 2/3 yet Deck, 4, beats deck 1, with probability , 5/9 ------------------------------------------------------- Set Number, 30, is as follows Deck , 1, has denominations , 1, 1, 6, 6, 6, 6 Deck , 2, has denominations , 5, 5, 5, 5, 5, 5 Deck , 3, has denominations , 3, 4, 4, 4, 4, 4 Deck , 4, has denominations , 2, 2, 2, 2, 7, 7 It is readily seen that Deck , 1, beats deck , 2, with probability , 2/3 Deck , 2, beats deck , 3, with probability , 1 Deck , 3, beats deck , 4, with probability , 2/3 yet Deck, 4, beats deck 1, with probability , 5/9 ------------------------------------------------------- Set Number, 31, is as follows Deck , 1, has denominations , 1, 1, 6, 6, 6, 6 Deck , 2, has denominations , 5, 5, 5, 5, 5, 5 Deck , 3, has denominations , 4, 4, 4, 4, 4, 4 Deck , 4, has denominations , 2, 2, 2, 3, 7, 7 It is readily seen that Deck , 1, beats deck , 2, with probability , 2/3 Deck , 2, beats deck , 3, with probability , 1 Deck , 3, beats deck , 4, with probability , 2/3 yet Deck, 4, beats deck 1, with probability , 5/9 ------------------------------------------------------- Set Number, 32, is as follows Deck , 1, has denominations , 1, 1, 6, 6, 6, 6 Deck , 2, has denominations , 5, 5, 5, 5, 5, 5 Deck , 3, has denominations , 4, 4, 4, 4, 4, 4 Deck , 4, has denominations , 2, 2, 3, 3, 7, 7 It is readily seen that Deck , 1, beats deck , 2, with probability , 2/3 Deck , 2, beats deck , 3, with probability , 1 Deck , 3, beats deck , 4, with probability , 2/3 yet Deck, 4, beats deck 1, with probability , 5/9 ------------------------------------------------------- Set Number, 33, is as follows Deck , 1, has denominations , 1, 1, 6, 6, 6, 6 Deck , 2, has denominations , 5, 5, 5, 5, 5, 5 Deck , 3, has denominations , 4, 4, 4, 4, 4, 4 Deck , 4, has denominations , 2, 3, 3, 3, 7, 7 It is readily seen that Deck , 1, beats deck , 2, with probability , 2/3 Deck , 2, beats deck , 3, with probability , 1 Deck , 3, beats deck , 4, with probability , 2/3 yet Deck, 4, beats deck 1, with probability , 5/9 ------------------------------------------------------- Set Number, 34, is as follows Deck , 1, has denominations , 1, 2, 6, 6, 6, 6 Deck , 2, has denominations , 5, 5, 5, 5, 5, 5 Deck , 3, has denominations , 4, 4, 4, 4, 4, 4 Deck , 4, has denominations , 3, 3, 3, 3, 7, 7 It is readily seen that Deck , 1, beats deck , 2, with probability , 2/3 Deck , 2, beats deck , 3, with probability , 1 Deck , 3, beats deck , 4, with probability , 2/3 yet Deck, 4, beats deck 1, with probability , 5/9 ------------------------------------------------------- Set Number, 35, is as follows Deck , 1, has denominations , 1, 4, 4, 4, 4, 4 Deck , 2, has denominations , 3, 3, 3, 3, 7, 7 Deck , 3, has denominations , 2, 2, 6, 6, 6, 6 Deck , 4, has denominations , 5, 5, 5, 5, 5, 5 It is readily seen that Deck , 1, beats deck , 2, with probability , 5/9 Deck , 2, beats deck , 3, with probability , 5/9 Deck , 3, beats deck , 4, with probability , 2/3 yet Deck, 4, beats deck 1, with probability , 1 ------------------------------------------------------- Set Number, 36, is as follows Deck , 1, has denominations , 1, 5, 5, 5, 5, 5 Deck , 2, has denominations , 4, 4, 4, 4, 4, 4 Deck , 3, has denominations , 3, 3, 3, 3, 3, 7 Deck , 4, has denominations , 2, 2, 2, 6, 6, 6 It is readily seen that Deck , 1, beats deck , 2, with probability , 5/6 Deck , 2, beats deck , 3, with probability , 5/6 Deck , 3, beats deck , 4, with probability , 7/12 yet Deck, 4, beats deck 1, with probability , 7/12 ------------------------------------------------------- Set Number, 37, is as follows Deck , 1, has denominations , 1, 5, 5, 5, 5, 5 Deck , 2, has denominations , 4, 4, 4, 4, 4, 4 Deck , 3, has denominations , 3, 3, 3, 3, 7, 7 Deck , 4, has denominations , 2, 2, 2, 6, 6, 6 It is readily seen that Deck , 1, beats deck , 2, with probability , 5/6 Deck , 2, beats deck , 3, with probability , 2/3 Deck , 3, beats deck , 4, with probability , 2/3 yet Deck, 4, beats deck 1, with probability , 7/12 ------------------------------------------------------- Set Number, 38, is as follows Deck , 1, has denominations , 1, 5, 5, 5, 5, 5 Deck , 2, has denominations , 4, 4, 4, 4, 4, 4 Deck , 3, has denominations , 3, 3, 3, 3, 7, 7 Deck , 4, has denominations , 2, 2, 6, 6, 6, 6 It is readily seen that Deck , 1, beats deck , 2, with probability , 5/6 Deck , 2, beats deck , 3, with probability , 2/3 Deck , 3, beats deck , 4, with probability , 5/9 13 yet Deck, 4, beats deck 1, with probability , -- 18 --------------------------------------------------- --------------------------------------------------- This ends this article, that took, 28.937, seconds to generate.