All the Suckers's Bets Decks with, 3, decks, with deck sizes, 4, 4, 4, using , 12, different denominations from 1 to, 12 with DISTINCT denominations By Shalosh B. Ekhad Up to obvious cyclic symmetry, there are, 13, distinct such collection of decks where Deck , 1, beats deck, 2 Deck , 2, beats deck, 3 but surprise, surprise Deck, 3, beats deck 1 Here there all are ------------------------------------------------------- Set Number, 1, is as follows Deck , 1, has denominations , 1, 3, 10, 11 Deck , 2, has denominations , 2, 7, 8, 9 Deck , 3, has denominations , 4, 5, 6, 12 It is readily seen that Deck , 1, beats deck , 2, with probability , 9/16 Deck , 2, beats deck , 3, with probability , 9/16 yet Deck, 3, beats deck 1, with probability , 5/8 ------------------------------------------------------- Set Number, 2, is as follows Deck , 1, has denominations , 1, 3, 10, 12 Deck , 2, has denominations , 2, 7, 8, 9 Deck , 3, has denominations , 4, 5, 6, 11 It is readily seen that Deck , 1, beats deck , 2, with probability , 9/16 Deck , 2, beats deck , 3, with probability , 9/16 yet Deck, 3, beats deck 1, with probability , 9/16 ------------------------------------------------------- Set Number, 3, is as follows Deck , 1, has denominations , 1, 4, 10, 11 Deck , 2, has denominations , 2, 7, 8, 9 Deck , 3, has denominations , 3, 5, 6, 12 It is readily seen that Deck , 1, beats deck , 2, with probability , 9/16 Deck , 2, beats deck , 3, with probability , 9/16 yet Deck, 3, beats deck 1, with probability , 9/16 ------------------------------------------------------- Set Number, 4, is as follows Deck , 1, has denominations , 1, 4, 10, 11 Deck , 2, has denominations , 3, 6, 8, 9 Deck , 3, has denominations , 2, 5, 7, 12 It is readily seen that Deck , 1, beats deck , 2, with probability , 9/16 Deck , 2, beats deck , 3, with probability , 9/16 yet Deck, 3, beats deck 1, with probability , 9/16 ------------------------------------------------------- Set Number, 5, is as follows Deck , 1, has denominations , 1, 4, 10, 11 Deck , 2, has denominations , 3, 7, 8, 9 Deck , 3, has denominations , 2, 5, 6, 12 It is readily seen that Deck , 1, beats deck , 2, with probability , 9/16 Deck , 2, beats deck , 3, with probability , 5/8 yet Deck, 3, beats deck 1, with probability , 9/16 ------------------------------------------------------- Set Number, 6, is as follows Deck , 1, has denominations , 1, 6, 8, 11 Deck , 2, has denominations , 4, 5, 7, 10 Deck , 3, has denominations , 2, 3, 9, 12 It is readily seen that Deck , 1, beats deck , 2, with probability , 9/16 Deck , 2, beats deck , 3, with probability , 9/16 yet Deck, 3, beats deck 1, with probability , 9/16 ------------------------------------------------------- Set Number, 7, is as follows Deck , 1, has denominations , 1, 7, 8, 9 Deck , 2, has denominations , 3, 5, 6, 12 Deck , 3, has denominations , 2, 4, 10, 11 It is readily seen that Deck , 1, beats deck , 2, with probability , 9/16 Deck , 2, beats deck , 3, with probability , 9/16 yet Deck, 3, beats deck 1, with probability , 5/8 ------------------------------------------------------- Set Number, 8, is as follows Deck , 1, has denominations , 1, 7, 8, 9 Deck , 2, has denominations , 4, 5, 6, 11 Deck , 3, has denominations , 2, 3, 10, 12 It is readily seen that Deck , 1, beats deck , 2, with probability , 9/16 Deck , 2, beats deck , 3, with probability , 9/16 yet Deck, 3, beats deck 1, with probability , 5/8 ------------------------------------------------------- Set Number, 9, is as follows Deck , 1, has denominations , 1, 7, 8, 9 Deck , 2, has denominations , 4, 5, 6, 12 Deck , 3, has denominations , 2, 3, 10, 11 It is readily seen that Deck , 1, beats deck , 2, with probability , 9/16 Deck , 2, beats deck , 3, with probability , 5/8 yet Deck, 3, beats deck 1, with probability , 5/8 ------------------------------------------------------- Set Number, 10, is as follows Deck , 1, has denominations , 1, 7, 8, 10 Deck , 2, has denominations , 3, 5, 6, 12 Deck , 3, has denominations , 2, 4, 9, 11 It is readily seen that Deck , 1, beats deck , 2, with probability , 9/16 Deck , 2, beats deck , 3, with probability , 9/16 yet Deck, 3, beats deck 1, with probability , 9/16 ------------------------------------------------------- Set Number, 11, is as follows Deck , 1, has denominations , 1, 7, 8, 10 Deck , 2, has denominations , 4, 5, 6, 11 Deck , 3, has denominations , 2, 3, 9, 12 It is readily seen that Deck , 1, beats deck , 2, with probability , 9/16 Deck , 2, beats deck , 3, with probability , 9/16 yet Deck, 3, beats deck 1, with probability , 9/16 ------------------------------------------------------- Set Number, 12, is as follows Deck , 1, has denominations , 1, 7, 8, 10 Deck , 2, has denominations , 4, 5, 6, 12 Deck , 3, has denominations , 2, 3, 9, 11 It is readily seen that Deck , 1, beats deck , 2, with probability , 9/16 Deck , 2, beats deck , 3, with probability , 5/8 yet Deck, 3, beats deck 1, with probability , 9/16 ------------------------------------------------------- Set Number, 13, is as follows Deck , 1, has denominations , 1, 7, 8, 11 Deck , 2, has denominations , 4, 5, 6, 10 Deck , 3, has denominations , 2, 3, 9, 12 It is readily seen that Deck , 1, beats deck , 2, with probability , 5/8 Deck , 2, beats deck , 3, with probability , 9/16 yet Deck, 3, beats deck 1, with probability , 9/16 --------------------------------------------------- --------------------------------------------------- This ends this article, that took, 0.757, seconds to generate.