All the Suckers's Bets Decks with, 3, decks, with deck sizes, 3, 3, 3, using , 9, different denominations from 1 to, 9 with DISTINCT denominations By Shalosh B. Ekhad Up to obvious cyclic symmetry, there are, 5, 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, 5, 9 Deck , 2, has denominations , 3, 4, 8 Deck , 3, has denominations , 2, 6, 7 It is readily seen that Deck , 1, beats deck , 2, with probability , 5/9 Deck , 2, beats deck , 3, with probability , 5/9 yet Deck, 3, beats deck 1, with probability , 5/9 ------------------------------------------------------- Set Number, 2, is as follows Deck , 1, has denominations , 1, 6, 8 Deck , 2, has denominations , 3, 5, 7 Deck , 3, has denominations , 2, 4, 9 It is readily seen that Deck , 1, beats deck , 2, with probability , 5/9 Deck , 2, beats deck , 3, with probability , 5/9 yet Deck, 3, beats deck 1, with probability , 5/9 ------------------------------------------------------- Set Number, 3, is as follows Deck , 1, has denominations , 1, 6, 8 Deck , 2, has denominations , 4, 5, 7 Deck , 3, has denominations , 2, 3, 9 It is readily seen that Deck , 1, beats deck , 2, with probability , 5/9 Deck , 2, beats deck , 3, with probability , 2/3 yet Deck, 3, beats deck 1, with probability , 5/9 ------------------------------------------------------- Set Number, 4, is as follows Deck , 1, has denominations , 1, 7, 8 Deck , 2, has denominations , 3, 5, 6 Deck , 3, has denominations , 2, 4, 9 It is readily seen that Deck , 1, beats deck , 2, with probability , 2/3 Deck , 2, beats deck , 3, with probability , 5/9 yet Deck, 3, beats deck 1, with probability , 5/9 ------------------------------------------------------- Set Number, 5, is as follows Deck , 1, has denominations , 1, 7, 8 Deck , 2, has denominations , 4, 5, 6 Deck , 3, has denominations , 2, 3, 9 It is readily seen that Deck , 1, beats deck , 2, with probability , 2/3 Deck , 2, beats deck , 3, with probability , 2/3 yet Deck, 3, beats deck 1, with probability , 5/9 --------------------------------------------------- --------------------------------------------------- This ends this article, that took, 0.053, seconds to generate.