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



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                         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



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                         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



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                         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



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                         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



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                         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



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        This ends this article, that took, 0.053, seconds to generate.