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
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This ends this article, that took, 28.937, seconds to generate.