

A121228


Number of ways to write the numbers 1 through 3n on the faces of three nsided dice, so that the 1st die beats the 2nd with probability > 1/2, the 2nd beats the 3rd with probability > 1/2 and the 3rd beats the 1st with probability > 1/2.


0



0, 0, 5, 13, 1732, 10705, 697733, 6539451, 320055263, 3757649717, 159846296757, 2168151028368, 84710946309286, 1271782693566515, 46887132021495098, 758979280972648162, 26825721979648877998, 460233727565745799839, 15752977776622170172890, 283061660420599350271338
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OFFSET

1,3


REFERENCES

M. Gardner, "Mathematical Games: The Paradox of the Nontransitive Dice and the Elusive Principle of Indifference." Sci. Amer. 223, 110114, Dec. 1970.


LINKS

Table of n, a(n) for n=1..20.
Shalosh B. Ekhad and Doron Zeilberger, A Treatise on Sucker's Bets, arXiv preprint arXiv:1710.10344 [math.CO], 2017.
Lee J. Stemkoski, Nontransitive Dice
Eric Weisstein's World of Mathematics, Efron's Dice.


EXAMPLE

a(3)=5:
Set 1:
Die 1: 1 5 9
Die 2: 3 4 8
Die 3: 2 6 7
Set 2:
Die 1: 1 7 8
Die 2: 4 5 6
Die 3: 2 3 9
Set 3:
Die 1: 1 7 8
Die 2: 3 5 6
Die 3: 2 4 9
Set 4:
Die 1: 1 6 8
Die 2: 4 5 7
Die 3: 2 3 9
Set 5:
Die 1: 1 6 8
Die 2: 3 5 7
Die 3: 2 4 9


CROSSREFS

Sequence in context: A153374 A247789 A012032 * A201260 A012173 A009143
Adjacent sequences: A121225 A121226 A121227 * A121229 A121230 A121231


KEYWORD

more,nonn,changed


AUTHOR

Mikhail Dvorkin (dvorkin_m(AT)yahoo.com), Dec 11 2006


EXTENSIONS

a(1) corrected by Jon E. Schoenfield, May 19 2007
a(6) and a(7) from Jon E. Schoenfield, May 19 2007
a(8) from Jon E. Schoenfield, May 23 2007
Further terms from the EkhadZeilberger paper added by N. J. A. Sloane, Dec 26 2017
a(13)a(20) from Bert Dobbelaere, Feb 24 2021


STATUS

approved



