A121228 Number of ways to write the numbers 1 through 3n on the faces of three n-sided 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, 5, 13, 1732, 10705, 697733, 6539451, 320055263, 3757649717, 159846296757, 2168151028368, 84710946309286, 1271782693566515, 46887132021495098, 758979280972648162, 26825721979648877998, 460233727565745799839, 15752977776622170172890, 283061660420599350271338

Offset

1,3

References

M. Gardner, "Mathematical Games: The Paradox of the Nontransitive Dice and the Elusive Principle of Indifference." Sci. Amer. 223, 110-114, 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

nonn

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 Ekhad-Zeilberger paper added by N. J. A. Sloane, Dec 26 2017

a(13)-a(20) from Bert Dobbelaere, Feb 24 2021

Status

approved