A121228 - OEIS

%I #20 Aug 26 2023 21:00:34

%S 0,0,5,13,1732,10705,697733,6539451,320055263,3757649717,159846296757,

%T 2168151028368,84710946309286,1271782693566515,46887132021495098,

%U 758979280972648162,26825721979648877998,460233727565745799839,15752977776622170172890,283061660420599350271338

%N 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.

%D M. Gardner, "Mathematical Games: The Paradox of the Nontransitive Dice and the Elusive Principle of Indifference." Sci. Amer. 223, 110-114, Dec. 1970.

%H Shalosh B. Ekhad and Doron Zeilberger, <a href="https://arxiv.org/abs/1710.10344">A Treatise on Sucker's Bets</a>, arXiv preprint arXiv:1710.10344 [math.CO], 2017.

%H Lee J. Stemkoski, <a href="http://home.adelphi.edu/~StemkosL/mathematrix/dice.html">Nontransitive Dice</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EfronsDice.html">Efron's Dice</a>.

%e a(3)=5:

%e Set 1:

%e Die 1: 1 5 9

%e Die 2: 3 4 8

%e Die 3: 2 6 7

%e Set 2:

%e Die 1: 1 7 8

%e Die 2: 4 5 6

%e Die 3: 2 3 9

%e Set 3:

%e Die 1: 1 7 8

%e Die 2: 3 5 6

%e Die 3: 2 4 9

%e Set 4:

%e Die 1: 1 6 8

%e Die 2: 4 5 7

%e Die 3: 2 3 9

%e Set 5:

%e Die 1: 1 6 8

%e Die 2: 3 5 7

%e Die 3: 2 4 9

%K nonn

%O 1,3

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

%E a(1) corrected by _Jon E. Schoenfield_, May 19 2007

%E a(6) and a(7) from _Jon E. Schoenfield_, May 19 2007

%E a(8) from _Jon E. Schoenfield_, May 23 2007

%E Further terms from the Ekhad-Zeilberger paper added by _N. J. A. Sloane_, Dec 26 2017

%E a(13)-a(20) from _Bert Dobbelaere_, Feb 24 2021