Abstract
We construct irreducible balanced nontransitive sets of n -sided dice for any positive integer n . One main tool of the construction is to study so-called fair sets of dice. Furthermore, we also study the distribution of the probabilities of balanced nontransitive sets of dice. For a lower bound, we show that the winning probability can be arbitrarily close to 1 / 2 . We hypothesize that the winning probability cannot be more than 1 / 2 + 1 / 9 , and we construct a balanced nontransitive set of dice whose probability is 1 / 2 + 13 − 153 / 24 ≈ 1 / 2 + 1 / 9.12 .
Highlights
Note that nontransitive sets of dice are first introduced by Gardner [1], further studied in [2, 3], and have been generalized in several directions
Main idea of the construction in [14] is to combine several balanced nontransitive sets of dice. erefore, the sets of dice that are constructed in [14] are reducible, and Schaefer and Schweig [14] question whether there exist balanced irreducible nontransitive sets of n-sided dice for all n
We study fair sets of n-sided dice for any positive integer n
Summary
Note that nontransitive sets of dice are first introduced by Gardner [1], further studied in [2, 3], and have been generalized in several directions (see [4,5,6,7,8,9,10,11,12,13,14]). One main purpose of the paper is to construct irreducible nontransitive sets of n-sided dice for any positive integer n. We study fair sets of n-sided dice for any positive integer n. We first explicitly calculate and prove that both the probability P(A > B) in our paper and the one in [14] are less than (1/2) + (1/9) We provide another new construction of a balanced nontransitive set of dice such that P(A > B) ≈ (1/2) + (1/9.12), which is less than (1/2) + (1/9), and is, as far as we know, the maximum among known constructions of balanced nontransitive sets of dice.
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