Related papers: Possible probability and irreducibility of balance…

For a positive integer $n$, an $n$-tuple of dice $(A_1,A_2,\dots,A_n)$ is called balanced if $P(A_1<A_2) = P(A_2<A_3) = \cdots = P(A_n<A_1)$ and nontransitive if $P(A_1<A_2), P(A_2<A_3), \dots, P(A_n<A_1)$ are each greater than…

We study triples of labeled dice in which the relation "is a better die than" is non-transitive. Focusing on such triples with an additional symmetry we call "balance," we prove that such triples of $n$-sided dice exist for all $n \geq 3$.…

We study the phenomenon of intransitivity in models of dice and voting. First, we follow a recent thread of research for $n$-sided dice with pairwise ordering induced by the probability, relative to $1/2$, that a throw from one die is…

We further study sets of labeled dice in which the relation "is a better die than" is non-transitive. Focusing on sets with an additional symmetry we call "balance," we prove that sets of $n$ such $m$-sided dice exist for all $n,m \geq 3$.…

We consider $n$-sided dice whose face values lie between $1$ and $n$ and whose faces sum to $n(n+1)/2$. For two dice $A$ and $B$, define $A \succ B$ if it is more likely for $A$ to show a higher face than $B$. Suppose $k$ such dice…

Two possibly unfair $n$-sided dice, both labelled $1, 2, \ldots, n$, are rolled, and the sum is recorded. How should the dice's sides be weighted so that the resulting sum is closest to the uniform distribution on $2, 3, \ldots, 2n$? We…

We settle a version of the conjecture about intransitive dice posed by Conrey, Gabbard, Grant, Liu and Morrison in 2016 and Polymath in 2017. We consider generalized dice with $n$ faces and we say that a die $A$ beats $B$ if a random face…

Answering a pair of questions of Conrey, Gabbard, Grant, Liu, and Morrison, we prove that a triplet of dice drawn from the multiset model are intransitive with probability $1/4+o(1)$ and the probability a random pair of dice tie tends…

In this paper, we give a direct construction for a set of dice realizing any given tournament $T$. The construction for a tournament with $n$ vertices requires a number of sides on the order of $n$, which appears to be the best general…

An $n$-sided die is an $n$-tuple of positive integers. We say that a die $(a_1,\dots,a_n)$ beats a die $(b_1,\dots,b_n)$ if the number of pairs $(i,j)$ such that $a_i>b_j$ is greater than the number of pairs $(i,j)$ such that $a_i<b_j$. We…

A generalized $N$-sided die is a random variable $D$ on a sample space of $N$ equally likely outcomes taking values in the set of positive integers. We say of independent $N$ sided dice $D_i, D_j$ that $D_i$ beats $D_j$, written $D_i \to…

A generalized $N$-sided die is a random variable $D$ on a sample space of $N$ equally likely outcomes taking values in the set of positive integers. We say of independent $N$ sided dice $D_i, D_j$ that $D_i$ beats $D_j$, written $D_i \to…

How many fair coin tosses to choose 1 of $n$ options with uniform probability? Although a probability problem, the solution is essentially number-theoretic, with special roles for Mersenne numbers, Fermat numbers, and the haupt exponent. We…

We determine the exact probabilities of the different isomorphism classes of tournaments that result from random sets of three and four independent dice drawn from the balanced uniform model of 3-sided dice.

Intransitive dice $D^{(1)}, \ldots, D^{(\ell)}$ are dice such that $D^{(1)}$ has advantage when played against $D^{(2)}$, dice $D^{(2)}$ has advantage when played against $D^{(3)}$ and so on, up to $D^{(\ell)}$, which has advantage over…

In this article, a new method for characterizing nontransitive dice is de- scribed. This new method is then used to describe the "Nontransitive Identities" (NI) that are possible for 3 dice with 3, 4 and 5 sides each as well as for 5 dice…

A fair sack is a finite set of independent dice, not required to be fair and allowed to have any number of sides, for which all totals are equally likely. These have been studied for over 60 years. Most results restrict the possible orders…

The prediction of the final state probabilities of a general cuboid randomly thrown onto a surface is a problem that naturally arises in the minds of men and women familiar with regular cubic dice and the basic concepts of probability.…

We consider a two player simultaneous-move game where the two players each select any permissible $n$-sided die for a fixed integer $n$. A player wins if the outcome of his roll is greater than that of his opponent. Remarkably, for $n>3$,…

Nontransitive dice are dice beating one another in a cyclic way: die A wins die B, B wins C, and C wins A (like in a rock-paper-scissors game). In this article, it has been shown that a structure of mutual wins of 3 nontransitive dice (with…