Using a SAT Solver to Find Interesting Sets of Nonstandard Dice

Abstract We describe a family of Boolean satisfiability (SAT) problems for which each solution corresponds to a unique set of nonstandard dice. We show that we can control the relationships between the dice in each solution by imposing a set of cardinality constraints on the variables in the corresponding SAT problem. We then present examples of interesting sets of nonstandard dice that we found by solving such problems. In particular, we describe a set of 19 five-sided dice that realize the Paley tournament on 19 vertices. Furthermore, we show that this set of dice is minimal in the sense that no set of 19 dice with less than five sides can realize the Paley tournament on 19 vertices.


INTRODUCTION.
Nonstandard dice are the subject of a well-known paradox in probability theory that is best explained in terms of a simple two-player game. In this game, the first player chooses one die from a pool of three dice. The second player then chooses one of the two remaining dice. The players then roll their chosen dice and compare the resulting values. The winner of the game is the player whose value is greater. We will call this game the intransitive dice game.
If the dice used in the intransitive dice game are all standard six-sided dice, then this is a fair game. That is, there is no strategy that either player can follow that will give them an advantage over the other player. There exist sets of nonstandard dice, however, for which the game is unfair.
Consider the set C = {A, B, C} of three three-sided dice described in Table 1. This set of dice, which we will call Condorcet dice, was described in the context of comparing teams of players with differing skill levels in [15]. Rolling these nonstandard dice generates independent uniform random samples from the set of values on the faces of each die. If X is a d-sided die then we write X = (x 1 , x 2 , . . . , x d ) where x 1 < x 2 < · · · < x d and say that {x i } are the faces of X. We will write R X to denote a uniform random sample drawn from the set of faces of a die X. Notice that, of the nine equally-probable possible outcomes for a trial that involves rolling  This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/ ), which permits unrestricted use, distribution, and reproductionin any medium, provided the original work is properly cited. dice A and B, the five outcomes (R A , R B ) ∈ {(5, 3), (5,4), (9,3), (9,4), (9,8)} yield results where the value sampled from die A is larger and the four outcomes (R A , R B ) ∈ { (1,3), (1,4), (1,8), (5,8)} yield results where the value sampled from die B is larger. That is, we have P{R A > R B } = 5/9 > 1/2. In this case we say that die A beats die B and write A B. Similarly, we see that B C. Surprisingly, however, we see that C A. So, if Condorcet dice are used in the intransitive dice game, then the second player can ensure that they will have an advantage over the first player.
We can represent the relationships between the dice in a set as a simple weighted digraph. If S is a set of dice then we let G S be the directed graph with vertex set S and edge set {(X, Y ) ∈ S | X Y } where the edge (X, Y ) has weight P{R X > R Y }. We will call G S the relationship graph of S and say that S realizes the graph G S . Figure 1 depicts the relationship graph G C of Condorcet dice.
For a slightly larger example, consider the set E = {A, B, C, D} of four six-sided dice described in Table 2. This set of dice, commonly known as Efron's dice, was discovered by Bradley Efron and famously described in [9]. Efron's dice have P{R A > R B } = P{R B > R C } = P{R C > R D } = P{R D > R A } = 2/3, P{R C > R A } = 5/9 and P{R B > R D } = 1/2. Consequently, neither (B, D) nor (D, B) is an edge of G E . Figure 2 Figure 2: The relationship graph of Efron's dice.
In this paper, we describe a method for finding sets of dice that realize a given relationship graph. The original motivation for this work was to address the asymmetry in the relationship graph of Efron's dice. We were hoping to find a set of four six-sided dice S = {A, B, C, D} such that P{R A > R C } = P{R B > R D } = 1/2 and P{R A > R B } = P{R B > R C } = P{R C > R D } = P{R D > R A } = 2/3. As it happens, we found that no such set of dice exists. In the process of proving that fact, however, we found several other interesting sets of nonstandard dice that demonstrate the effectiveness of our search method and that may be of independent interest.
Most notably, we found a set M of 19 five-sided dice that can be used to play a four-player variant of the intransitive dice game. In such a game, the players take turns choosing a die from a shared pool of dice. Each pair of players then roll their chosen dice and compare the resulting values. The winner of each pairwise comparison is the player whose value is greater. If M is used to play this game, then the last player to choose their die can ensure that they have an advantage over each of their three opponents. That is, the relationship graph G M of M has what is known as the Schütte property S 3 . It is known (see [4]) that if S is a set of n d-sided dice with the property S 3 then we must have n ≥ 19. We show that M is minimal in the sense that if S is a set of 19 d-sided dice with the property S 3 then d ≥ 5.
Our approach to the problem involves characterizing a set of dice that realizes a given relationship graph as the solution to a Boolean satisfiability (SAT) problem.
Recall that a SAT problem poses the question of whether there exists an assignment of values to a set of Boolean variables such that a given Boolean formula evaluates to "True". For example, given the Boolean variables x, y, z and the Boolean formula (x ∧ y) ∨ z we see that if z = 1 then the formula evaluates to "True". Thus we say that the formula is satisfiable or SAT. If instead we consider the Boolean formula (x ∧ z) ∧ (y ∧ ¬z) then because we require both x = 1 and z = 1 to satisfy the first clause, and y = 1 and z = 0 to satisfy the second clause, there is no setting of the variables such that the formula will evaluate to "True". We say that such a formula is unsatisfiable or UNSAT.
Characterizing a set of nonstandard dice as a solution to a SAT problem allows us to use existing implementations of algorithms to solve SAT problems, known collectively as SAT solvers, to find suitable sets of dice or prove that no such set of dice exists. In our work we used the SAT solver MiniCARD (see [14]). MiniCARD is an extension of the SAT solver Minisat (see [7,8]) which handles cardinality constraints natively rather than translating such constraints into conjunctive normal form (CNF) and handling them as if they were generic Boolean constraints. We used the Python package PySAT (see [11]) to describe our problems, interface with the SAT solver, and process the resulting output.

RELATED WORK.
The literature addressing the question of which relationship graphs can be realized by a set of nonstandard dice began in earnest with a series of papers (see [25][26][27]) by Steinhaus and Trybuła in the 1960s. In these papers, the authors establish some fundamental constraints on the edge weights in a relationship graph of a set of n independent random variables (1) } and π n = sup min{ξ 1 , ξ 2 , . . . , ξ n } where the supremum is taken over all such sets of random variables, then 3 4 − 3 n(n + 4) ≤ π n < 3 4 .
They also computed numerical approximations to π n for 1 ≤ n ≤ 30 and showed that we have π 4 = 2/3. Notice that Efron's dice achieve this bound. These results were refined by Savage in 1994 (see [21]). In this paper, the author proves that we have π 3 = ( √ 5 − 1)/2 and a construction for a set of nonstandard dice that approximate this bound with arbitrary precision. This construction makes use of the fact that if F i is the ith Fibonacci number then lim i→∞ F i /F i+1 = π 3 . By carefully allocating values to the faces of a set of three F i -sided dice for sufficiently large i, the author produces a set of dice with min{ξ 1 , ξ 2 , ξ 3 In 2021, Komisarski provided a geometric proof of a formula to compute the value of π n for all n (see [13]). In this paper, the author shows that π n = 1 − 1 4 cos 2 π n + 2 .
According to the author, the same formula appeared in an earlier work related to nontransitive probabilities but for which no English translation is available (see [3]). Curiously, this formula has also appeared in other contexts such as determining the minimum number of edges needed for an arbitrary graph to contain a given subgraph (see e.g., [16]). The bounds {π n } apply to any family of independent random variables. Because π n is irrational for all n = 4, however, no set of dice with finitely many faces will be able to achieve these bounds. In this paper the author also provides weaker bounds that can May 2023] NONSTANDARD DICE VIA SAT be realized by sets of dice. He shows that if {X (i) } n i=1 is a set of dice with {ξ i } as defined above, then there exists a d ∈ N such that min{ξ 1 , ξ 2 , . . . , ξ n } ≤ Several papers restrict their attention to sets of three dice where ξ 1 = ξ 2 = ξ 3 . In 2016, Schaefer and Schweig called such sets of dice balanced (see [23]). In their paper the authors show that there exist balanced sets of three d-sided dice for all d ≥ 3. Furthermore, they show that S = {X, Y, Z} is a balanced set of three d sided dice if and only if That is, S is balanced if and only if the face-sums of the three dice are equal. Finally, they show that there exist balanced sets of d-sided dice such that lim d→∞ π 3 − ξ i = 0. We note that this result disproves Conjecture 5.1 of [10].
The works discussed above are primarily concerned with establishing bounds on the values of the cyclic probabilities {ξ i }. Another thread of inquiry addresses questions about which tournaments, i.e., directed graphs in which every pair of vertices are connected by a directed edge in one of the two possible orientations, can be realized by sets of nonstandard dice. Note that many of these works treat relationship graphs as unweighted digraphs. That is, given two dice X and Y , they are interested only in whether X Y or Y X rather than the value of the probability P{R X > R Y }. We will say that a set of dice S realizes the unweighted digraph An early example of work in this vein was done by Moon and Moser in 1967 (see [15]). In this paper the authors consider teams created by partitioning a set of individuals of differing skill levels. They then define a relation on the resulting set of teams where they say that T 1 T 2 if the probability that a random individual chosen from team T 1 is more skilled than a random individual chosen from team T 2 . Comparing teams in this way is equivalent to comparing sets of nonstandard dice. In the language of nonstandard dice, the main result of this paper is that it is possible to realize any unweighted tournament with n vertices using a set of n λ(n)-sided dice where λ(n) ∈ (n 2 / log n).
Angel and Davis (see [1]) provide a construction for sets of dice that realize arbitrary tournaments. They show that for any unweighted tournament T on n vertices, there exists a set of n (n + 1)-sided dice that realizes T . These results are strikingly similar to those of Shaefer (see [22]) who shows that for any tournament T there exists a balanced set of n n-sided dice that realize T . These results were published in quick succession and appear to be the product of independent investigations.
Buhler, Graham, and Hales (see [5]) provide a fascinating alternative construction for sets of dice that realize arbitrary tournaments. Their construction involves rolling each die in a given set multiple times, adding up the results, and comparing the resulting sums. That is, if X is a die, then the authors define X[n] to be the sum of n independent rolls of X. They provide a construction of a set of dice X = {X (i) } k i=1 such that for every unweighted tournament T on k vertices there exist infinitely many n such that X [n] = {X (i) [n]} k i=1 realizes T . A pair of papers by Bednay and Bozokói (see [2,4]) focus on sets of dice that realize a class of tournaments which they call Paley tournaments. The Paley tournament on p vertices, denoted P p where p = 4m + 3 ≥ 7 is prime, is defined by its incidence matrix M of size p × p where Paley tournaments have two properties that make them interesting objects of study in the context of nonstandard dice. First, they exhibit a kind of rotational symmetry. That is, each row of the incidence matrix M is a rotation of some standard template. In [2] the authors exploit this symmetry to reduce the number of faces required for a set of dice to realize P p . Second, some Paley tournaments have what is known as the Schütte property S k . If G is a directed graph on the vertex set V , we say that G has the property S k if for every subset W ⊂ V with |W | = k, there exists some element As per Theorem 2 of [4], it can be shown that P 19 is the smallest tournament that has the property S 3 and P 67 is the smallest tournament that has the property S 4 . This is significant because a set of dice S whose relationship graph G S has the property S k can be used to play a (k + 1)-player variant of the intransitive dice game. In such a game, the players take turns choosing a die from a shared pool of dice. Each pair of players then roll their chosen dice and compare the resulting values. The winner of each pairwise comparison is the player whose value is greater. If G S has the property S k then the last player to choose their die can ensure that they have an advantage over all of their opponents. That is, for every set of dice Notice that this is a statement about pairwise comparisons between dice and does not ensure that the last player to choose their die can ensure that they have an elevated probability of winning an n-way comparison between dice for n > 2.
There is a related body of work that concerns how common sets of intransitive dice are. In 2016, Conrey et al. (see [6]) proposed several conjectures related to the probability that a random set Y of three nonstandard dice drawn from the so-called "multiset" model has a intransitive cycle. In particular, they conjectured that the probability that G Y is intransitive is 1/4 + o (1). Subsequently, a collaborative Polymath project (see [20]) proved a theorem similar to the conjecture of Conrey et al. for sets of dice drawn from the so-called "balanced sequences model." These results suggest that, for some ensembles of nonstandard dice, intransitive dice are actually quite common.
We have found few descriptions of how constraint programming tools can be used to find sets of nonstandard dice. In a conference presentation in 2010 (see [18]), O'Sullivan showed how to encode the relationship graph for a set of dice and Z X as a constraint programming problem. A blog post by Kjellerstrand (see [12]) provides a concrete implementation of O'Sullivan's approach using the constraint modeling language MiniZinc (see [17]). In a set of lecture notes from 2017 (see [24]), Soliman shows how to encode a similar problem where the solution is a set of dice that achieves the sup min bound on the probabilities {ξ i } discussed above. In all of these examples, nonstandard dice were used as a simple introductory example to constraint programming rather than as an object of independent interest. As such, the authors did not consider sets of dice with more complicated relationship graphs. We are unaware of any work describing how constraint programming can be used to find novel sets of nonstandard dice that realize arbitrary relationship graphs.
3. DICE TO SAT. One fundamental challenge in using a SAT solver to find sets of nonstandard dice with a given relationship graph lies in encoding such a set of dice as the solution to a Boolean satisfiability problem. Doing so requires us to define a collection of Boolean variables and an appropriate set of constraints on the values those variables can take.
Assumptions. Let S be a set of dice and F = X∈S X be the set of faces on all of the dice in S. We will assume that no face appears on more than one die in S. That is, We assume that the values of the faces are consecutive integers starting with 1. That is, Variables. We define a collection of Boolean variables that can be used to describe the relationships between the faces of a set of dice. To do so, we consider a set of dice L for which we know the names of the dice and the number of sides on each die, but not the values on the faces. Here we use the notation L instead of S to emphasize that the values on the faces are unknown.
Let d : L → N be the function that maps each die to the number of sides on that die. For each pair of dice (X, Y ) ∈ L we define a set of d(

Constraints.
Given a collection of variables as described in the preceding paragraph, we derive a set of constraints that restrict the values the variables can take from the assumptions we have made about the relationships between the faces of a set of dice. We describe five types of constraints: reflexive, sorting, converse, transitive, and cardinality constraints.
Reflexive constraints. The reflexive constraints enforce our assumptions on the relationships between the faces of a single die. We have x i ≥ x j if and only if i ≥ j . These requirements can be expressed as literals of the form [X, X] i,j for all X ∈ L and i ≥ j , and ¬[X, X] i,j for all X ∈ L and i < j.
Converse constraints. The converse constraints enforce the requirement that relationships between the faces are antisymmetric. These constraints capture the implications that Sorting constraints. The sorting constraints enforce requirements on the faces of different dice due to the fact that the faces of each die are distinct and that we always list the faces in ascending numerical order. There are two types of sorting constraints, horizontal constraints and vertical constraints.
Horizontal constraints capture the implications that if x i > y j then Vertical constraints capture the implications that if x i > y j then x i+1 > y j . That is, Transitivity constraints. The transitivity constraints enforce requirements on the faces of different dice due to the fact that they are integer valued and therefore well-ordered. These constraints capture the implications that if x i > y j and y j > z k then These implications can be expressed as disjunctive clauses of the form Notice that this ordering of the faces is a property of the integers and does not imply that the relation between dice is transitive.
Cardinality constraints. The cardinality constraints enforce requirements on the relationships between dice by specifying the weights assigned to each edge of the relationship graph for a set of dice. Given a set of dice L, let w : L × L → N be a function that maps every pair of dice to a positive integer. A cardinality constraint for X and Y imposes a constraint on the variables [X, Y ] of the form Notice that, because w is an integer-valued function, cardinality constraints are not Boolean constraints. Rather, they are what is known as pseudo-Boolean constraints. A pseudo-Boolean constraint is a constraint that, given a collection of Boolean variables, specifies how many of those variables can take the value "True".
There are a multitude of ways for a SAT solver to incorporate pseudo-Boolean constraints into a problem specification. Traditionally, pseudo-Boolean constraints are translated into formulas written in conjunctive normal form which can then be handled as if they were generic Boolean constraints. MiniCARD, however, uses a watchedliteral scheme for cardinality constraints which can be more efficient. See [14] for details.
For the purposes of this discussion, it suffices to understand that pseudo-Boolean constraints can be incorporated into the formulation of a SAT problem. The Python package PySAT provides an interface that abstracts away much of the complexity of the manipulations required to do so. See [11] for details.

SAT TO DICE.
In Section 3 we described a way to represent a set of dice as a collection of Boolean variables and derived some necessary constraints on the values those variables can take. In this section, we show that those constraints are sufficient to allow us to reconstruct a set of dice.
In what follows, we say that V is a solution to the nonstandard dice problem φ (L, d, w) if V is a collection of Boolean variables as described in Section 3 which satisfies the constraints described in Section 3. φ(L, d, w) and for every X ∈ L and 1 ≤ i ≤ d(X) we let

Theorem 1. If V is a solution to the nonstandard dice problem
then S = {(x 1 , x 2 , . . . , x d(X) ) | X ∈ L} is a set of dice such that for all X, Y ∈ L, Sketch of proof. We first identify a unique pair (X, i) such that [X, Y ] i,j = 1 for all Y ∈ L and 1 ≤ j ≤ d(Y ). We then remove all variables involving that maximal pair and show that the remaining variables are a solution to a smaller nonstandard dice problem. So, we can repeat the above procedure to identify maximal pairs and then eliminate variables involving those pairs until every variable has been eliminated. This shows that S as defined by (2)  Theorem 1 shows that there is a one-to-one correspondence between solutions to nonstandard dice problems and sets of dice. If V is a solution to the nonstandard dice problem φ (L, d, w) and S is the corresponding set of dice, then L determines the number of dice in S, d determines how many faces are on each die X ∈ S, and w determines the weights of the edges in the relationship graph G S .

SEARCHING FOR SETS OF NONSTANDARD DICE.
We now describe how we used the techniques described in Section 4 to search for sets of nonstandard dice and a few of the more interesting things that we found. In the remainder of this discussion, we will only consider homogeneous sets of nonstandard dice where all of the dice have the same number of sides. For such sets of dice, we will treat the function d that maps each die to the number of faces on that die as an integer.
Simple cyclic dice. Recall that our original motivation was to address the asymmetry in Efron's dice where each die in the set has a different spectrum of relationships with the other dice. We hoped to find a set of four six-sided dice We quickly realized that no such set of dice exists. In fact, a simple exhaustive search shows that Efron's dice are the unique balanced set of four six-sided dice with P{X (i) > X (i+1) } = 2/3. We confirmed this result by characterizing a set of dice with the desired properties as a solution to the nonstandard dice problem φ (S, d, w) where and used a SAT solver to determine that the problem is unsatisfiable for d ≤ 60. We have not found any d for which the problem is satisfiable.   Motivated by this failure, we expanded our search to look for balanced sets of nonstandard dice S = {X (i) } n i=1 such that there exists some ξ > 1/2 such that P{R X (i) > R X (j ) } = ξ S if j − i ≡ 1 (mod n) and P{R X (i) > R X (j ) } = 1/2 otherwise. We call these sets of dice simple cyclic dice. As described above, we were unable to find a simple cyclic set of four dice with ξ = 2/3. We were able, however, to find many simple cyclic sets of four dice with bias 1/2 < ξ < 2/3. For example, a set Q of four six-sided dice with ξ Q = 11/18 is described in Table 3. Figure 3 depicts the relationship graph G Q of this set.
Rotationally symmetric dice. We can further expand our search by relaxing our requirement that P{R X (i) > R X (j ) } = 1/2 for all j − i ≡ 1 (mod n). Instead, we can require only that P{R X (i) > R X (j ) } is a function of j − i (mod n). We call such sets of dice rotationally symmetric dice. Notice that the class of simple cyclic dice is a subclass of the rotationally symmetric dice. Table 4 describes a rotationally symmetric set P = (A, B, C, D, E) = (X (1) , X (2) , X (3) , X (4) , X (5) ) of five six-sided dice. For all We call these dice pentagram dice. Figure 4 depicts the relationship graph G P of pentagram dice. Notice that X (1) has a big advantage against X (2) , a small advantage against X (3) , a small disadvantage against X (4) and a big disadvantage against X (5) . Notice that there are strict subsets of pentagram dice, for example {A, C, D}, that form three-long intransitive cycles. Consider also the set H = (A, B, C, D, E, F ) = (X (1) , X (2) , X (3) , X (4) , X (5) , X (6) ) of six six-sided dice described in Table 5. We call these dice hexagram dice. Figure 5 depicts the relationship graph of hexagram dice. For all X (i) , X (j ) ∈ H we have Notice that the relationship graph of hexagram dice contains many interlocking cycles. This structure leads to complex relationships between the dice. For example, consider the six-long cycle A B C D E F A. At each step with X Y , we have P{R X > R Y } = 7/12. This corresponds to a path around the perimeter of the graph of G H as depicted in Figure 5 in the clockwise direction. We also have the three-long cycle A E C A. Once again, at each step with X Y we have P{R X > R Y } = 7/12. This cycle, however, corresponds to a path through the interior of the graph of G H as depicted in Figure 5 in the counterclockwise direction.
Both pentagram and hexagram dice are examples of sets of nonstandard dice that were quite easy to find using a SAT solver but difficult to find using other known strategies. For example, a naive simple exhaustive search for hexagram dice involves searching through all 36! ≈ 2 138 possible permutations of the face-value assignments. Some simple refinements make this (absurd) computational task somewhat more manageable. For example, because we always sort the faces on each die, we only need to check the 36 6,6,6,6,6,6 ≈ 2 81 possible allocations of faces to dice. We also observe that if we insist that the smallest face is assigned to die A then we only need to check 1 6 36 6,6,6,6,6,6 ≈ 2 78 possibilities. So, a search strategy that requires exhaustively checking all possible face-value assignments will necessarily be extremely clever, extremely computationally expensive, or both. In contrast, finding hexagram dice using the techniques we describe is simple and efficient. Our non-optimized single-threaded implementation of SAT search finds a set of hexagram dice in less than 100ms using commodity hardware.
Paley dice. Consider the set O of seven three-sided dice described in Table 6. This set was introduced by Oskar van Deventer and described in [19]. Oskar dice have the property that for every pair of dice X, Y ∈ O, there exists a third die Z ∈ O such that Z X and Z Y . That is, G O has the Schütte property S 2 . So, in a game where each player chooses a die and then each pair of players compares their chosen dice as in the two-player game, the last player to choose can ensure that they will have an advantage over both of the other players. Figure 6 depicts the relationship graph G O of Oskar dice. Recall that we write P p to denote the Paley tournament on p vertices and notice that O realizes P 7 .
As mentioned in Section 2, the smallest tournament that has the Schütte property S 3 is the Paley tournament P 19 . A natural generalization of Oskar dice, then, would be a set of 19 dice that realize P 19 . One such set M of 19 five-sided dice is described in Table 7. Let Q be the set of quadratic residues modulo 19. That is, Q = {1, 4, 5, 6, 7, 9, 11, 16, 17}. For all X (i) , X (j ) ∈ M we have    Table 6: The faces of Oskar dice. Figure 6: The relationship graph of Oskar dice.
The relationship graph G M is depicted in Figure 7. The dice in M have fewer sides than those in sets of dice created using existing algorithms. The most efficient of these were presented in [1] and [2]. Both constructions produce sets of 19 nineteen-sided dice. In [2], the authors claim to have found a set of 19 nine-sided dice that realize P 19 but provide neither a description of that set nor the algorithm that they used to construct it.
By modifying the form of the cardinality constraints that we used, we were able to show that no set of 19 d-sided dice with d < 5 realizes P 19 . Recall that in Section 3 we described a set of cardinality constraints that prescribe the values of [X, Y ] i,j for all X and Y . The converse constraints ensure that we have [Y, X] j,i . So, we see that half of the cardinality constraints are redundant. We can remove all of the redundant cardinality constraints by discarding those that specify the value of [Y, X] j,i for all X, Y with Y X. We can then replace the remaining cardinality constraints with pseudo-Boolean constraints that only specify lower bounds. More precisely, we can specify that w * (X, Y ) ≥ d(X)d(Y )/2 + 1 for all X, Y with X Y . Letting L be a set of labels with |L| = 19, we can consider the nonstandard dice problems φ(L, 3, w * ) and φ(L, 4, w * ). We found that both problems are unsatisfiable. As such, we see that M is minimal in the sense that if S is a set of 19 d-sided dice with the property S 3 then d ≥ 5.
May 2023] NONSTANDARD    6. FUTURE WORK. The most pressing question left unanswered in the course of this discussion is whether there exists a simple cyclic set of four d-sided dice with P{X (i) > X (j ) } = 2/3. In Section 5 we claimed that any such set must have d > 60, but we have said nothing about the case where d > 60. Our experience, however, prompts us to make the following conjecture: (1) , X (2) , X (3) , X (4) is a set of nonstandard dice such that for all X (i) , X (j ) ∈ S then ξ < 2/3.
Notice that our current approach is not well suited to proving this conjecture. This would require showing that the problem is unsatisfiable for all d and our current approach treats d as a parameter that is fixed in advance. We do not see a way to modify our approach to handle the case where d is an integer-valued variable whose value is determined as part of the solution to some constraint programming problem.
Recall that in Section 5 we showed that there exists a set of 19 five-sided dice that can be used to play a four-player variant of the intransitive dice game. A natural question that we did not address here is whether there exist sets of dice that can be used to play an n-player variant for n ≥ 5. For n = 5, it suffices to find a set of dice whose relationship graph realizes P 67 . The generic constructions described in [1], [2], and [4] provide a means for creating such sets of dice. As we have shown, however, these constructions can be inefficient. That is, they can produce sets of d-sided dice where d is larger than necessary. We have tried to apply our techniques to find sets of dice with fewer sides than those produced by these generic constructions but have thus far been largely unsuccessful. We have found that no such set of dice exists for d = 3 or d = 4 but have been unable to determine whether the corresponding nonstandard dice problems for d ≥ 5 are satisfiable.
One last interesting thread of inquiry stems from the following variant of the twoplayer intransitive dice game described in Section 1. In this variant, the first player chooses two dice from a shared pool of five dice. The second player then chooses two of the remaining three dice. Each player rolls their dice and combines the result of their rolls using some aggregation function. Natural choices for this aggregation function include the mean, maximum, and minimum of the values of the two values. The players then compare the resulting aggregate values. The winner of the game is the player whose aggregate value is greater. Notice that the relationship graph that describes the relationships between disjoint pairs of dice would necessarily realize an orientation of the Petersen graph. It would be interesting to explore this and other settings involving comparisons between more than two dice at a time.   L, d, w) and (X, d(X)) is the unique maximal pair whose existence is guaranteed by Lemma 3. If for all Z ∈ L we let for all Y, Z ∈ S we let and we let V = {[Y, Z] j,k | Y, Z ∈ L, 1 ≤ j ≤ d (Y ), 1 ≤ k ≤ d (Z)}, then V is a solution to the nonstandard dice problem φ(L, d , w ).
Proof. Because the reflexive, sorting, converse, and transitivity constraints apply to every subset of variables, it remains only to show that V satisfies the cardinality constraints for φ (L, d , w ). That is, it remains to show that for all Y, Z ∈ L we have We consider the two cases Y = X and Y = X separately.
where the final inequality follows from the maximality of (X, d(X)).
where again the final inequality follows from the maximality of (X, d(X)).
So in all cases, V satisfies the cardinality constraints for φ(L, d , w ).

Theorem 5.
If V is a solution to the nonstandard dice problem φ(L, d, w) and for every X ∈ L and 1 ≤ i ≤ d(X) we let Proof. Lemmas 3 and 4 together imply that for every 1 ≤ m ≤ X∈L d(X) there exists a unique (X, i) such that x i = m. The reflexive constraints imply that for all X ∈ L we have x i ≥ x j if and only if i ≥ j . So, S is a set of dice. Furthermore, the transitivity constraints guarantee that for all X, Y ∈ L, 1 ≤ i ≤ d(X), and 1 ≤ j ≤ d(Y ), we have x i ≥ y j if and only if [X, Y ] i,j = 1.