A Fair Shake: How Close Can The Sum Of n-Sided Dice Be To A Uniform Distribution?

Abstract

Two possibly unfair n-sided dice, both labeled $1,2,\dots ,n$, 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 $2,3,\dots ,2n$? We answer this question by explicitly identifying the optimal pair of dice. This resolves a question raised by Gasarch and Kruskal in 1999 in a surprising way. We present additional results for the case of more than two possibly unfair n-sided dice and for the hypothetical case where the weights on each die are permitted to be negative, but must still sum to one.

MSC:

• 00A08

ACKNOWLEDGMENT

We thank Frank Farris and Chi Hoi Yip for their helpful feedback on the manuscript. We are also grateful to the two anonymous referees who helped us enhance the clarity of our work.

DISCLOSURE STATEMENT

No potential conflict of interest was reported by the authors.

Notes

1 Interestingly for $m=2$, when this Python code uses an initial guess where the vectors p and q are identical (that is, $p=q$), it sometimes produces a mistaken minimum for D that is a saddle point where p and q are identical. Further, when this happens, it reproduces the same numerical results given in [GK99]. But the smallest perturbation of the dice’s initial guess to be nonidentical, even when the initial guess is located quite close to the saddle point, corrects this issue, leading the code to produce the p and q that correspond to the true minimum for D.

2 The code works with probabilities $p_i,q_i,$ and $r_i$ for the three dice (labelled ${\mathtt{p}}_{\mathtt{i}}$, ${\mathtt{q}}_{\mathtt{i}}$, and ${\mathtt{r}}_{\mathtt{i}}$ in the code), where $i=1,2,\dots ,n$. The code is easily extended to more than three dice—or contracted for the case of two dice. To remove the restriction that each $p_i,q_i,$ and $r_i$ must be between 0 and 1, we can simply remove the phrase “bounds = bounds” from the minimize command in code block [6]. However, our open questions in this section consider actual probabilities, so we keep the restriction of being between 0 and 1.

Additional information

Notes on contributors

Shamil Asgarli

SHAMIL ASGARLI received his Ph.D. in mathematics from Brown University in 2019. Following a three-year postdoctoral fellowship at the University of British Columbia, he joined Santa Clara University as an assistant professor. His current research interests are in algebraic, arithmetic, and finite geometry.

Department of Mathematics and Computer Science, Santa Clara University, 500 El Camino Real, USA 95053 [email protected]

Michael Hartglass

MICHAEL HARTGLASS is currently an associate professor in the Department of Mathematics and Computer Science at Santa Clara University. Prior to joining the faculty at SCU, he received his Ph.D. in mathematics from UC Berkeley in 2013, held a one year postdoc at the University of Iowa, followed by a three year postdoc at UC Riverside. His area of research is operator algebras and noncommutative probability.

Department of Mathematics and Computer Science, Santa Clara University, 500 El Camino Real, USA 95053 [email protected]

Daniel Ostrov

DANIEL OSTROV received his Ph.D. in applied mathematics from Brown University in 1994. After a year of post-doctoral studies at Brown, he joined the Department of Mathematics and Computer Science at Santa Clara University, where he is currently a professor. His research areas include optimization, financial mathematics, and partial differential equations.

Department of Mathematics and Computer Science, Santa Clara University, 500 El Camino Real, USA 95053 [email protected]

Byron Walden

BYRON WALDEN received his Ph.D. from Yale University in 1992. After teaching at Sacred Heart University and the University of New South Wales, he came to Santa Clara University and is currently an associate professor. His research interest is complex analysis.

Department of Mathematics and Computer Science, Santa Clara University, 500 El Camino Real, USA 95053 [email protected]