Arthur T. Benjamin: Solving the Race in Backgammon


Backgammon is perhaps the oldest game that is still played today. It is a game that combines luck with skill, where two players take turns rolling dice and decide how to move their checkers in the best possible way. It is the ultimate math game, where players who possess a little bit of mathematical knowledge can have a big advantage over their opponents.  Players also have the opportunity to double the stakes of a game using something called the doubling cube, which—when used optimally—leads to players winning more in the long run. Optimal use of the doubling cube relies on a player's ability to estimate their winning chances at any stage of the game.

When played to completion, every game of backgammon eventually becomes a race, where each player attempts to remove all of their checkers before their opponent does. The goal of our research is to be able to determine the optimal doubling cube action for any racing position, and approximate the game winning chances for both sides. By calculating the Effective Pip Count for both players and identifying the positions' Variance Types, we arrive at a reasonably simple method for achieving this which is demonstrably superior to other popular methods.

Arthur Benjamin is the Smallwood Family Professor of Mathematics at Harvey Mudd College. In 2020, he won the inaugural American Backgammon Tour Online (ABTO) with the best overall performance in a series of 17 national tournaments.  In 2022, he was inducted into the American Backgammon Hall of Fame. He has written several books that present mathematics in a fun and magical way.

Published Sep. 6, 2022 1:19 PM - Last modified Sep. 6, 2022 1:19 PM