The Winnability of Klondike Solitaire and Many Other Patience Games

Abstract

Our ignorance of the winnability percentage of the solitaire card game ‘Klondike’ has been described as “one of the embar-rassments of applied mathematics”. Klondike, the game in the Windows Solitaire program, is just one of many single-player card games, generically called ‘patience’ or ‘solitaire’ games, for which players have long wanted to know how likely a particular game is to be winnable. A number of different games have been studied empirically in the academic literature and by non-academic enthusiasts. Here we show that a single general purpose Artificial Intelligence program named ‘Solvitaire’ can be used to determine the winnability percentage of 73 variants of 35 different single-player card games with a 95% confidence interval of ± 0.1% or better. For example, we report the winnability of Klondike as 81.945% ± 0.084% (in the ‘thoughtful’ variant where the player knows the rank and suit of all cards), a 30-fold reduction in confidence interval over the best previous result. The vast majority of our results are either entirely new or represent significant improvements on previous knowledge. Solvitaire uses depth-first search and exploits a number of AI techniques including transposition tables, symmetry breaking, dominances, and streamliners. We give the first correctness proofs of two key dominances for patience games.