Charlie Blake, Ian P. Gent
Our ignorance of the winnability percentage of the solitaire card game `Klondike' has been described as ``one of the embarrassments 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 $\pm$ 0.1% or better. For example, we report the winnability of Klondike as 81.945% $\pm$ 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.