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The Winnability of Klondike Solitaire and Many Other Patience Games

Charlie Blake, Ian P. Gent

TL;DR

Here it is shown that a single general purpose Artificial Intelligence program, called "Solvitaire", can be used to determine the winnability percentage of 45 different single-player card games with a 95% confidence interval of +/- 0.1% or better.

Abstract

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 +/- 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.

The Winnability of Klondike Solitaire and Many Other Patience Games

TL;DR

Here it is shown that a single general purpose Artificial Intelligence program, called "Solvitaire", can be used to determine the winnability percentage of 45 different single-player card games with a 95% confidence interval of +/- 0.1% or better.

Abstract

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 +/- 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.

Paper Structure

This paper contains 44 sections, 5 theorems, 2 equations, 2 figures, 7 tables.

Table of Contents

  1. Authors' Note:
  2. Introduction
  3. Terminology of Patience Games
  4. History of Solving Patience Games
  5. Results Summary
  6. Results for Previously Studied Games
  7. Results Only Obtained Using Solvitaire
  8. Results on Variants of Klondike and Freecell
  9. Exhaustive Search using AI Methods in Solvitaire
  10. Exhaustive Depth-First Search
  11. Configurable Rule-Sets
  12. Transposition Tables
  13. Symmetry
  14. Dominances
  15. Safe Moves To Foundations
  16. ...and 29 more sections

Key Result

Theorem 1

In a game of the type described above, a winnable instance is also winnable with the restriction that when any card currently in the tableau/free cell/reserve is safely buildable, the next move must be the move of a safely buildable card to foundation.

Figures (2)

  • Figure 1: (Left) Sample layout of the game of Klondike part way through play. (Right) The same layout illustrating some terminology from \ref{['terminology']} with general areas of the layout outlined in black, and specific features outlined in red.
  • Figure 2: Klondike variations: draw size from stock (x-axis) plotted against (left) percentage winnability and (right) percentage of winnable games for which worrying back is critical

Theorems & Definitions (6)

  • Definition 1: Safely buildable
  • Theorem 1
  • Corollary 2
  • Corollary 3
  • Theorem 4
  • Theorem 5