We finally have an answer to a surprisingly tricky mathematical problem: what are the chances of winning a game of solitaire? It was once called an 鈥渆mbarrassment鈥� of mathematics that this couldn鈥檛 be solved, but now a computer program is homing in on the solution.
Solitaire, also known as patience, is a single-player card game that involves sorting cards using certain rules. Probably the most famous variant is called Klondike, which is often installed on Windows operating systems.
Klondike is hard. Renowned mathematician Irving Kaplansky once played 2000 games and won only 36.6 per cent. Later, computers won more than 80 per cent, but there was still a huge amount of uncertainty about the true odds.
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To address this, Charlie Blake and Ian Gent at the University of St Andrews, UK, wrote a computer program to compute the approximate odds of winning any version of solitaire.
For Klondike, the program dealt 1 million random hands and computed the best strategy for each. It had to examine 20 billion partially played positions, and calculate sequences as long as 2274 moves to find that you have a roughly 82 per cent chance of winning the game.
For other variants, the odds ranged from nearly 100 per cent for Freecell, which has also often appeared on Windows, to around 16 per cent for Trigon, which is similar to Klondike, but has stricter rules on when cards can be moved.
The pair鈥檚 program can鈥檛 prove mathematically what the odds are, but with a random sample of a million hands, they have reduced the remaining uncertainty by a factor of 30, so that the estimates should be within 0.1 per cent of the true number.
鈥淲e can certainly be less embarrassed now that we not only know the winning probability but have a winning program that generalises to other games,鈥� says Prasad Tadepalli at Oregon State University.
Reference: arXiv,
