Introduction
I
started playing Klondike, the traditional form of Solitaire, when I was a small
boy. Over the years the game has helped fill many an idle moment, and today I
can even play on my telephone. I bet
Alexander Graham Bell never saw that coming. At some point, many Solitaire
players, like me, will wonder if they are completing as many games as they
should be. A bit of Google-ing
will uncover the following statement on the WIKIPEDIA
Solitaire page:
However, the theoretical odds of winning* a standard game of non-Thoughtful Klondike are currently unknown. It has been said that the inability for theoreticians to calculate these odds is "one of the embarrassments of applied mathematics”.
OK, no theoretical answer, but what about a statistical or brute-force solution based on playing lots and lots of games? Well, after much more Google-ing I was unable to find anyone anywhere who had published significant statistical results that gave a good answer to this.
So, back in 2005 I wrote a computer program that would play Solitaire using a predefined set of rules. The intention was to use the tool to play lots and lots of games, and to optimize a strategy that would on average move all cards to the Ace piles in more games - and ultimately come up with the answer to how frequently this should happen. Early versions of this program were slow and it took quite some time to run the iterations to come up with the best play strategy. The latest version is blisteringly quick in comparison, and I am now able to play millions and millions of randomly dealt games very quickly to determine if one move decision is better than another.
I recognized certain truths about this approach, and these need to be emphasized:
- I am an imperfect programmer and am unable to create a program that will play every game as well as an experienced human player could.
- Equally, people in general are also imperfect, and make mistakes. It is therefore highly likely that any results based on actual human play are going to be less than the theoretical maximum.
The bottom line here is that the statistics that I have come up with are just an indication of how many games could go all out. The exact average all out rate, were it ever possible to calculate it, would be higher. This is a personal opinion (based, frankly, on how hard I've tried to improve the results) but I don't think that it would be that much higher.
* The term "Winning" in Klondike Solitaire is ambiguous. If one considers the betting aspect of the game, then "Winning" can mean moving 11 cards or more to the Ace piles, which returns at least $55 on a deck which cost $52, and making a net profit on the deck. "Winning" in many peoples terminology also means moving all cards to the Ace piles, also referred to as "Going all out", or "Clearing the table". On this website I will not use the word "Win" because of this ambiguity.