Storehouse Solitaire

April 14, 2013

- Instead of dealing cards from the stock three-at-a-time, we deal them one-at-a-time, but we are limited to three passes through deck.
- Instead of building the tableau down in alternate colors, we build it down in the same color.
- The game starts with all four two's dealt to the foundation, instead of with one random card.

The effect of the first two changes is to eliminate virtually all need to make intelligent choices while playing the game. (The third change does nothing much except reduce the deck size a bit.) You can't go wrong by playing every card you can play, except maybe late in the game when the reserve has been emptied, and a space opens up on the tableau. Then you need to decide if you want to play the top waste card into that space, or deal another card from the stock, and there is some chance that whether you win or lose will depend on your choice.

I found that 71% of Canfield games were solvable, so one might suppose that that game is much easier. It isn't. Any player who is vaguely awake can win 44% of Storehouse games, while even the most ingenious Canfield player would be lucky to get as high as 35%. The percentage of games that can be solved is not really a very good predictor of how hard a game is.

An average of 31.6 cards could be removed to the foundation in each hand. The histogram below shows the full distribution of the numbers of cards which could be removed. Note that our counts of removed cards include the four cards that were dealt to the foundation, so it is never less than four. I also omitted the "52 card" column from the graph, because it is too tall.

For each number of shuffles from one to ten I ran the solver on one million games. The first game used a fully randomized deck, and after that the procedure above was followed to produce the deck for the next game.

The resulting win rates are shown in the graph below:

Red line is win rate with fully randomized deck

This graph follows a pattern that is growing very familiar as I test different solitaire variations. If you shuffle just a few times, you have an elevated chance of winning. (In this case, with just one shuffle, we won every game, and with two shuffles we lost only one games in five hundred.) But four shuffles actually leaves you with a slightly lower chance of winning than you would have with a fully randomized deck, though, of course, with enough shuffles things converge to the same odds as a fully randomized deck.

Red line is win rate with fully randomized deck

As with many other games I have studied, if we shuffle only a couple times, then we win a lot, regardless if the deck was sorted in increasing or decreasing order. With 3 to 7 shuffles, we see we do much better if our deck was originally sorted in decreasing order than if it was in increasing order. And as the numbers of shuffles gets large, we converge, of course to the fully randomized deck.

Note that after seven shuffles, the player who started with a reversed deck still has a 3% better chance of winning than the player who started with a ordered deck. This is not as spectacular a disproof of the "seven shuffles rule" as some others that have been pointed out, but it is an indication that that phenomenon continues to have some influence even in more complex games.

The ordered deck graph looks very similar to the gathered deck graph, even though the win rate is under 50%. However, even in lost games we tend to get a substantial number of cards off, so likely this contributes.