Experimental Analysis of
Trigon Left Solitaire

Jan Wolter
April 10, 2013

The Game

Trigon Left is a variation of Trigon, which has previously been analyzed here. It differs in the way empty spaces are filled. In Trigon Left, they are automatically filled from by the longest movable sequence from the next tableau column. If there is no next tableau column, or the next column is empty too, then any King may be moved in. You can read the rules or play the game on politaire.com.

This game is even more mindless than Trigon. The only decision points in Trigon occurred when you were deciding which King to move into an empty space. In Trigon Left, empty spaces are filled for you until you clear out the last column, which usually doesn't happen until late in the game, at which point it generally make very little difference what you do.

The game is won more often than Trigon. As we discussed in the analysis of Trigon, it is fairly easy for unsolvable columns to appear there. In Trigon Left you can often break up those unsolvable columns simply by emptying out the previous column.

Solver

The solver for Trigon Left is a simple variation on the depth first search system used in the Trigon solver. 98% of winnable games were won without ever backtracking, but sometimes 9 backtracks were needed, slightly more than for Trigon. I think that by the time an empty column becomes available, late in the game, there tend to be a lot of kings exposed and available to move in. Trying out different orders of moving them into empty spaces adds up to a few more backtracks than we see in Trigon, but it's still a trivial number, so the solver needs hardly any intelligence at all.

Random Deals

I ran the solver on the first million Politaire games, seeds zero through 999999. All cards were removed for 272,416 games, so 27% of games were solvable, a bit more than in Trigon, where only 16% were solvable.

An average of 20.15 cards could be removed to the foundation in each hand. In 12,336 games (1.2%), no cards could be removed, about the same number as in Trigon. The histogram below shows the full distribution of the numbers of cards which could be removed.

Numbers of Removable Cards in One Million Random Trigon Left Games

This looks much like the distribution for Trigon, except for the higher number of solvable games.

Natural Shuffling of Gathered Cards

As with Trigon, we ran an experiment in which, instead of generating completely randomized decks for each hand, we picked up the cards from the previous game in the order they lay on the table, applied n riffle shuffles to the deck, and then cut the deck once, before redealing.

We ran a million iterations of this procedure for each value of n from one to ten, and found the following win rates:

Trigon Win Rate with n Shuffles of Cards Gathered from Previous Game
Red line is win rate with fully randomized deck

So players shuffling only one to three times would tend to experience noticeable higher win rates with Trigon Left. Note that the less shuffled decks never give us lower win rates, as they sometimes did with Trigon.

Natural Shuffling of Ordered Decks

As in my experiments with Trigon, I also ran a series of test where for each game we started with a sorted deck, either in increasing order or decreasing order, and then shuffled that the requisite number of times before cutting the deck and dealing the game.

Again, a million hands were played each way, with the following results:

Trigon Left Win Rate with n Shuffles of a Ordered/Reversed Deck
Red line is win rate with fully randomized deck