# Experimental Analysis of

Trigon Left Solitaire

Jan Wolter

April 10, 2013
[General Introduction]

## 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

The general pattern of these results is similar to those for Trigon.