# Experimental Analysis of

Acme Solitaire

Jan Wolter

April 19, 2013
[General Introduction]

## The Game

Acme Solitaire is another variation of
Canfield and.
Storehouse.
As in Storehouse, we start with four cards on the foundation, deal cards one
at a time, and build in the same color on the tableau.
Unlike Storehouse, we cannot move stacks of cards, and are allowed only
two passes through the stock.
It also differs in that the foundations build up from ace instead of two,
but that makes no real difference to the game.
You can read the rules or
play the game on politaire.com
or many other places.
This game is obviously harder to win than Storehouse, and the player has
to make more difficult choices.
It isn't always clear if playing a card to the tableau is a good idea,
because each card added to a stack makes it harder to move that stack.

There is a variant called Coyote which allows stack
moves, making the game much easier, to the point of being rather simple-minded,
since you can play by just making every possible move. Only after the reserve
is empty and empty space open up in the tableau is there any need for thought.

According to Pretty Good
Solitaire Acme is a century old game, but it's not wildly popular now.
According to the same site's score lists,
Canfield was played 12 times as often and
FreeCell was play played 580 times as often.

## Solver

The depth-first-search solver that I built for
Storehouse was easily adapted to Acme.
There are many more choice points, but the game is still very constrained,
so it runs pretty quickly.
The solver for Coyote was simpler still.
## Random Deals

I ran the solver on the first million Politaire Acme games,
seeds zero through 999999.
All cards were removed for only 36,401 games, 3.6% of the total.
An average of 16.9 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.

**Numbers of Removable Cards in
One Million Random Acme Games**
I also solved the first million Coyote games, and found that 19.1% of games
could be won, removing an average of 23.2 cards.
The distribution of the number of removable cards is shown below.

**Numbers of Removable Cards in
One Million Random Coyote Games**
## Natural Shuffling of Gathered Cards

If the game is played with real cards instead of on a computer,
then it is unlikely that the cards will be completely randomized
between games.
So we consider the case where, when a game has been completed,
we pick up the cards one stack at a time, reversing the stacks that were
faceup, shuffle *n* times, cut once, and deal the next hand.
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:

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

Well, here's a game that seems to be a heck of a
lot easier with hand shuffled cards.
If we only did one shuffle, pretty much every game was winnable.
Only three in a million were unsolvable,
and those were the first three games played,
before we got the deck sorted at all.
The win rate was barely lower with two shuffles.
With three shuffles and a cut, we get down around 13%.
After that it converges pretty quickly to the same value as a fully randomized
deck.

It seems quite possible that this game was substantially more enjoyable
without computer shuffling, but only if people were doing fairly bad shuffling.

## Natural Shuffling of Ordered Decks

As usual, in my next test, I started each hand with
a sorted deck (each suit in increasing order),
or a reversed deck (each suit in decreasing order),
then shuffled that deck *n* times, and cut once before
dealing.
I again had the computer play a million games
for each value of *n*.
**Acme Win Rate with ***n* Shuffles of a
Ordered/Reversed
Deck
Red line is win rate with fully randomized deck

In this instance, the ordered and reversed deck aren't really dramatically
different.