# Experimental Analysis of

Agnes Sorel Solitaire

Jan Wolter

April 23, 2013
[General Introduction]

## The Game

Agnes Sorel Solitaire is one of two similar
Klondike variations traditionally known as "Agnes".
David Parlett gave it a last name to distinguish it from the other.
You can read the rules or
play the game on politaire.com
or many other places.
The game uses a tableau similar to the Klondike tableau, but there is no waste.
Instead, when no other moves can be made (or sooner if we wish) we deal cards
from the stock onto each tableau pile, as in Spider.

The traditional rule in this game is that blank spaces in the tableau may not
be filled from another tableau pile.
These days many people allow blank spaces to be filled by any card or stack of
cards from another tableau pile.
I'll call the older, harder version "Classic Agnes Sorel" and the newer, easier
version "Relaxed Agnes Sorel".

## Solver

The depth-first-search solver that I built for
Trigon was easily adapted to Agnes Sorel.
For the classic game, it runs very fast,
In the relaxed game,
there are lots of choices about what to do when an empty space appears,
and we end up with much larger search trees to explore.
## Random Deals

I ran the solver on the first million Politaire Agnes Sorel games,
seeds zero through 999999.
For the classic game only 45 games out of the million were solvable,
which is about one in twenty-thousand.
(Game number 28115
is one that is solvable.)
On the average only 5.40 cards could be removed.

**Numbers of Removable Cards in
One Million Random Classic Agnes Sorel Games**
Note the long tail on this graph.
It is possible to be stuck with as few as two
cards left on the tableau, and that happened for
seed 967427.
The relaxed version of the game, where any card may be moved to an empty space,
is substantially easier.
Out of a million games tested, 78,304 were solvable, about 7.8%.
We were able to remove 9.48 cards on average.

**Numbers of Removable Cards in
One Million Random Relaxed Agnes Sorel Games**
The distribution looks much the same, except for the much higher number of wins,
and the fact that we can't get stuck with so few cards left on the tableau.
## 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. Note that the vertical
scale on this graph is from 0% to 1%, not 0% to 100%.
Because the numbers of winnable games are so low, and because we have magnified
the vertical scale a lot, this graph is trifle noisy.

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

Clearly, even a modest amount of shuffling suffices to make this game
just as difficult as full randomization.

Here's the same graph for the relaxed version of the game, with a vertical
scale that goes all the way to 100% this time:

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

## 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*.
**Classic Agnes Sorel Win Rate with ***n* Shuffles of a
Ordered/Reversed
Deck
Red line is win rate with fully randomized deck

An ordered deck doesn't seem to help a whole lot, but
an reversed deck seems helpful so long as we hardly shuffle it at all.

**Relaxed Agnes Sorel Win Rate with ***n* Shuffles of a
Ordered/Reversed
Deck
Red line is win rate with fully randomized deck

Note that the line for the reversed deck case dips slightly below the
red line for 5 shuffles or more. The same pattern appears more prominently
in many other solitaire games.