Experimental Analysis of
Agnes Sorel Solitaire

Jan Wolter
April 23, 2013

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