Solitaire Statistics Revisited

I opened a thread on this subject just over a year ago. It sparked an interesting debate… Chronos and Dex were really into it for a while.

Then it died, because no one could answer my question.

I would like to bring it up again; lots and lots of smart people have signed up since then and maybe someone new can help us out.

I play solitaire (basic “klondike” game) on my Palm Pilot. It tracks statistics, and, a year ago, I was [sup]22[/sup]/[sub]154[/sub] for a winning percentage of 13%. I remarked then that my percentage had never dropped below 12% or gone above 15%.

My record is now [sup]67[/sup]/[sub]432[/sub], or 15%. Again, in the entire year that has passed, I’ve never dropped below 12 or 13% and never risen above a brief stop at 17%.

I don’t spend a lot of time thinking about my game, I just play. I use a straightforward and consistent strategy. I believe that the games I do not win are simply un-winnable.

What I want to know is:

Given 52! possible combinations of cards in a shuffled deck, might it be true that only 15% or so of all possible games are winnable?

In other words… is my score a measure of my skill, or is it… in the cards?

By way of providing a second data point:

I got 125/740, or 16% won.

I wouldn’t call September 14, 2000 over a year ago.

Since then, I am embarrassed to admit, I have played a number of games fairly carefully. 34 of 94 were wins for 36%. As described in the original thread, you need to plan ahead by knowing where every card is in the hand is, and before playing a card be aware of what cards will become available for the next pass through the hand.

I was putting in more thought than most people when they play this game, but there was plenty of room for more planning if only I had more patience.

From circumstantial evidence in Manlob’s post, it sounds like he’s playing the three-card version of Klondike, so I’ll ignore that for now. David Parrott in A History of Card Games notes that one-card Klondike is a fiendishly inflexible game, with little opportunity for skill. A brief analysis of the game shows why: unlike most solitaires, there are very few choices to be made in the average game of Klondike. Once in a while you may have the choice of, say, putting a red seven on a choice of two black eights, but that’s about it. Otherwise, the game is mechanical in a way that most solitaires are not–other solitaires give you choices of discard piles, greater flexibility in building foundations or win piles, etc. (One wonders, as did Parrott, how Klondike became the most popular form of solitaire when it’s so mechanical. My guess: it’s precisely its mechanical nature that makes it popular, since you can play it without thinking too much.)

So my answer: winning at Klondike is in the cards, not in the player’s skill.

Here’s an idea: you can make the problem much more tractable by simplifying it. For example, in a modified Klondike where the deck has only 2 suits where each suit has, say, 5 cards, and there are only 2 piles it shouldn’t be too hard to figure out the generic scenarios which make the game absolutely unwinnable (and then proceed to enumerate them).

Now, that’s probably not going to give you a very good approximation to the percentage of unwinnable games in real Klondike, but with those results in hand it might be possible to figure out what happens as you relax the constraints.

I’m too lazy to do it myself, but that’s what I’d do if I were a Real Mathematician.

It’s an idea, Bobort, but for an accurate model, you do need all four suits, at least. Most of the player decisions in single-draw Klondike (as opposed to three-draw, where you want to keep seeing the right cards come up) depend on their being two possible opposite-color cards on which to play a given card. If you reduce it to two suits, then the only decision is whether to move a card to the aces or leave it on the stacks for further development, a decision which often doesn’t even come up in a game at all.

Just coming back to my thread to correct myself–the author of A History of Card Games is David Parlett, not David Parrott. A two o’clock mistake.

Yes, I was referring to drawing 3 cards at a time, as was sdimbert if you read the thread referenced in the OP. This is using the rules of Microsoft solitaire (allows cards to be played off the aces piles and partial sequences to be moved).

The winning percentage of sdimbert’s lackadaisical strategy indicates he is not even winning half of what a more careful approach to the game would give. Being thoughtful also helps in the version of the game with dealing one card at a time.

I used the same program I used in the original thread and added various strategies to see the effect. This program played at least a few hundred thousand games per case. Starting with the basic lazy strategy of always playing a card from the hand when possible and always moving the leftmost sequence when there is a choice, it wins 29.4%. Adding the condition that the first cards played from the hand are those that allow another sequence to be immediately played on it (i.e. a red 7 won’t be played from the hand unless there is a black six at the foundation of one of the sequences) raises the winning percentage to 33.1%. Always moving the rightmost sequence when there is a choice raises it to 42.1%. If when there is a choice, the sequence with the most facedown cards below it is moved, winning percentage rises to 43.2%.

Although these are simple strategy rules that were chosen for the ease in which they can be added to my program, each one shows a gradual improvement in winning percentage. There are many other things that can be considered if making a full effort at winning. When I carefully play manual games dealing one at a time, I win 65%. So there is still lots of improvement possible over a the most basic strategy.

Looking at sdmibert’s low win percentage, I was sure he was playing the one-card version. Actually, looking again, his win percentage would be high for the one-card game: my dad has played a couple hundred games on Bicycle Solitaire and has won about 5%. Note that this is one-card, once through. No way in heck could anyone be getting 65% on one-card, once through.

What Parlett said about Klondike is inadmissble for the three-card version of the game. I totally agree that it’s possible, through card counting and “perfect play”, to raise sdimbert’s win percentage dramatically.

One last thing. The Microsoft rules (cards can be played off the ace piles, partial foundations can be moved) dramatically tilt the balance in favour of the player. I assume that sdimbert’s version does not allow these moves, which in a strict sense I would consider “cheating.” Without these moves, player choice in Klondike is much more limited, and I doubt sdimbert would be able to get anywhere near the win percentages claimed by Manlob.

I am playing the draw-3 flavor of Solitaire! :frowning:

I guess Manlob has settled it.

I just suck.


I have played 4656 sessions of 3 card klondike and won 765 time for a 16.4 % average.

I have been playing Spider Solitaire for a number of years and my statistics show I settle at about a 17% win rate. It usually starts out a little higher, but soon falls into the 16-18% range. I have reset my stats dozens of times with almost the exact same results. I assumed it was because of my skill level, but it may just be inherent in the game itself. Interesting.

Dealing one at a time, once through the deck, which I consider to be ‘real’ Klondike, results in wins around the 5% mark. No way can Manlob win 65% of the time. Nuts. The idea of ‘carefully playing’ is idiotic. One at a time klondike is almost completely chance. Skill has nothing to do with it.

Note this is a 10 year old zombie which refers to an 11 year old zombie.

I’ll say! :eek:

I opened this thread thinking, “Hey, a thread about playing cards! I should look at it…oh, half the posts are by me. Man, have I been here a long time.”

Interesting that the win rates seem to settle between 15-18% for three-card, and at about 5% for one-card, even ten years on. I have no idea of the skill set of individual Dopers (and my dad, who’s still playing), but I feel confident in saying that skill has little effect on win rates, and that thus Klondike is a largely mechanical exercise.

I totally fell for that at first. I was like, “his PalmPilot is still going strong after more than a decade, that’s impressive.”

This may be a zombie, but what the hell.

According to Cracked, the statistics of solitaire is one of the 8 simple questions science can’t answer.

It’s the way you play. I waste a lot of time with spider solitaire and my win percentage is 53%. However, that’s misleading, since there’s a bug in the way it calculates that percentage and it should be significantly higher.

My version is one that came with my Linux system. However I’ve also wasted time with the Windows version and had a similarly high percentage.

I read that article. I thought, “That’s a simple question?” There are only 52! (i.e. 8 times 10[sup]67[/sup]) ways the cards can be dealt, after all…