OddThinking

A blog for odd things and odd thoughts.

The Finer Points of Klondike


Introduction

If Casey can write about his relationship with Tetris, I can write about my relationship with Klondike

When I was a kid we couldn’t afford all of these fancy-pants solitaire games. We only knew one single-player gard game. We couldn’t even afford to give a ten-dollar name, like “Klondike”, “Canfield” or “Fascination”. We didn’t even know it by the name by which it is now famous – “Microsoft Solitaire”. We simply called it “Patience”.

We didn’t play no fandangled casino version of the game, either. You turned over three cards at a time, as often as you like until you got stuck. There were no stinkin’ points or pesky time-limits. Either you won, or you lost – it was that simple.

Single-Player Game

Klondike, as I now call it – I can afford the ten dollar name now – is a solitaire game; it is only made for one player, and it only works with one player. Don’t play Klondike with anyone near you, or you will both get frustrated.

I subscribed to the old theory that it was a safety device for travellers – if you ever get totally lost in the desert, pull out a pack of cards and start to play Klondike. Within minutes someone will find you, just to hiss “Put the black 7 on the red 8!”

By the time I reached adolescence, I started to get a chip on my shoulder about people starting to play the game near me. Perhaps I am being unreasonable, but I still see it as a rude act. Starting a game of Klondike directly next to someone is sending out the message “I don’t want to talk to you. In fact, I so much want to avoid interacting with you, I will choose a second-rate, boring one-player card game over any of the far more interesting two-player games that are in existence.”

How shallow is Klondike?

Part of my attitude problem was that I understood the game to be pretty simplistic. Oh, sure, I had there were a few basic strategies to work on, but it was too much simple pattern-recognition and luck.

Then, one day, I was playing it on my PC. Why? Because the game has exactly one thing going for it. It is small and I could play it while I simultaneously waited for my software to compile and my files to download. I was, by the way, alone, so I had no philosophical objection to the game.

I was hit by a piece of inspiration – it was better to work backwards up the stock, rather than forwards – I’ll explain shortly, but suffice to say I no longer immediately play an Ace from the stock the moment I see it (as just one example). If I was foolish enough to play in front of someone now, their hiss to put the black 7 on the red 8, without the context of the rest of the stock pile would be uninformed and even more annoying.

This was a bit of a relevation to me – to have new insight into a game I have known for most of my life gave me a bit more respect for the game. It put it in a different class of games.

If I learnt something new tomorrow about Backgammon, I wouldn’t be at all surprised – I understand that Backgammon is a moderately deep game, and I haven’t fathomed its depths. However I thought Klondike was much more shallow – that I had pretty much got the basic strategy for Klondike by the time I was 13 years old – maybe just a little tweaking was required here and there to get to optimal play. So to come up with a significant change to the way I played was a great surprise to me.

Let me not mislead you – I still lose more often than I win. It is still a boring game, and not as fun as, say, Battlefield 2. In fact, playing my new strategy makes the game even more boring. It is still pattern recognition and luck, but the level of strategy increased significantly.

Am I inspired or just stupid?

A horrid thought was still in the back of my mind. Was this a bold new strategy, raising me into the ranks of the Klondike Kings, or was it just something that most 13 year olds worked out for themselves and I was just slower than everyone else?

To test this theory, I chose an arbitrary page from the selection that Google offered: ChessAndPoker.com’s Solitaire Strategy Guide by James Yates. I figured if he put the effort into solving solitaire that he put into Chess or Poker, I was on a winner. [This may be considered anecdotal, rather than research - I quickly skimmed a couple of other pages, and found quite different suggestions but nothing that undermined the analysis below.]

Playing the Yates Way

Yates has a list of nine rules. Some of them are obvious:

1. Always play an Ace or Deuce wherever you can immediately.

The justification is easy: There is nothing to lose by doing this.

Some of the rules are more subtle, but I think most people would hit on them by the age of 18.

5. Don’t clear a spot unless there’s a King IMMEDIATELY waiting to occupy it.

The justification for this is easy too: There is nothing to gain by clearing a spot prematurely, but not doing it keeps your options open.

Some Yates’ rules are real, but nonetheless minor, optimisations – like the focus on “smoothness” and the formalisation of the “5,6,7 or 8″ rule. These are the tweaks that help make you a better player, but I am not entirely convinced they make a huge difference from the heuristics I was using subconsciously in the past.

Yates also seems to be silent on whether you should ever simply place a card from the stock to the hand just because you can. If a black 7 is showing on the stock, a red 8 is open and the other black 7 is already in play, is there any reason not to make the move? I would answer yes. Yates says nothing.

It seems that my optimisation is either wrong, or obscure enough to be beyond the typical player. That’s good news!

My Optimisation

I assume that you are playing the version of the game where three stock cards are turned over simultaneously, and you can go through the stock cards ad infinitum with no time limit.

Imagine it is early in the game, and there are no spaces available for a king yet.

Imagine the top 9 cards are:

  1. 7 ♣
  2. 2 ♥
  3. A ♥
  1. A ♠
  2. Q ♥
  3. A ♣
  1. 2 ♣
  2. K ♣
  3. K ♥

A typical “greedy” approach would be to turn up the first 3 cards, and play the A ♥ followed by the 2 ♥. Then turn up the next three cards, and play the A ♣. That’s it. The next run through the stock will reveal the Q ♥; and the K ♥, and leave nowhere to go.

This time, lets reset the deck and take a more relaxed approach. First zip through the cards once, and don’t play any cards. Simply remember the last card that could be played. In this case, it is the A ♣.

[Actually, where there are two similar cards that could both only be played to the tableau, you ignore the second one. For example, if there are two black sevens that could be played, ignore the second one.]

Turn over the waste pile to create a new stock and start working through it again. This time, skip straight to your remembered card and play that. Proceed to the end of the deck using the greedy approach if any further cards are now playable.

Turn over the waste pile again and start from the top of the algorithm. This time, you will notice the 2 ♣ as the last legal play. That’s one we didn’t see before! Yay! It is working!

Play that, and away we go again.

This time through, we add another complication.

The stock looks like this now:

  1. 7 ♣
  2. 2 ♥
  3. A ♥
  1. A ♠
  2. Q ♥
  3. K ♣
  1. K ♥

The last legal play is now the A ♥, but if we play that, we will reveal the 2 ♥. Should we do a double play? That is, play both of the cards in succession?

The answer depends on what the very next card in the stock would be. As it is the A ♠, and that is immediately playable, we choose not to play the 2 ♥ now. Instead we work through the rest of the stock. When we get another chance to go through it we will find the A ♠ is now available. We wouldn’t have seen that if we did a double-play the first time. However, there is no reason not to do it this time.

So this play revealed two more cards than the simple greedy approach would have found.

Play Summary

  • The typical greedy approach would work forwards through the stock, constantly playing the first legal card that could be played.

  • The new reverse approach works backwards through the stock, constantly playing the last distinct legal card that can be played.

    • If two similar cards are available (same rank and colour), competing for the same position on the tableau, the second one is ignored.

    • Only accept a double play (or a triple for that matter) if the next hidden card in the stock would not be playable next time through the deck.

Benefit

The benefit of this strategy is that the cards nearer the bottom of the stock are rotated more often – there is more opportunity to find the elusive cards that you need, and not have them get blocked behind other cards.

This approach doesn’t guarantee a win. It doesn’t even guarantee to do no worse than the greedy approach on a given deal. [There are some deals where greedy play would reveal combinations of cards in the correct order, where this approach shuffles the order they appear, making them unplayable.]

However, I am confident it generally improves the chances of a game coming out, by reducing the risk of critical cards from being blocked behind others in the stock.

You will notice that my logic is informal – I haven’t proven that I am right, nor have I established that the idea is original. I’ll leave that to the Klondike historians! Right now, I’m going back to Battlefield 2!


Comments

  1. I noticed this in Solitaire in my teens, but never followed it through as you did. I had realised that there were cases where playing a card (or not) earlier in the stack would have ramifications on the next time through, and the thing that frustrated me was that this showed easy corner cases that were completely insoluble: a card you require can be trapped at the very top of the stack under a card that can’t yet be played. Sure, it’s not as annoying as clock patience, but it’s still pretty bad.

    I’m disappointed that you haven’t run some kind of statistics about these choices! I’ll put it on my list of Games To Ruin By Simulating, such as dominos or Bejeweled :-)

  2. To everyone else, but Casey:

    Read about Casey’s ruination-by-simulation of Bejeweled and, while you are there, urge him to get around to the dominoes write-up.

    To Casey,

    Your comment provoked me to read a couple of the papers referred to by the Wikipedia entry on Klondike.

    Usman Latif is another game-analysis-geek-after-my-own-heart. His specialty is Minesweeper, but he has a page on Klondike. He has performed a Monte Carlo analysis of the chance that a game is not merely unwinnable, but hasn’t even got an initial move, and estimates it as 0.25%. He guesses that the total number of unwinnable hands would be between 2.5 and 10%.

    This isn’t very satisfactory for a couple of reasons.

    The first is that I don’t accept that the analysis of the unplayable hand is so difficult, that Monte Carlo is the best technique. Perhaps I will spend a bit of time thinking about this one, and seeing if I can prove that. [Stop Press: I spent a short time on this, and it got harder than I expected very quickly. I haven't given up yet, but my first approach was a wash-out.]

    The second reason is that his method of estimating unwinnable is not at all rigorous.

    That brings us to an academic paper by Yan et al (in press) with an interesting quote:

    It is one of the embarrassments of applied mathematics that we cannot determine the odds of winning the common game of solitaire. Many people play this game every day, yet simple questions such as What is the chance of winning? How does this chance depend on the version I play? What is a good strategy? remain beyond mathematical analysis.

    They go on to produce an implementation of a game-solver for a variant of Klondike. The variant is called Thoughtful Solitaire, and all the cards are cards are dealt face-up. That removes any unknowns after the initial deal.

    So the number of games that they can solve should be much higher than regular Klondike. It provides an upper-bound on what a lucky Klondike player, who always guesses the best move, might achieve.

    Of course, their implementation might not be optimal – and indeed their roll-outs (iterations) were limited due to processing time, so they could do better still with the same algorithm. The results they get (70% wins) compare very well to their nearest expert human.

    If a computer processing for 1 hour and 45 minutes cannot do better than solve 70% of the hands, even with all of the hidden information revealed, what chance do we have playing Klondike?

  3. [Editor’s note: This comment was originally submitted on a different post. I moved it here where I think it is more likely to be read.]

    Interesting site. Especially the puzzle solving stuff.

    I am working on an interesting problem at the moment, or rather what started out as one problem and turned out to be a string of other problems, most of which are just as interesting.

    The problem *was*: What are the odds that any game of Klondike Solitaire is winnable?
    Obviously the answer requires a reliable way to work out all the winnable combinations. It raises questions like:

    1. If you knew all the positions of the cards (ie no face-down cards), is it possible with a perfect strategy to win all the time? My intuition says no, but I want to prove it logically.

    2. Is it possible to determine at any stage of the game whether you can win from that position or not?

    These and tons more ideas have been floating though my brain for a few days now.

    Taking a programmer’s approach to this, the game is actually a *very* lousy sorting algorithm, where you have a bunch of queues that have different rules about moving and adding items, and four final LIFO queues that only accept the cards by suit in order.

    A friend suggested something which sounds very helpful (may turn out not to be) and that is to start with the game already solved and try work backwards to achieve the game state at start (or whatever point during play).

    My programmer / semi-logical half of the brain says that working with a deck of 52 cards is going to melt my brain. The number of possible deals with a deck that size is more than 8 x 10^64. If I start with 16 cards (4 per suit) the total possible set is about 21 billion, which sounds more reasonable.

    If I reduce all the possible plays, while keeping the concepts of the game there (intermediate stack, final 4 stacks and the deck) and ignore the 3 cards at a time rule in favour of one card and unlimited loops through the deck, I should get a much better idea of the problem with far fewer probable outcomes.

    I just wanted to weigh in quickly on your idea that NP-complete games are uninteresting. Well, maybe you didn’t actually say that, you said they hold no interest for you because you enjoy cracking the algorithms that will “solve” the puzzle.

    I think that NP-completeness is probably a key factor in games like Solitaire holding our interest. The element of chance always stays there. The interest (for players) lies in making choices within limits (facedown cards, only allowed to move whole stacks at a time) and not knowing for sure what the consequences of the choice will be.

    If you reduced Solitaire to a 5 step algorithm that, if played by those rules, would solve the game perfectly in a reasonable amount of time, the game would lose the interest of players. Similar to your analysis of the card game that boiled down to randomly picking a card from the deck.

    Part of what I want to achieve is to find out whether the variant of Solitaire I described is NP-complete or not. But if it turns out to be NP-complete, there are still interesting experiments to be done, like find out which rules make it NP complete. With no rules, you would just flip through the deck, lay the cards out in numerical order by suit, and you’re done! The game is 100% solvable. Adding each constraint to the game, and finding out at what point it becomes NP-complete, that will hold my interest for a while.

    Sorry for the record post length.

  4. Bork,

    What are the odds that any game of Klondike Solitaire is winnable?

    See the comment above referring to the estimates of Latif and of Yan et al, and of the difficulty calculating it.

    1. If you knew all the positions of the cards (ie no face-down cards), is it possible with a perfect strategy to win all the time? My intuition says no, but I want to prove it logically.

    Again, the above two papers help here. Yan describes exactly this game as “Thoughtful Solitaire”, and only gets 70% success, while Latif shows that some games don’t even get as far as a first move around 0.25% of the time.

    More thoughts to follow.

  5. game is actually a *very* lousy sorting algorithm

    Wow! That is a lateral way of looking at it. I never would have thought of it. I would be interested in hearing if this approach leads to any interesting insights into a solution.

    The number of possible deals with a [52 card deck] is more than 8 x 10^64.

    How did you arrive at that figure?

    I started with 52! unique deals = ~8 x 10^67.

    Then I divided that by 4! to acknowledge that, while the suit is important, the suits are not ranked in anyway and could be substituted for each other, to get ~3 x 10^66.

  6. Bork,

    Your point that NP-complete games may still be interesting – if only as a rule-base that can be modified into non-NP-complete games – is well taken.

    I am not at all convinced, however, that Klondike is NP-complete.

  7. I hope as you said that “thoughtful solitaire” (I read the paper as well – gave me many insights) is probably not NP-complete.

    I’m not sure about dividing the total possible deals by four. Once the game has progressed to an obviously winnable state, the suits don’t matter any more, but during play, the sorting uses different rules (ie black suits on red suits) and only at the end are they separated. The suits do have a bearing on the sorting process.

    I do have a feeling that the order of all four suits is a factor here. Like I said, I am working on intuition at the moment. I am going to start experimenting by building a very simple (Ie non-graphical) simulator, using some textgrid controls and arrays of “card” objects. I will probably put this up on SourceForge or Codeplex – it would be great to get additional people’s input on this.

  8. Hmmm… I was wrong. Dividing by 4! is not correct. (Note: “4!” = factorial(4) = 4 x 3 x 2 x 1.)

    Let’s take a particular deal, and find all the fives. Can I switch them around randomly with each other without affecting the outcome? For the Clockwork Patience, the answer would be Yes, and I would divide the total number of deals by 413. However, for Klondike the answer is No.

    Now, let’s find all the clubs and hearts. Could I switch every club with every corresponding heart without affecting the outcome? For Spider Solitaire, the answer would be Yes, and I would divide the total number of deals by 4!. However, for Klondike the answer is No. I wrongly claimed this was true in the comment above.

    Now, let’s find all the clubs and spades. Could I switch every club with every corresponding spade without affecting the outcome? For some games, such as Bridge, the answer would be No, but for Klondike the answer is Yes. Equally, I could swap hearts with diamonds. I could even swap both black suits with both red suits.

    The complete list of valid suit substitutions is:

    1. H<–>D
    2. C<–>S
    3. C<–>S,H<–>D
    4. S<–>H,C<–>D
    5. S<–>D,C<–>H
    6. S–>D–>C–>H–>S
    7. S–>H–>C–>D–>S
    8. No change

    That’s a total of 8 different suit swap combinations which are permitted.

    So, I should only divide 52! by 8 to find the unique Klondike deals.

  9. Interesting analysis here guys. Glad my humble little guide helped spark such a lively discussion. I’ve since produced a dominoes guide and the Vegas-style Klondike guide as well. Great site Julian.

    James Yates
    ChessandPoker.com

  10. Well, I threatened to do it, and I’ve finally started.

    Based on the inspiration, mainly from this blog post, and partly from other sources on the web, I’ve started my Cracking Solitaire project. So far it consists of a new blog, but I will soon open up a Codeplex or Sourceforge project with all the little experimental programs that are going to be produced.

    Thanks for an inspiring blog, Julian!

  11. I have been working on a Klondike Solitaire for quite some time. You might have a look.

  12. Sorry, the link didn’t show in the previous post.

    http://www.codeplex.com/deeplogic

  13. That first post should read, “Klondike Solitaire Solver” (i.e. a program that simulates and solves Klondike Solitaire games)

  14. I believe that Yates’s strategy refers to the’Klondike version, where one card at a time is turned over from the deck and a player is allowed to go through the deck as many times as they like.’
    Your strategy refers to the version where at one time three cards from the deck are turned over at the same time.
    Obviously then, your strategy is not compatible with Yates’s.

  15. Drats,

    Tom is absolutely right about Yates.

    Yates also writes about the Vegas version where you have three cards flipped, but limited run through the cards.

    So where does that leave us? It doesn’t undermine my strategy. It does undermine my claim that the strategy is non-obvious, leaving the claim unproven.

    Oh well.

  16. Talking about “Thoughtful Solitaire”, It’s worth mentioning that this is actually a very fun game to play. Yes, with a deck of cards and everything.

    I usually deal a regular game of klondike, but with all cards face up. Then I also deal the rest of the deck in groups of three, so I can see the entire deck from start. The rules for how to play cards from the hand are more complex than most regular games of solitaire, so I will leave them as an exercise for the card players out there.

    There is also a very neat computer game for Windows called K+ Solitaire. It is “Thoughtful Solitaire” with all cards showed from start, and it also has a really quick solution finder. Find it at http://www.greatgamesexperiment.com/game/kplus

    In this game, the rules for which cards are allowed to be played can be very confusing (although the computer handle it for you). Take for example the following first cards:

    1. 7 club
    2. 2 heart
    3. 1 hearts

    4. 1 clubs
    5. 5 diamonds
    6. 1 spades

    The ace of hearts and the ace of spades will be playable from the start. If you now play the ace of hearts (simulating in a regular game of Klondike that you draw three cards and play the ace of hearts), the 2 of hearts will now be playable. But also the ace of clubs since that would be playable in the next round (if you drew all the cards on the hand and then re-drew the first three).

    But if you play the ace of spades now, the 2 of hearts will not be playable any more, since playing the ace of spades simulates you drawing three more cards (leaving the two behind) and playing the ace.

    I don’t know if the solution finder will be able to find all solutions, but just clicking “new game” and then “search” will show that much more than 50% of all games have a solution.

    The latest numbers form the wikipedia article is from the “ICGA Journal”, September 2007 where Bjarnason, Tadepalli and Fern show an algorithm that can solve around 80% of all thoughtful solitaire games in less than 4 seconds. They also show that between 82% and 91.44% of all Klondike games have a possible solution.

  17. >> “They also show that between 82% and 91.44% of all Klondike games have a possible solution.”

    This is somewhat misleading as the rules are never clearly defined. They are using very “loose” rules allowing a large percentage of games to be solved. The number of solveable games must obviously depend on the rules.

  18. What are the odds of winning game of klondike (Draw 1, Unlimited Re-Deals) if one uses the Yates strategy (http://www.chessandpoker.com/solitaire_strategy.html)?

    I figure someone math geek must’ve answered this question, but 30 minutes on google hasn’t yielded an answer yet…

    My win percentage so far (only 50 games played) is about 34%.

  19. Also, here’s a lecture about solitaire by Stanford math prof Persi Diaconis. http://tinyurl.com/5o4g9x

  20. I got interested in the win percentage, and with computer Klondike, using the command-Z key to back up and correct bad moves, I can get to 71-73% solvable games. I am pretty patient at exploring all the options. So while I could believe that 75, or maybe 80% are solvable I do not believe claims of 98% solvable. No way.

    When playing with cards, my percentage is much lower, because there are moves where you have to make a choice that could be wrong statistically (you have two kings, which do you move off the stack. This would be very different for the Thoughtful Solitaire variant). I would guess that my percentage is 25% with cards. Also with cards it is a different game because half the time is spent shuffling. :)

    When playing on my cellphone, I seem to win many more games, 30-40%, and have wondered if they are all “solvable” games but I think this is too hard to program. Perhaps it is just that I can focus on the game and not how annoyed I am by shuffling.

  21. How about the chances of winning Klondike (all 52 cards placed in their suits) on a FIRST deal at one sitting, no “cheating,” three cards from the stock turned over at the same time?

    This has never been my experience until last night.

    No “big deal,” right?

  22. I play Windows 7′s game, 1 card draw, and with unlimited undo. I have won 609 out 732 games or about 83.2%. Not many seem to play this style–so it’s hard to say–but I think this rate pushes against the upper limit of winning. I found one player who claimed in excess of 90% over approximately 500 games, but, were that true it would mean I am giving up too soon on over 40% of my losses, which I truly do not think is the case.

  23. Well, I figured out how one can achieve a 90% or even 100% win rate in Windows 7′s game: when encountering an undesirable puzzle, simply Ctrl-Z back to the beginning then quit. It’ll destroy the puzzle and not penalize your streak. Why? Well it seems undo undoes everything, including resetting the “has not played” flag.

  24. Cute, Rob. I found a similar bug some time ago in Windows Hearts. Starting a second game before taking a move in the first one ensures that both matches have the same cards dealt to each player in the first hand. You can play one hand to lose, to find where all the cards are, and then play to win in the second game with secret knowledge.

  25. Klondike Solitaire is like a Rorschach test — it is a mirror of the projections from the player’s mind (values, attitudes, beliefs). The game is as profound as the player.

    Philosophy is reasoning, a product of the human mind. Logic, strategies and rigid rules are products of the human mind. Winning is a product of the human mind. The human mind is not very deep. The human mind is the world’s greatest reality-distortion mechanism.
    Computer-based solitaire in contrast to actual playing cards avoids illegal (rule breaking) moves, analogous to preventing violations of natural law.

    The game is a microcosm of the Cosmic Play. The journey is the thing. Serendipity and unexpectedness provide joy for every hand. A hand that is blocked by only one card of only two cards remaining in the stack or by only one remaining hidden card in the tableaus offers as much delight as a smooth complete transition from the stock pile to tableau piles and then to foundation piles. Laughing at yourself for being blocked by the smallest of obstacles is a reminder of daily life.

    To paraphrase a saying attributed to William Blake, “Solitiare [life] is not a problem to be solved but a mystery to be lived.”

    When a solitaire player evolves from the mind to the heart and then from the heart to the innermost center of being, the game becomes deeper and deeper and eventually transcends all limits. The game becomes a meditation.

  26. “Klondike Solitaire is like a Rorschach test”
    No, I am afraid it isn’t. Not in any meaningful way.

    “it is a mirror of the projections from the player’s mind (values, attitudes, beliefs). The game is as profound as the player.”
    The game is fairly simple, even as games go. That you think it is as profound as the player, I am afraid that reflects poorly on you.

    “Philosophy is reasoning, a product of the human mind. Logic, strategies and rigid rules are products of the human mind. Winning is a product of the human mind.”
    So Solitaire is like Philosophy, because it is was conceived by humans! So, by the same argument, solitaire is also like love, solidarity and Jackass: The Movie. The works of Shakespeare are like hard-core pornography. Truth is like Falsehood.

    “The human mind is not very deep. The human mind is the world’s greatest reality-distortion mechanism.”
    This reminds me of the query: “How deep is this question?” It is surprisingly deep. But it is a product of the human mind, which *itself* is the product of the human mind! So therefore, they are both deep, leading to a contradiction in your argument.

    “Computer-based solitaire in contrast to actual playing cards avoids illegal (rule breaking) moves, analogous to preventing violations of natural law.”
    Do you mean natural law in the sense of “an observable law related to natural phenomena”? So putting a 2 on a 4 is just like breaking a law of thermodynamics? Or do you mean natural law in the sense of unchanging moral principles that form the basis of human conduct? So, putting a 2 on a 4 is like savagely attacking an unarmed person?

    “The game is a microcosm of the Cosmic Play.”
    A microcosm is about representing a large and complex reality, with a smaller, simpler one. Except, Klondike doesn’t do that. (Am I wrong? What do the four foundation piles represent?)

    > The journey is the thing. Serendipity and unexpectedness provide joy for every hand.
    Unexpectedness? From a turned-over card? It is one of 13 ranks and 4 suits. How unexpected can you get?

    “Laughing at yourself for being blocked by the smallest of obstacles is a reminder of daily life.”
    You laugh while playing solitaire? This has gotten decidedly weird.

    “When a solitaire player evolves from the mind to the heart and then from the heart to the innermost center of being, the game becomes deeper and deeper and eventually transcends all limits. The game becomes a meditation.”
    I try to make sure commenters feel welcome here. I try to be polite. But then someone comes along and spouts utter, utter bullshit. It makes it more difficult to truly have both semi-serious discussions about trivialities like Klondike, and also serious discussions about true issues of the heart, when people who understand neither come and spout pseudo-philosophical, pseudo-spiritual, pseudo-scientific gibberish.

    If you have any real insights, I would be happy to hear them. If you have any trivial or funny comments, I would be happy to hear them. But when you have neither, there really is no reason to pipe up.

  27. I have removed a couple of comments from the same person as above. They purport to be an experimental result about the number of cards laid down after playing 104 games of Klondike. It contained a number of howlers, such as suggesting that because there was no known optimal strategy, that there are no possible heuristics, and intuition is the only play. The conclusions reached (e.g. that less than 7% are solvable) seems to contradict with other anecdotes mentioned here. Basically, it got more rigorous, but no more insightful than the last comment, which wasn’t addressed, so I removed it as noise.

  28. Klondike is a great game when you realise as the folks here do that it is quite deep. I developed my own rules to improve the chances of winning. One that I like is: make the first card of the draw-three playable. If say it is a red 4, then find a way to place a black 5. When all else fails, you have a black 5 which is the “key” that opens the red 4 “lock” and the rest of the reserve deck can be displaced by one card, opening up new possibilities. Another is: look for a “sequence filler”. If there are a red Jack and a red 9 showing then play a black 10 from the reserve if one becomes available. Then play the 9. This turns over the card under the 9 which is what you want. Attempt to exhaust the sequence fillers before trying a different strategy. Play from the back of the deck so as to minimise the displacement of the front of deck cards. They are not going anywhere. Train memory to recall what cards are in the deck and in what order so as to maximise potential plays. As I say this is not a trivial game. With discipline and concentration, I can get 33% wins using these rules plus quite a few others. This is without going back and refining the move. But I do allow any number of plays of the reserve and I can take a card back from the ace stack because the computer allows me to.

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  1. OddThinking » Analysing the first move of Klondike

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