Based on some references I got from another question I learned that:

While convalescing from an illness in 1946, Stan Ulam was playing solitaire. It, then, occurred to him to try to compute the chances that a particular solitaire laid out with 52 cards would come out successfully (Eckhard, 1987).

Interest piqued, I googled on an write up on what Ulam would have done but came up empty handed -- I found countless mentions of this episode but nothing that was step by step.

I found the original paper by Eckhard, here, but again it does not address the application to MCMC

So Question:

Does anyone have a pretty introductory write up on how Ulam applied MCMC to solitaire?


  • 1
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    – Glen_b
    Jul 8, 2015 at 1:46

2 Answers 2


As already said by Paparazzi, the paper is about Monte Carlo method, not Markov Chain Monte Carlo. MCMC is just a part of a broader family of Monte Carlo methods (i.e., informally, using simulation to solve statistical problems).

This is actually described in the referred paper

[...] I was convalescing from an illness and playing solitaires. The question was what are the chances that a Canfield solitaire laid out with 52 cards will come out successfully? After spending a lot of time trying to estimate them by pure combinatorial calculations, I wondered whether a more practical method than "abstract thinking" might not be to lay it out say one hundred times and simply observe and count the number of successful plays. [...]

So it's pure Monte Carlo: he simulated a number of plays and then counted the number of events of interest among the all simulated events. It was obtained using a direct simulation.

Actually MCMC was discovered later with Metropolis algorithm being the first MCMC algorithm as described in A Short History of Markov Chain Monte Carlo: Subjective Recollections from Incomplete Data paper by Robert and Casella (2011).


But there is no evidence he used Markov chain Monte Carlo. There is no reason to use a probability distribution. You just shuffle the deck and get a random deck - straight up Monte Carlo. See Fisher Yates for random shuffle.

  • $\begingroup$ I always thought that Ulam came to Monte Carlo, and not MCMC via the solitaire example. However I believe that Metropolis as in Metropolis-Hasting algo was at Los Alamos with Ulam . There is a wonderful article by Diaconnis called something like "THE MCMC revolution" which has a rather nice example of using MCMC to decode encrypted documents. $\endgroup$
    – meh
    Dec 28, 2016 at 20:52
  • $\begingroup$ I beg to differ. Perhaps OP has asked a question for which the technical answer is "there is no evidence". However, while tangential, I think my comments might be of interest to OP. Feel free to act on your theory and just not reply to this. $\endgroup$
    – meh
    Dec 28, 2016 at 21:12
  • $\begingroup$ @ Paparazzi Did you read my first comment ? It says explicitly, that Ulam used in your words "straight up" Monte Carlo. So my final comment to you is , meh. Feel free to continue posting, I'll read, but I am done. $\endgroup$
    – meh
    Dec 28, 2016 at 21:19
  • $\begingroup$ @aginensky Read my answer. MCMC is an algorithm not a place. If you are getting a shuffles with a probability distribution rather than random it is a crooked house. This is gone not where. $\endgroup$
    – paparazzo
    Dec 28, 2016 at 21:34
  • $\begingroup$ Perhaps you can go into more details or back up your answer with some references? Basically I agree, the paper is about how Ulam discovered Monte Carlo, not MCMC. MCMC is just a part of multiple different Monte Carlo algorithms and methods. $\endgroup$
    – Tim
    Dec 28, 2016 at 22:16

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