Solitaire fills time away from family, friends, and other activities.  For math geeks, it becomes kind of an obsession to consider the likelihood of outcomes.

Like crossword and Sudoku puzzles, Solitaire attracts “geeks” and the following information related to Solitaire is likely to attract the dork in you.


The Idiosyncrasy of Klondike

The Solitaire game, Klondike, intrigues those who like to analyze algorithms and consider the odds.  Actually, the occasional inability to move on or complete a game is a unique characteristic of this particular version of Solitaire.  Not all Klondike games can be solved, but why?

Klondike for Windows

Klondike, offered along with the Windows operating system, features seven stacks with a deck to draw cards.  A game is deemed unplayable when the following conditions occur in unison.

  • No aces appear throughout playable cards
  • No playable cards (in the rows) can be moved to a different stack
  • No playable cards in the card deck can be moved to a stack


Geeks quickly calculate the probability of the first instance to occur by considering the number of all combinations of 15 cards (taken from a deck with no aces) and dividing it by the combination of 15 cards (taken from a deck including aces).  Using Binomial coefficients, the result is calculated.

The calculation assumes almost twenty-five percent of games start with no aces among playable cards, and seventy-five percent of games have at least one ace appearing in the initial 15 playable cards.

Probability of Unplayable Games

The above does not help in calculating the probability of unplayable games of Solitaire.  There is no analytical way to calculate how to satisfy all three conditions as described above.  The number of all possible games is enormous and poses problems in using a computational approach.

Monte Carlo Simulation

A Monte Carlo simulation can help estimate the possible outcomes.  The simulation repeats an experiment a large number of times.  Simulating either a favorable or an unfavorable outcome, the number of times a favorable event occurs is divided by the number of events in total and gives an estimate of the positive outcome.

Scheme Program

A coded ‘scheme program’ simulated 10 million Klondike games and calculated a total of 24,893 unplayable games.  So, about one out of every four-hundred Klondike games are unplayable.

Unplayable games are unsolvable and a number of games that appear playable at first cannot be finished.  So, 2.5 to 10 percent of all Klondike games are ‘unsolvable.’

Are You a Geek?

Does the above surprise you or make you want to double check to the beat of your own calculations?  You may just be a geek!  Reading the information makes one assume most games are actually winnable.

Yet, the opposite happens most of the time.  Why?  Because a whole lot of guesswork is involved in a successful game and but a few miscalculations leads to an end in options.  Can you defy the odds regardless of the calculations?  Load a game of Solitaire and try your attempt at achieving amazing results!

Jared Hill is a computer scientist and avid gamer. Whenever he has a chance, he likes to post about technology and games online. His articles can be found on many tech, gaming and entertainment blogs.

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