Contemporary Development With Functional Programming

Schnapsen Log Archives

Columns for August, 2012

Make a Virtue of Necessity

Martin Tompa
August 2, 2012

The cake is gone, the coffee is refilled and steaming, and Peter still sits across from you at the Café Abeles, with Lore kibitzing. You couldn’t be more content. You and Peter have each shown a marriage and the trick point score is dead even….

Exit Cards, Stoppers, and Optimal Endplay

Martin Tompa
August 8, 2012

I am again going to break from our normal format, in order to tell you about a fascinating scholarly article that has a bearing on the Schnapsen endgame. The paper is very mathematical, developing a “calculus” of cards that allows you to look at the hand you were dealt, calculate the maximum number of tricks you can take, and determine a strategy for obtaining that maximum number of tricks. Sounds useful, doesn’t it? And, for the mathematically inclined, this calculus involves infinitesimals, which are “numbers” greater than 0 but less than any positive fraction such as 1/100 or 1/1000000000 that you can come up with. It’s all quite beautiful, in a way that only mathematics can be.

The paper is “Two-Person Symmetric Whist”, by Johan Wästlund, which was published in The Electronic Journal of Combinatorics in 2005. The author is a researcher in the Department of Mathematical Sciences at Chalmers University of Technology in Sweden….

A Theory Concerning the Last Trick

Martin Tompa
August 13, 2012

While we are on a break from our normal format, I’d like to tell you about another very elegant result by the mathematician Johan Wästlund, closely related to his results on the game of symmetric whist. Today’s topic is an optimal strategy for the sole purpose of winning the last trick. Since a reasonable fraction of Schnapsen outcomes are decided by who wins the last trick when the stock is exhausted, this question has a clear connection to Schnapsen strategy.

The paper of Wästlund that contains this result is “The Strange Algebra of Combinatorial Games”, written in 2009. This paper discusses several different sorts of games. The results we will discuss today are just from the paper’s Section 7.2, which deals with a version of the traditional Swedish card game Femkort (“Five-card”), in which the winner of the game is the player who wins the last trick.

The good news is that Wästlund’s results on Femkort are much simpler…


Martin Tompa
August 17, 2012

You are back at the Café Abeles, at your favorite table, playing cards with your buddy Peter. Outside it’s steaming hot and humid, but in here the ceiling fans and heavy stone walls keep things nice and cool. Cool enough that you’re enjoying your hot coffee while you play.

It’s been a close game so far and you’re in the endgame of a close deal where no one declared a marriage and the stock has just been exhausted….

Losing Tempo

Martin Tompa
August 20, 2012

It seems as though your coffee hasn’t even had time to cool at the Café Abeles when you find yourself once again in a quandary….

Order from Chaos

Martin Tompa
August 24, 2012

Lore has once again taken Peter’s place across the table from you at the Café Abeles. As friendly as Lore is, she’s a much more intimidating opponent than Peter is. You’ve noticed some surprising plays from her, and you suspect that there have been many more that were too subtle for you to notice….

About the Author

Martin Tompa

Martin Tompa (

I am a Professor of Computer Science & Engineering at the University of Washington, where I teach discrete mathematics, probability and statistics, design and analysis of algorithms, and other related courses. I have always loved playing games. Games are great tools for learning to think logically and are a wonderful component of happy family or social life.

Read about Winning Schnapsen, the very first and definitive book on the winning strategy for this fascinating game.


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Recent Columns

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