Life So Short, the Craft So Long to Learn

The Schnapsen Log

December 20, 2013

Waiting for Change (solution)

Martin Tompa

Dear Hans,

I have not written in a long while, because the recent times have been full of upsets. Peter asked me to tell you how I played the deal he laid out for you. He thinks it would be a good exercise for me to write out a complete mathematical analysis. I will try my best.

Peter has taught me that, when I am on lead at the last trick before the stock is exhausted, I should first consider closing the stock. Closing the stock was appealing, because I had trump control and would gain 3 game points if I succeeded.

Cashing my T would pull Peter’s last trump and bring my trick point total to 54. Now I had to consider the possibilities for which card was left in the stock. If it was the A, my T would be the master spade and would give me enough trick points. If it was either of the kings, I would eventually trump one of Peter’s big cards with my Q for enough trick points.

The tricky cases seemed to be if one of the two big clubs was in the stock. Let’s construct Peter’s hand if, say, ♣T was still in the stock and I had already pulled his trump:

Peter: (0 points)

♣ AK

Tibor: (54 points)
♣ QJ

This seems like a difficult case (and the case where ♣A is still in the stock is nearly identical). My black cards are all losers, and I cannot simply lead Q, because Peter would discard ♣K and I would only get to 61. I realized that the solution was to wait patiently to trump Peter’s T! All I had to do was lead T and let him cash as many black winners as he likes. Eventually he would have to lead T and I would get 67 trick points. The beauty of this play is that it ensures that Peter’s ♣K is used to win a trick for him rather than falling on my Q.

So that made 5 possible cards still in the stock that allow me to win 3 game points if I close. The remaining possible card was T, so that this would be the position when I closed the stock:

Peter: (0 points)

Tibor: (40 points)
♣ QJ

I don’t see how I can win in this last case. Whatever I do, Peter can cash his three big black cards, and all I will capture with my two trumps are his two kings, which don’t give me enough trick points.

I had better make sure that the same sequence of plays on my part works no matter which of the 5 winning cards (A, K, ♣ATK) is in the stock. I first play T to pull Peter’s trump (if he has it), then I lead T. If A is in the stock, I have won. Otherwise Peter wins the trick and eventually has to let me trump a big card. Since there are 5 possible cards in the stock that give me 3 game points and 1 (T) that causes me to lose 3 game points, my expected gain is ⅚(+3) + ⅙(−3) = 2 game points.

Before closing the stock, though, I had one more thing to consider: can I possibly gain more than 2 game points without closing the stock? It seemed unlikely, since I would have to prevent Peter from taking a trick. My T is the only possible lead I could make that prevents Peter from taking a trick with the stock open. That brings me to at least 54 trick points. If I then draw ♣A from the stock, or either of the kings (for a marriage), I will gain 3 game points. In the other three cases, Peter will discard ♣K on my T. I will have lost trump control and will not take another trick, losing 1 game point. For instance, if I were to draw A from the stock, I would have to lead from this position:

Peter: (0 points)

♣ AT

Tibor: (54 points)
♣ QJ

Whatever I lead, Peter will pull my trump and take the remaining tricks. Since there are 3 possible cards in the stock (K, ♣AK) that cause me to gain 3 game points and 3 cards (A, ♣T, T) that cause me to lose 1, my expected gain if I leave the stock open is ½(+3) + ½(−1) = 1. Better to close the stock, which I did, and expect to gain 2 game points.

I hope that I will see you again soon. That would make me very happy.

With kisses,
Your Tibor.

© 2013 Martin Tompa. All rights reserved.


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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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