Life So Short, the Craft So Long to Learn

The Schnapsen Log

March 27, 2013

How Much Do You Trust to Chance? (solution)

Martin Tompa

It’s looking promising to collect another 2 game points in this deal. You already have 50 trick points, and your two aces show promise for reaching 66. As usual, the first option you should consider, when on lead to the last trick before the stock is exhausted, is closing the stock. Closing the stock is appealing, because Peter would have to follow suit if you then lead ♣A. In fact, you would be guaranteed to win 2 game points unless the last face-down card in the stock is ♣T. In that one unlucky case, you would lose 2 game points, because your only trick would be A. Still, that one unlucky case occurs with probability only 1/6, since only 1 of the 6 cards you haven’t yet seen causes your play to fail.

Often, when I am in this sort of situation with probability of failure only 1/6 or 1/5, I happily take the plunge, assuming that I won’t be that unlucky. It’s often right to take this plunge, but I probably shouldn’t be doing it so happily. To see why, let’s compute the expected number of game points: you expect to gain ⅚(+2) + ⅙(−2) = 4/3 game points. This is really not so much greater than 1 game point, which should feel a little disappointing for a position where you have such a trick point lead and so much potential to win. Things are slightly worse if you happen to know one of Peter’s cards, for instance if he executed the trump exchange earlier in the deal. In that case, your expected gain is only ⅘(+2) + ⅕(−2) = 6/5 = 1.2 game points, because there are only 5 unseen cards, 1 of which is bad for you.

Actually, in the current situation, where Peter is only 1 more game point from winning the game, this expected game point calculation doesn’t make sense, because you cannot lose 2 game points. Instead, you will lose the whole game with probability 1/6, which should feel even more disappointing.

In most situations, despite what I’ve said, closing the stock will be your best prospect, and you have to just hope you won’t be unlucky. But you don’t have to do it happily.

In today’s deal, however, Peter is so far from 33 trick points that we really ought to consider whether you can win 2 game points with the stock left open. Your best lead in this case is J. Peter cannot afford to duck this lead, since you could then cash A for enough points. So let’s assume he wins the trick with A. This puts him on lead from this position:

Peter: (25 points)
♣ T

You: (50 points)

♣ AJ

Whatever Peter leads, you can win your two aces immediately and are guaranteed to collect 2 game points. Why would you want to close the stock and take even a small risk to gain 2 game points, when you can leave the stock open and be guaranteed to achieve the same goal? Leaving the stock open and leading J is a safety play (protecting you against the possibility that ♣T is the last card in the stock), a play we have seen a number of times before. However, this one has a new slant. Usually, safety plays arise when your opponent is on lead. In today’s deal, we have a safety play when you are on lead.

© 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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