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Contemporary Development With Functional Programming

The Schnapsen Log

April 9, 2016

Keep Calm (solution)

Martin Tompa

There is nothing you can do to prevent Rudi from winning a trick with the ♣Q he showed as part of the marriage at trick 1. If he is to fall short of 66 trick points, then, your only hope is to force him to trump J with that ♣Q, which would give him a total of 65. This means you must assume that he does not hold ♣A and that he does not hold two diamonds. These assumptions make what we are constructing a desperation play, where you assume whatever you must in order to have a chance of success.

After winning his T lead, then, you will want to tackle diamonds and knock out his ♣Q immediately, while you maintain control of the spade and heart suits. But this still leaves one question: does Rudi have one diamond or none? If he has one, you must start with A, but if he has none, you must start with J. How do you decide?

From the unseen cards at the end of trick 3, there are more 5-card hands with one diamond that Rudi could hold than 5-card hands with no diamond. But this isn’t the right way to think about the situation, because all those possible 5-card hands are not equally probable. After all, you have the additional information that Rudi closed the stock and led T. He would not have made that play from some of the possible 5-card hands. So what can you infer from his actions?

Surely Rudi does not hold KQ, or he would have declared that marriage and ended the game. Could he hold TK? Let’s construct such a hand for Rudi and look at it from his point of view, in what I earlier called a role reversal.

Concealed cards:
AKQ
AQ
♣ A
ATQJ

Rudi’s cards:
T
TKJ
♣ Q

Could Rudi have a hand containing TK such as this? Not likely, since from this hand he would lead K rather than T in order to establish his T as a winner if you hold A. Leading K wins the deal unless Rudi’s opponent holds ♣A, A, no Q, and either A or no spades at all.

We have eliminated two possible heart holdings Rudi could have, given his play. Can we infer more? Could Rudi hold Q? Let’s again construct such a hand for Rudi and look at it from his point of view.

Concealed cards:
AK
AKQJ
♣ A
AQJ

Rudi’s cards:
TQ
T
♣ Q
T

It seems unlikely that Rudi would lead T from such a hand. He is usually better off leading one of the red suits and hoping that you open up the spade suit, in case you hold AK. The inference that Rudi does not hold Q is not iron-clad; for instance, Rudi’s lead of T would be reasonable from a hand such as this:

Rudi’s cards:
TQ
TQ
♣ Q

Here Rudi could lead T, hoping that you either don’t have A or that you will open up the heart suit for him. Clearly, though, Rudi could just as well have chosen to lead T from this hand, so the probability that he holds this hand should be weighted by ½.

Let’s summarize. Rudi does not have ♣A, does not have Q with a few exceptions, and has none of the combinations TQ, KQ, and TK. Given these inferences, the only hands Rudi could hold that contain no diamond are these:

1 T TQJ ♣Q -
½ TQ TQ ♣Q -
1 TQ TJ ♣Q -

The first column shows the weight of the hand’s probability, weighted by ½ if the lead of T is one of two equal choices.

In contrast, the list of hands Rudi could hold that contain one diamond is as follows:

½ T TQ ♣Q T
1 T TQ ♣Q Q
½ T TJ ♣Q T
1 T TJ ♣Q Q
½ T KJ ♣Q T
1 T KJ ♣Q Q
½ T QJ ♣Q T
1 T QJ ♣Q Q

These lists show that there are the equivalent of 2.5 hands Rudi might hold containing no diamond, and 6 hands Rudi might hold containing one diamond.

Since it is more likely Rudi holds one diamond than none, you should win his T lead, then lead A (holding your breath until he follows suit), and finally lead J (holding your breath until he trumps with ♣Q). If Rudi doesn’t have ♣A up his sleeve, he will get no further than 65 trick points and you will win the game. Well inferred, maximizing your chances of defeating him on this deal!

© 2016 Martin Tompa. All rights reserved.


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About the Author

Martin Tompa

Martin Tompa (tompa@psellos.com)

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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