Psellos
Contemporary Development With Functional Programming

The Schnapsen Log

December 14, 2016

If Worse Comes to Worst (solution)

Martin Tompa

Concealed cards:

KJ
♣ TKJ
J

Your cards:
AT
Q
♣ AQ

Trump: J
Stock: 1 face-down card
Game points: Opponent 7, You 7
Trick points: Opponent 30, You 26
On lead: You

Do you remember the first thing you should consider when on lead to the last trick before the stock is exhausted? Right, you should consider closing the stock, because there are fewer cases for your analysis to take into account. Another reason to consider closing is that your opponent hasn’t reached 33 trick points yet.

You have three top tricks that are guaranteed winners, after which those two queen you hold don’t look too promising. How many trick points will you accumulate in those three tricks? In the worst case, your opponent will contribute three lowly jacks to them, bringing your trick point total to 26 + 11 + 10 + 11 + 2 + 2 + 2 = 64, not quite enough. Notice, though, that if your opponent is missing one of those jacks, you will reach 66 trick points with just those three tricks.

What is the probability that your opponent holds all three missing jacks? Surely there’s a good chance that one of those jacks could still be in the stock, right? Since there are 6 cards you have not seen and each one is equally probable to be face-down in the stock, the probability of success is ½: there are 3 cases in which a jack is in the stock, and 3 cases in which a king or ten is in the stock. Cashing your three winners gives you a 50% chance of gaining 2 game points and a 50% chance of losing 2 game points. Since you believe you are the better Schnapsen player, 50-50 is not what you want to achieve.

Is it time to abandon the plan of closing the stock? No, not yet. You should first think about what your opponent would be holding if one of those 3 “unfortunate” cards (K, ♣T, ♣K) is face-down in the stock. Maybe something good will come of it.

K is an easy case, because if that card is in the stock, your Q is another winner and you will accumulate plenty of trick points by leading it. That leaves the two clubs ♣TK to consider. They are both equivalent, so let’s look at the position if ♣T is in the stock:

Opponent: (30 points)

KJ
♣ KJ
J

You: (26 points)
AT
Q
♣ AQ

Do you see how to win from this position with the stock closed? Take your time before reading on.

You need a few more trick points than your top 3 tricks will collect. These points could only come from winning your ♣Q. Your ♣AQ opposite opponent’s ♣KJ hopefully suggests an elimination play to you. Start by cashing your two trumps, in order to eliminate these as enries back into your hand. Assuming that your opponent contributes J and J to these tricks, you have reached the following position:

Opponent: (30 points)

K
♣ KJ

You: (51 points)

Q
♣ AQ

Now lead your Q. If K is in the stock, Q is a winner and ♣A will give you enough points. If K is not in the stock as in the diagrammed position, your opponent will be thrown in and must open up the club suit for you. (Your opponent could be holding ♣TK, but that is a case we have already considered, when a jack is face-down in the stock.)

© 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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