Psellos
Contemporary Development With Functional Programming

The Schnapsen Log

January 17, 2017

The Battle and the War (solution)

Martin Tompa

Concealed cards:
Q
AKJ
♣ AKQ
J

Your cards:
AK
TQ
♣ —
A

Trump: J
Stock: 3 face-down cards
Game points: Opponent 1, You 2
Trick points: Opponent 7, You 32
On lead: You

Even though you only have 32 points in your tricks, and even though there are still 3 face-down cards left in the stock, did you consider closing the stock? You really should, since you need only 2 game points to win the whole game and your opponent is still below 33 trick points.

If you close the stock, you have three certain winners (AK and A) in your hand. Add their trick points to the 32 you already have in your tricks for a total of 32 + 11 + 4 + 11 = 58. This means that, if your opponent contributes at least 8 trick points to these three tricks, you will have enough to reach 66. If either of the concealed Jacks is in the stock, or if both of the concealed Queens are in the stock, your opponent must contribute at least 8 trick point and you win the game.

Do you also have other chances of succeeding with the stock closed? Your TQ opposite opponent’s possible AK should make you think of the possibility of an elimination play. Unfortunately, though, you are missing a key ingredient for such a play: you have no way to exit your hand without opening up the heart suit yourself. So there is no elimination play possible.

However, if either of those key cards AK is still in the stock, you have another sure winner. After closing the stock, pull one round of trumps with A, cash A, and lead Q. If your opponent holds only one of the cards AK, your remaining T will be a winner, and your carefully retained trump is the way back into your hand to cash it. (Leading T instead of Q works just as well.)

To review, then, if you close the stock, you will succeed if any of the following conditions hold:

  1. Either Jack is still in the stock.
  2. Both Queens are still in the stock.
  3. Either A or K is still in the stock.

This seems intuitively like a lot of chances for success. Let’s work out the probability to be sure.

In order for you to fail, your opponent must hold exactly AKJ, J, and either Q or ♣Q. That is, there are only 2 possible five-card hands your opponent can hold that will defeat you. With 8 cards in total that you have not seen, how many five-card hands are possible for your opponent to hold? Exactly

(8 ⋅ 7 ⋅ 6) / (3 ⋅ 2 ⋅ 1) = 56.

(It is easiest to see this if you think of the number of possibilities for the 3 face-down cards in the stock. There are 8 possibilities for the top card, 7 for the middle card, and 6 for the bottom card. But this product 8 ⋅ 7 ⋅ 6 counts each of your opponent’s possible hands 3 ⋅ 2 ⋅ 1 times, since that is the number of ways you could permute the 3 face-down cards in the stock without affecting which 5-card hand your opponent holds.)

Therefore, your probability of failing is 2/56 = 1/28, so your probability of success is 27/28. You cannot complain about those odds, even against a lucky opponent. In all likelihood, you will win the battle and the war.

© 2017 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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Thoughtful Actions, Mar 25
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Carpe Diem, Feb 10
The Glass is 9/10 Full, Feb 2
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