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Contemporary Development With Functional Programming

The Schnapsen Log

December 5, 2020

What’s in the Stock? (solution)

Martin Tompa

Concealed cards:
J
KJ
♣ KJ
Q

Your cards:


♣ AT
AKJ

Trump: K
Stock: Open, 1 face-down card
Game points: Tibor 5, You 5
Trick points: Tibor 27, You 34
On lead: You

The first thing you should consider, when on lead to the last trick before the stock is exhausted, is whether to close the stock. If you do, you need Tibor to hold both missing clubs and the missing diamond in order to reach 66 trick points. There is only a 50% chance that he does, since any of the 6 missing cards could be in the stock and only 3 of these 6 give you a success. Therefore, if you close the stock, the expected number of game points you gain is ½(+2) + ½(−2) = 0.

Let’s see if you can do better than this with the stock open. But what is the right card to lead with the stock open? It seems confusing, particularly if Tibor discards some mystery card on this trick and you draw the mystery face-down card from the stock. There are too many combinations of these two mystery cards to keep straight. Let’s approach this differently, by going through the possibilities for the last face-down card in the stock, visualizing Tibor’s hand in each case and figuring out how he will play.

  1. Suppose there is a club in the stock. No matter what you lead, Tibor can discard his only club on it and you will draw the last club. After that, all you will win is one diamond trick, and you will end up losing 1 game point.
  2. Suppose Q is in the stock. Tibor can see that, if he ducks your lead, you can still win another club trick and then declare the diamond marriage, which will give you plenty of trick points and 2 game points. Therefore, he will trump your lead with J.
  3. Suppose J is in the stock. There isn’t much Tibor can do in this case, and Q is a good play for him whatever you led.
  4. Finally, suppose there is a heart in the stock. In this case, what Tibor plays depends on your lead.
    • If you lead A and Tibor ducks, he can see that you will win three tricks in clubs and diamonds and get plenty of trick points. Therefore he will trump your A with J, limiting your gain to 1 game point.
    • If you lead ♣A and J is in the stock, Tibor will discard Q, which prevents you from winning any diamond tricks. Now the only trick left for you is ♣T, and you will lose 1 game point. This case demonstrates why it is critical that you lead a diamond rather than a club.
    • Finally, suppose you lead ♣A and K is in the stock. Now Tibor cannot discard Q, because that will give you 48 trick points, and then cashing ♣T and K will bring you to exactly 66 and you will win 2 game points. Therefore he will trump your ♣A, limiting your gain to 1 game point.

Having gone through those cases, we can now compute your expected game point gain if you lead either of your aces. If you lead ♣A, your expected gain is ½(+1) + ½(−1) = 0 game points. (The second term corresponds to the cases when a club or J is in the stock.) If you lead A, your expected gain is ½(+1) + ⅙(+2) + ⅓(−1) = 1/2 game point. (The second term correponds to the case when J is in the stock and the third term to the case when a club is in the stock.) A is superior to ♣A, and also superior to closing the stock.

Are we done? No, we’re not because, with the stock open, there is a bizarre lead that we haven’t considered: J. I have already said what Tibor will play if a spade, club, or diamond is in the stock, so consider how it will look from his point of view if, say, J is in the stock.

Concealed cards:

J
♣ AT
AKJ

Tibor’s cards:
J
K
♣ KJ
Q

Trump: K
Trick points: Tibor 27, You 34
Lead: J

If Tibor ducks J by discarding ♣J, that gives you 38 trick points. Cashing your two aces will bring you to 38+11+11+4+3 = 67 trick points, and you will win 2 game points. Therefore Tibor must win this trick. But whether he wins it with Q or J, he will not have reached 33 trick points. You will draw K from the stock and Tibor will be on lead from this position. (I will assume Tibor won the trick with Q; if he trumped, the situation is similar.)

Tibor: (32 points)
J
KJ
♣ KJ

You: (34 points)
K

♣ AT
AK

No matter what Tibor leads, you can win a trump and two club tricks, for a total of 34+4+11+10+2+2+4 = 67 trick points. You will win 2 game points in this case of a heart in the stock. The same is true if J or Q is in the stock. Therefore your expected gain is ⅔(+2) + ⅓(−1) = 1 game point. The unexpected lead of J is your winning play!

© 2020 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.

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What's in the Stock?, Dec 5
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