Psellos
Contemporary Development With Functional Programming

The Schnapsen Log

December 29, 2013

Exit Plans (solution)

Martin Tompa

The first thing Peter considers, as he should, is closing the stock. He has 3 certain winners, ♣AQ and A, but cashing them will not get him to 66 trick points, even if the lowest unseen card, J, is face-down in the stock.

When he looks at the arrangement of the diamond suit, though, it does seem that an elimination play may be possible. He would have to coerce Apu to open up the suit, leading away from his TQ. Does Apu have safe exit cards that need to be eliminated before Peter throws him in with a heart? Yes, he does, if Peter still holds any trumps at that point. In that case, any heart would be a safe exit for Apu. So Peter must begin by playing both his trumps, on which Apu will, in the worst case for Peter, discard KJ. This will leave Peter on lead in this position:

Unseen cards:

AT
♣ —
TQ

Peter: (41 points)

Q
♣ —
AK

With his trumps gone, it is now safe for Peter to lead Q, throwing Apu in. If Apu has only diamonds left at this point, he is forced to open up the suit and Peter will take both remaining tricks for a total of 69 trick points.

Peter notices, though, that Apu needs to hold both diamonds for this play to succeed: capturing T alone would bring Peter’s total to only 62 trick points. Since either of two diamonds in the stock causes Peter’s play to fail, Peter’s expected gain is ⅔(+2) + ⅓(−2) = 2/3 game points.

That’s not bad, but Peter is not ready to close the stock yet. He first considers whether he can do better than 2/3 game points by leaving the stock open. The facts that Apu is still under 33 trick points and Peter has absolute trump control suggest that it’s possible he can do better.

With the stock left open, the best way to keep the diamond endplay option available and Apu’s trick point total below 33 is by playing a trump, say ♣A. Apu will discard J if he has it, and Peter’s trick point total will be 34. Peter can next pull Apu’s newly drawn ♣J by leading ♣Q, for a total of 39 trick points.

Peter’s A is guaranteed eventually to bring his trick point total to at least 53. This means that, if Peter draws any of the 4 cards AK or TQ from the stock, he will get 66 trick points before Apu wins another trick, and Peter will gain 2 game points.

There are two other possible cards Peter can draw from the stock. If he draws T, this will be the position after he pulls Apu’s last trump for the elimination play:

Apu: (25 points)

AK
♣ —
TQ

Peter: (39 points)

TQ
♣ —
AK

This is a charmingly symmetric position in which Apu can be endplayed in either diamonds (if Peter leads a heart) or hearts (if Peter cashes A and exits with K). In either case, Peter will end with 67 trick points and Apu with 53, so Peter will win 1 game point.

The final case is that J is the last face-down card in the stock. In this case, Apu will discard K on Peter’s ♣A at trick 5, and Peter will be on lead from this position when the stock is exhausted:

Apu: (25 points)

AT
♣ J
TQ

Peter: (36 points)

QJ
♣ Q
AK

There is again an elimination play, but this time Apu can only be endplayed in one suit. Peter eliminates Apu’s one safe exit card by pulling trump, reaching 41 trick points, and then throws Apu in with a heart. Apu can cash his other high heart, but eventually must open up the diamond suit, presenting Peter with another 28 trick points for a total of 69. Peter’s gain is again 1 game point.

To summarize, with the stock left open, there are 4 cards Peter can draw from the stock that will give him 2 game points and 2 cards that will give him 1 game point. Therefore, his expected gain is ⅔(+2) + ⅓(+1) = 5/3 game points. Peter compares this to his expected gain of 2/3 game points if he closes the stock, and realizes that he is much better off leaving the stock open!

© 2013 Martin Tompa. All rights reserved.


Comments

blog comments powered by Disqus

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.

Subscribe

Getting Started

Links for Schnapsen and Sixty-Six

Links in German

Links in Hungarian

Recent Columns

December
Singleton Tens, Dec 1
November
Complete Information, Nov 28
September
Gamble, Sep 30
April
Homework on Expected Values, Apr 26
March
Thoughtful Actions, Mar 25
Noble Sacrifice, Mar 4

Archives

2018
2017
2016
2015
2014
2013
2012