Life So Short, the Craft So Long to Learn

The Schnapsen Log

February 16, 2013

On a Wing and a Prayer (solution)

Martin Tompa

Do you think your prospects look bleak? If so, I agree with you. You are likely to lose this deal.

Let’s start, as we should, by seeing what would happen if you close the stock. The problem is that you stand to gain only 1 game point, but to lose 2 game points. Your only hope of winning is if one of the two missing trumps is in the stock, for otherwise the only trick you will win is A. The probability that one of the missing trumps is still in the stock (assuming that you didn’t see Peter exchange J for one of them earlier in the deal) is 1/3, since there are 6 unseen cards and you need either of 2 of them to be the remaining card in the stock. Therefore, your expected gain if you close the stock is ⅓(+1) + ⅔(−2) = −1 game point. That is, you expect to lose 1 game point if you close the stock. If you saw Peter exchange the trump earlier in the deal, your expected loss is even more.

You’re not going to do any worse than losing 1 game point if you don’t close the stock, so we may as well assume that you leave it open. Let’s consider the possible leads you can make with the stock left open.

If you lead J, Peter will probably win with a higher heart and will then declare the royal marriage. Is there any chance Peter would duck your J? He would only consider ducking if he didn’t already hold the royal marriage. And even in that case, he would realize that ducking is a mistake. If (say) Q was the last card in the stock, the situation after discarding (say) J would look something like this:

Peter: (46 points)
♣ —

You: (51 points)


From this position, Peter knows that you would force him by leading ♣K, and that would be the last trick he would take.

If, instead of leading J, you lead a club, Peter will probably trump and then cash his A for enough points. It would again be a mistake for Peter to duck, even if you lead the lowly ♣K. This trick would give you at least 53 trick points, and then cashing A would give you enough.

So far, every possibility we’ve considered leads to the loss of 1 game point. It’s looking bleak. The only lead we didn’t consider is A, which would bring your trick point total to at least 60. What can you hope to draw from the stock that will allow you to make another trick? Only K, the master trump. You won’t win another trick if A is in the stock, because Peter will discard Q on your A. Similarly, if K is in the stock, Peter will discard J on your A. Leading A is a desperation play, in which your only hope is to draw the particular card K from the stock. Since there is only a 1/6 chance of success, your expected gain is ⅙(+1) + ⅚(−1) = −2/3 game point. That is, you expect on average to lose 2/3 game point by making this play. Since, with every other play, you lose 1 full game point, A is your best option. Not only is your expected loss slightly less than if you were to close the stock, but you also don’t risk giving Peter 2 game points as you would if you closed the stock. Since you’re likely to lose, when the deal is over you’d rather that the game point score be 4-2 than 3-2. So lead out your A and pray for good luck.

© 2013 Martin Tompa. All rights reserved.


blog comments powered by Disqus

About the Author

Martin Tompa

Martin Tompa (

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.


Getting Started

Links for Schnapsen and Sixty-Six

Links in German

Links in Hungarian

Recent Columns

Sidestep a Few Landmines, Sep 2
Two Last-Trick Problems, Jun 27
More Extremes of Luck, May 21
Grasping at Straws, Apr 4
A New Scheme for Remembering Cards, Mar 23
As Luck Would Have It, Sep 9