# The Schnapsen Log

## Danger at Every Turn (solution)

#### Martin Tompa

The first thing you should consider, when on lead at the last trick before the stock is exhausted, is closing the stock. Since Peter has 54 trick points, if you close the stock you would have to collect enough points without giving up a trick to him. All you could do is cash ♥A and ♠A, hoping that one of those queens is in the stock and you will collect Peter’s ten. Catching one of those tens would bring you to 68 trick points. If you failed to catch a ten, your only other chance is if ♥T is in the stock, in which case cashing either ♥K or ♣A would give you enough.

But remember that both of you have passed the 33 trick point threshold so, if you didn’t close the stock, the most you could possibly lose on this deal would be 1 game point. The odds that one of those 3 good cards is in the stock are not high enough to be worth risking the loss of 2 game points in order to win only 1. You probably already believe this intuitively, but let’s work through the expected game points if you were to close, just to verify your intuition. There are only 5 cards you haven’t seen. (Peter must have ♦K in his hand from the marriage he declared.) If any of ♥T, ♥Q, or ♠Q is in the stock, you will win 1 game point. If either of ♠T or ♦J is in the stock, you will lose 2 game points. Therefore, your expected gain is ⅗(+1) + ⅖(−2) = −1/5 game points, slightly negative. Notice that the formula gives precision to the intuition I stated beforehand: the chance of winning 1 game point is overshadowed by the risk of losing 2 game points.

All right, then, you probably shouldn’t close the stock, so let’s see how to proceed with the stock open. With Peter at 54 trick points, almost any card you lead will hand him the win. He must have at least one trump in hand, so you cannot lead either black ace. He must have at least one of the missing tens, so you cannot lead ♠J either. This leaves only a trump lead. You might as well lead the master trump ♥A as a form of desperation play, hoping to draw something from the stock that will give you the win.

Whatever is in the stock, you will be able to cash ♥A and ♠A, which will bring your trick point total to at least 60 (if Peter discards ♦J on your ♥A). If you draw either of the missing trumps from the stock, that extra trick will certainly give you enough. You cannot draw ♦K, because you know that Peter is already holding that card. The only other draw that helps you is ♠Q:

Peter:(54 points)

♠ T

♥ TQJ

♣ —

♦ K

You:(46 points)

♠ AQJ

♥ K

♣ A

♦ —

From this position, Peter’s ♠T will fall on your ♠A and your trick point total will be 67. Of the 5 unseen cards, then, 3 of them (♥T, ♥Q, ♠Q) will give you 1 game point and 2 of them (♠T, ♦J) will cost you 1 game point. Therefore, the expected number of game points you will win is ⅗(+1) + ⅖(−1) = 1/5. This slighly positive expected score seems like a good result for you, given how close Peter got to 66 trick points. And it’s better than the negative expected score you get by closing the stock.

© 2013 Martin Tompa. All rights reserved.