# The Schnapsen Log

## Change of Hands (solution)

#### Martin Tompa

Tibor was far behind in trick points, but made up for it with his three aces. He said that the first thing he considered was closing the stock. His three aces plus three queens from me would bring his trick point total to 61, so he knew he needed to win a trick with his ♥K. An elimination play looked promising if I held ♥TQ. He could cash his black aces, and I would have to hold on to both of my hearts to protect ♥T. If, say, ♦T was still in the stock, this would be the resulting position:

Hans:(40 points)

♠ —

♥ TQ

♣ —

♦ A

Tibor:(47 points)

♠ —

♥ AK

♣ —

♦ J

Now Tibor would throw me in with a diamond and I would be forced to open up the heart suit, giving Tibor another 28 trick points for a total of 75.

Tibor realized the problem with this play, though. If either of the two unseen hearts was still in the stock, I would get to 66 before he did. He had to eliminate his ♣A and ♠A entries before endplaying me. If, for instance, ♥Q was the last card in the stock, this would be the position when Tibor then tried to throw me in:

Hans:(40 points)

♠ —

♥ T

♣ —

♦ AT

Tibor:(47 points)

♠ —

♥ AK

♣ —

♦ J

Even though my ♥T would give Tibor enough points to win, he wouldn’t know that it was going to fall to his ♥A. When he instead tried to throw me in with a diamond, I could cash ♦AT for 67 trick points and Tibor would lose 2 game points.

This is actually an interesting position to think about. If I have discarded ♠Q and ♦Q on Tibor’s black aces, it puts him in an absolute quandary. If either diamond is in the stock, then he wants to lead a diamond for the endplay. But if either heart is in the stock, he wants to instead cash ♥AK for enough points. He has no clue which way to proceed, and he might as well toss a coin to make the decision.

Anyway, the upshot of all this is that, if Tibor closed the stock, he would win 1 game point with probability 3/5 (neither heart in the stock) and lose 2 game points with probability 2/5 (either heart in the stock). Remember that ♦Q couldn’t be in the stock, as I had declared the diamond marriage earlier. Therefore, the expected number of game points Tibor would gain by closing the stock was ⅗(+1) + ⅖(−2) = −1/5.

Tibor realized this was probably not his best outcome, so he turned his attention to what would happen if he left the stock open. It didn’t seem appealing to him to lead an ace, since the aces would be useful for controlling their suits once the stock was exhausted. So what about leading ♦J, Tibor wondered.

If he led ♦J, he could see it would do me no good to win the trick, for I would be on lead from this position:

Hans:(53 points)

♠ Q

♥ TQ

♣ —

♦ TQ

Tibor:(19 points)

♠ A

♥ AK

♣ AJ

♦ —

Tibor has four winning cards, so he is guaranteed to capture one of my tens. That would give him 73 trick points in total, plenty to win 1 game point.

I should instead duck his ♦J lead, discarding ♠Q if I have it. This would bring Tibor’s trick point total to 24. Whatever he draws from the stock, his three aces eventually will fetch ♣J and two queens, bringing his trick point total to 65, just one point shy of winning. This means that drawing any winner from the stock, ♠Q, ♥T, ♥Q, or ♦A, would give Tibor 1 game point. That leaves just the two smaller diamonds to consider.

If he drew ♦Q, he would be on lead from this position:

Hans:(40 points)

♠ —

♥ TQ

♣ J

♦ AT

Tibor:(24 points)

♠ A

♥ AK

♣ A

♦ Q

His three aces are guaranteed to collect one of my tens, bringing his trick point total to 72 and earning 1 game point.

Finally, if he drew ♦T, he would be on lead from this position:

Hans:(40 points)

♠ —

♥ TQ

♣ J

♦ AQ

Tibor:(24 points)

♠ A

♥ AK

♣ A

♦ T

In this last situation, Tibor’s original elimination comes back into play. He cashes ♣A and ♠A in that order, bringing us to this position:

Hans:(40 points)

♠ —

♥ TQ

♣ —

♦ A

Tibor:(51 points)

♠ —

♥ AK

♣ —

♦ T

Now his lead of ♦T endplays me: I have to open up the heart suit, and Tibor wins 1 game point.

So it was a small expected loss if Tibor closed the stock, or a guaranteed gain of 1 game point if he left it open and led ♦J. An easy choice, Tibor thought. What he liked about this example was the idea of leading a “loser” (♦J) rather than either closing the stock or cashing an ace.

© 2014 Martin Tompa. All rights reserved.