# The Schnapsen Log

## Pressured Out (solution)

#### Martin Tompa

This is how Tibor explained his thinking to me. The first thing he considered is closing the stock, since that simplifies the possibilities. (You see how faithfully I have passed on your lessons to him.) With the stock closed, he can certainly win tricks with his ♦AQ and ♠T (using his ♠K to knock out my ♠A while he still has a trump to regain the lead). He must collect at least 2+3+4 = 9 trick points from me, all of which will bring his trick point total to at least 27 + 24 + 9 = 60.

Now, if any of those “small” unseen cards ♠QJ or ♥K is still in the stock, he will in fact collect at least 15 trick points from me, which would bring his total to at least 66. Likewise, if ♠A is still in the stock he will have no problem.

But the part of Tibor’s analysis that most impressed me was what would happen if ♥T is still in the stock. This is the position that Tibor spread on the table, but he had worked it all out in his head during the play:

Me:(20 points)

♠ AQJ

♥ K

♣ T

♦ —

Tibor:(27 points)

♠ TK

♥ Q

♣ —

♦ AQ

Tibor said he would play it exactly the same way. First he would lead ♠K to knock out my ♠A. I cannot cash my ♥K, because I need that “small” card as a discard on one of his 3 winners to keep him from reaching 66. So I must exit with ♠J. He then cashes ♦A, on which I discard ♠Q, reaching this position:

Me:(35 points)

♠ —

♥ K

♣ T

♦ —

Tibor:(53 points)

♠ —

♥ Q

♣ —

♦ Q

When Tibor now cashes his last trump, I am squeezed: to keep him from reaching 66, I must discard ♥K, which establishes his ♥Q as a new winner. That is, if I don’t discard ♣T on ♦Q, I will be forced to discard it on ♥Q one trick later!

Very neat, isn’t it? I was surprised to see that a squeeze is possible even though Tibor must let me in with my ♠A before the squeeze. After all, I could capture his squeeze card ♥Q with my ♥K once I am on lead, breaking up his squeeze play. But in that case I consign myself to donating my valuable ♣T to his tricks later! I find it fascinating.

Returning to the deal with the stock closed, the only card he hadn’t yet considered that might be in the stock is ♣T, but he saw that in this one case he could not win. I took out a pen and paper to show him how to work out the expected number of game points he would gain: ⅚(+2) + ⅙(−2) = 4/3 game points. He seemed to understand the concept.

This is as far as Tibor got in his analysis, and he closed the stock and proceeded as planned. I think he did extremely well. The only avenue he didn’t pursue, because he hadn’t known about expected game points, was to check whether he could possibly do better than 4/3 game points leaving the stock open. I showed him how to do this and will include this last part on his behalf.

With the stock open, the only lead he could make that would prevent me from crossing the 33 trick point threshold immediately is a trump. But then, unless he draws ♠A from the stock, he could not reach 66 without first letting me reach 33. (If true, his expected gain could not possibly be greater than 4/3 game points: the most it could be is ⅚(+1) + ⅙(+2) = 7/6.) For instance, if he led ♦A, I discarded ♠J, and he drew ♠Q from the stock, he would be on lead from this position:

Me:(20 points)

♠ A

♥ TK

♣ T

♦ J

Tibor:(40 points)

♠ TKQ

♥ Q

♣ —

♦ Q

From here he could cash ♦Q and declare his marriage, but that would leave him just shy of winning at 65 trick points when I cross the threshold.

Tibor was quite interested to learn probabilistic expectation as a mathematical means to compare two alternative plays. It feels good to carry on the brotherly tradition of sharing these ideas. You will be very proud of Tibor when you next play against him.

With kisses from both of us,

Peter.

© 2013 Martin Tompa. All rights reserved.