The Schnapsen Log

The Last Place You Look (solution)

Martin Tompa

“The key to analyzing this particular endgame,” Hans explains, “is to imagine adding the ♥Q to your hand and imagine all the unseen cards being in Emmi’s hand. As though you were each holding six cards and drew the face-up ♥Q in advance of losing the fifth trick. This is often the right way to approach the endgame if the decision will come down to who wins the very last trick.”

Hans reconstructs these synthetic six-card hands on the table for everyone to study:

Emmi: (20 points)
♠ AQ
♥ J
♣ Q
♦ AK

You: (31 points)
♠ K
♥ KQ
♣ A
♦ TQ

“Now, of course, you can’t start out by declaring the royal marriage, because you’re not really holding ♥Q in your hand yet. No, the point of this exercise is to count entries into both hands, for the purpose of working out a play to win the final trick. In this case, you have three entries, ♥KQ and ♣A, and Emmi also has three, ♠A and ♦AK. Since you are on lead and Emmi is the first one to use up one of her entries, this suggests that you should be able to win the final trick. Your general strategy is to lead a loser and use up one of Emmi’s entries every time you are on lead. That means starting out with either a spade or a diamond.”

“But I did start out with a spade,” you object immediately.

“Yes, my dear, you’re right. You did, and it didn’t work out as well as we might have hoped. The reason is that Emmi wasn’t required to follow suit. If she had been required to follow suit, you would have won the final trick in this imaginary scenario where you each start with six cards. But we have the extra complication of having to consider the possibility that Emmi will duck trick 5, possibly discarding from some other suit. That is exactly what happened when you led ♠K.

“So, you must make it more expensive for Emmi to duck your lead. Among your losers, the lead that will make it most expensive for her to duck is ♦T. That’s probably the last card you would have imagined leading out voluntarily, isn’t it? Let’s just see what would happen if you do, though.

“If Emmi were to win this trick, the first thing to check is that she couldn’t get to 66 trick points. Her three winners would only bring her trick point total to 63, so we’re fine on that point. We’ve already decided that she cannot win the final trick by taking your ♦T, as long as you lead a loser whenever you are on lead. That means Emmi must duck, otherwise she is certain to lose 1 game point. Suddenly it makes a lot of sense to lead the surprising ♦T, because that 10-point card is going to end up in your own tricks!

“All right, then, Emmi ducks your ♦T. Her best discard is the one she chose, ♣Q, because that denies you the possibility of cashing ♣A as long as she holds a trump. This trick will bring your trick point total up to 44. You are guaranteed 6 more trick points later when you win your ♥K. That’s 50. Now we come to the cumbersome part of enumerating the possible cards you might draw from the stock.

“Three of the possibilities are easy. If you draw either of your marriage partners, ♠Q or ♦K, that’s a quick win of 70 trick points and 2 game points. If you draw ♥J, you can run your trumps and cash ♣A for 2 game points.

“Suppose next that you were to draw ♠A. This would leave you on lead from this position.” Hans deftly rearranges the cards on the table to illustrate.

Emmi: (20 points)
♠ Q
♥ QJ
♣ —
♦ AK

You: (44 points)
♠ AK
♥ K
♣ A
♦ Q

“In order to win the last trick, you want to lead a loser,” Hans continues. “But not ♦Q, remember, because you don’t want to lead a suit in which your opponent has more cards than you do. You want to leave such suits to your opponent to open up. What you should do instead is cash ♠A and then lead your losing ♠K in order to force Emmi to trump. This puts her on lead from this position.” Hans removes a few cards from the table.

Emmi: (27 points)
♠ —
♥ J
♣ —
♦ AK

You: (58 points)
♠ —
♥ K
♣ A
♦ Q

“Now Emmi would like to force you back in diamonds in order to win the final trick with her last trump. But she can’t. Do you see why not?”

There is a long pause before you see the answer. “Because trumping her ♦K with my ♥K would give me exactly 66 trick points!”

“Very good, my dear,” Hans says with a nod. “She cannot play diamonds, so her only other choice is to play her trump, which would allow you to cash ♣A. It’s a counterforce play, and it endplays Emmi at trick 8. The reason the counterforce works is because you won ♦T at trick 5: that put enough points in your tricks for the counterforce. Emmi had better cash her ♦A at trick 8, in order to limit your win to 1 game point.

“If you draw ♦A from the stock, the situation is nearly identical.” Hans rearranges the cards on the table.

Emmi: (20 points)
♠ AQ
♥ QJ
♣ —
♦ K

You: (44 points)
♠ K
♥ K
♣ A
♦ AQ

“This time you cash ♦A, bringing your trick point total to 59, and force her with ♦Q. Emmi is once again endplayed by a counterforce play, and must yield 1 game point to you.

“The last draw to consider is ♣Q, which we assumed until now was Emmi’s discard at trick 5. In this case, she will probably have discarded ♠Q instead, and you will be on lead in this position.”

Emmi: (20 points)
♠ A
♥ QJ
♣ —
♦ AK

You: (44 points)
♠ K
♥ K
♣ AQ
♦ Q

“♣Q is the sole losing draw for you,” Hans continues. “The only trick you will take from this position is your master trump. The remainder, and 1 game point, go to Emmi.

“Let’s finish by computing your expected gain if Emmi ducks trick 5,” Hans says. You are relieved that he is finally drawing to a close. “Three draws, ♠Q, ♦K, and ♥J, give you 2 game points. Two draws, the aces, give you 1 game point. And the last draw, ♣Q, causes you to lose 1 game point. Therefore, your expected gain if Emmi ducks trick 5 is ½(+2) + ⅓(+1) + ⅙(−1) = 7/6. Since you are guaranteed to win 1 game point if she doesn’t duck, this means you expect to gain about 1 game point whether she ducks or not.”

“But, Hans,” you object. “I only needed 1 more game point to win the game.”

“Oh, right you are, my dear,” Hans admits, somewhat embarassed. “I’d forgotten that. In this case, my conclusion should have been that, if Emmi ducks, you will win the game with probability 5/6 and lose 1 game points with probability 1/6.”

© 2013 Martin Tompa. All rights reserved.

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