The Schnapsen Log

Close Quarters (solution)

Martin Tompa

“The key card on which I think you should have focused,” Hans explains, “is the ♦T. If I am holding that card, you will be able to win it, probably by trumping, and that trick is key. You can actually play out the hand in almost any way, as long as you reserve a trump in your hand for the purpose of winning the ♦T.

“Probably the most straightforward way to proceed is to cash just one trump, say ♣A, on which I discarded ♦Q. Then you can cash your ♠A and give up a spade trick, let’s say to my ♠T. I would then be on lead.” Hans rearranges the cards on the table to show the position.

Unseen cards:
♠ K
♥ A
♣ —
♦ T

Tibor’s cards:
♠ —
♥ K
♣ K
♦ —

Trick points: Hans 31, Tibor 54

Hans continues. “I can cash the ♥A, if I have it, but as long as one of my cards is ♦T, you will win it and accumulate enough trick points.”

“What if the ♦T is in the stock?” Tibor asks.

“Then you’re going to lose 2 game points, and nothing to be done about it,” Hans answers. “In that case, you cannot stop me from winning both ♥A and ♠T, and you will collect only the three small cards ♠KJ and ♦Q for a total of 62 trick points. But if any other card is the last face-down card in the stock, you will gain 2 game points. This means that your expected gain is ⅚(+2) + ⅙(−2) = 4/3 game points.”

“All right,” Tibor concedes, “that would have been a better way for me to play it. But you said there was a second improvement as well.”

“Yes, that’s right,” Hans agrees. “With an expected gain of 4/3 game points, you should ask yourself whether you can do better leaving the stock open. You would have to make a lead that wins the trick, because otherwise I can cash the ♥A and limit you to 1 game point at most. Let’s try leaving the stock open and leading ♣A. You are guaranteed to win tricks with ♣AK and ♠A, and I must contribute at least two jacks and a queen, so that already gives you, let’s see, 27 plus 26 from your hand plus 7 from mine, for 60 trick points. If you draw ♠T, ♥A, or ♦T from the stock, that’s an extra trick for you that will put you well over the top, for 2 game points. If you draw ♠K, the marriage will likewise give you 2 game points. That leaves just the two smallest cards to consider.

“Let’s start by drawing ♠J. This would be the position.” Hans again rearranges the cards on the table.

Hans: (18 points)
♠ TK
♥ A
♣ J
♦ T

Tibor: (41 points)
♠ AQJ
♥ K
♣ K
♦ —

“You will need to set up a second spade trick in order to accumulate the extra trick points. That means you cannot afford to pull the last trump, because you need your trump to regain the lead. So you play ♠A followed by ♠J, putting me on lead from this position.” Hans removes a few cards from the table.

Hans: (30 points)
♠ —
♥ A
♣ J
♦ T

Tibor: (56 points)
♠ Q
♥ K
♣ K
♦ —

“Should you be concerned about losing trump control by a forcing defense? No, because you have endplayed me with a counterforce play. The only card I could use to force you to trump is the ♦T, but trumping that would give you 70 trick points. I will cash my ♥A to get over 33, but after that the only lead that doesn’t end the deal is ♣J, which does you the favor of pulling trumps for you. You finish the deal by cashing your established ♠Q and catching my ♦T for 75 trick points, winning just 1 game point.

“The only draw from the stock we have not considered is ♦Q. That would leave you on lead from this position.” Hans rearranges the cards one final time.

Hans: (18 points)
♠ TK
♥ A
♣ J
♦ T

Tibor: (40 points)
♠ AQ
♥ K
♣ K
♦ Q

“This is the only draw from which you cannot win. Neither of us can reach 66 trick points, but I have three tricks, ♠T, ♥A, and ♦T, to your two, so I will win the last trick and 1 game point.

“Putting this all together, if you leave the stock open your expected gain is ⅔(+2) + ⅙(+1) + ⅙(−1) = 4/3 game points.” Hans is finished and stops speaking.

“Wait,” Tibor objects. “You said this was an improvement, but my expected gain is exactly the same whether I close the stock or not, 4/3 game points. How is leaving the stock open an improvement?”

“Ah,” Hans replies. “Leaving the stock open is an improvement because I needed only 2 game points to win the entire game. Closing the stock risks losing those 2 game points, with probability 1/6. By leaving the stock open, you have the same expected gain, but you do not run the risk of losing the game. The most you will lose is 1 game point, again with probability 1/6.”

Tibor looks at his big brother with admiration. “After five years of playing together here, Hans, I am still learning from you.”

© 2014 Martin Tompa. All rights reserved.

Comments