Old Position, New Twist

One of my favorite bridge stories occurred at the San Antonio bridge club several years ago. I was playing a contract against a very sweet but completely clueless little old lady. I needed to play AT9 of clubs opposite Kxx for no losers so I stripped the hand and put her in to break the suit for me. Without a care in the world she put the queen of clubs on the table. Ordinarily, one should play for this to be from Qxx based on restricted choice (the defense must exit an honor or you have no chance to guess the suit). In this case however, I was confident that this lady had the QJx otherwise she would never exit with an honor. I played accordingly and went down! She had found the Q play from Qxx. When I congratulated her for her excellent play she had no idea what I was talking about. She found it to be nothing more than routine.

Aaron Haspel, as an unfortunate result of being a good friend of mine, has had to suffer through that sad story dozens of times. I'm sure it was on his mind when, playing against me, he wound up in 4 with these cards:

KQ53
K75
AQ6
842

AJ972
AT9
J52
Q9

In the auction I had made a takeout double of 1 on his left and then they had a free run. On the ace of clubs lead his RHO played the jack, and I continued with king and another club which he ruffed. I had shown up with the AK of clubs and had to have the king of diamonds for my bid. Aaron now saw that the best possible way home would be to strip everything but hearts and put me in with a diamond (seem familiar?). He started by drawing two rounds of trumps, everybody following. Now he finessed against the diamond king, cashed the ace, and exited with his last one. I was in with the king and produced...the queen of hearts! Putting yourself into Aaron's seat, what would you do now? Before immediately deciding to play for split honors based on the previous story, let's think about this a little deeper.

There is a key difference between the hand in my story and the one Aaron was playing. Here there is not restricted choice on the Q, assuming I am a good defender. As I have only shown up with AK, K and have followed to 2 spades, I know that declarer has already placed me with the Q for my bid. Accordingly, with QJ I MUST play the queen, otherwise Aaron will know that I still have it in my hand. This means it's not clear what the percentage play is in hearts. To figure it out we need to calculate the number of possible QJx(x) combinations and weigh it against the number of Qxx(x) combinations. If I have 3 hearts, there are 4 QJx's possible and 6 Qxx's possible. If I have 4 hearts there are 6 QJxx's possible and 4 Qxxx's possible! Based on the club and diamond cards played, Aaron felt that I was most likely to be 2344 or 2335 making it percentage to play for Qxx(x) if I would also always exit with the Q from that holding too. But would I?

We've established that I should clearly play the Q from QJx(x), but that is not so clear from Qxx(x). Remember, I'm not looking at the T or 9 in dummy. Therefore, if Aaron had started with ATx of hearts I could get out with a low one and guarantee a set. Only getting out with the queen would give away the contract. I would have to know that Aaron has AT9 specifically to know that it's right to exit with the Q. Amazingly, I should know just that. If he had had ATx of hearts his only shot at the contract would be to find me with the QJ of hearts, and exit with 3 rounds of hearts instead of 3 rounds of diamonds. If I did have the QJ of hearts, I would be forced to lead away from the K. For Aaron's line to make any sense, he must have AT9 specifically.

After weighing these considerations carefully, Aaron did play for split honors and was right. After I gave him kudos on his nice play, he told me that it was routine.

UPDATE: It has come to my attention that my math on the number of possible QJx(x) combinations was wrong. In fact, it was perfect if there were SIX missing hearts, but there are actually 7. Brilliant! This means there are more Qxx combinations than QJx which makes Aaron's play mathematically correct even if he could not read my shape.