Play the King less than a third of the time

by Jonathan Mestel

How randomly do you play? I'm asking you - not your partners. And how might one play a card a third of the time? Well, you could hum a waltz under your breath. If declarer plays on the first beat of a bar you play one card, otherwise another. This whimsical notion derives from the play problem given last issue. You are playing 4s from the West hand. The defence cash three diamonds and exit with a heart. You win in dummy and run the sQ. How do you continue (a) if the queen wins the trick, and (b) if South covers the queen?

S A5432
D 432
C K2
W        E
H K43
D 765
C AQ43

When can we avoid losing a trick with sQJ9 opposite sAxxxx? We can forget about 4-1 breaks. Furthermore, we need the K onside, but there is nothing we can do against K10x. There are three possible holdings for South when we can succeed: K10, Kx and Kxx. Holdings Kx and Kxx are each 3 times as likely as K10, as there are 3 possible small cards. If South holds K10, he must cover the Q with the K, and the winning strategy is then to play the J, dropping the 10. With Kx, if he covers we win and finesse the 9 on the second round, while if he ducks, we lead the 9 to the ace, as his king falls. With Kxx, if he does not cover we lead the J next round, pinning North's 10.

This is all very well, but of course we can't see South's cards, and in practice we have a guess to make whether South plays a small card or the K on the first round. Before reading further, I suggest you decide what you would do in each case. Assume the opposition is good and plays impassively in tempo.

First, consider South's problem. If he holds K10 he must cover the Q and hope we misguess. Likewise, holding Kxx, it would be foolish for South to cover, as when we win and lead towards the J9 the 10 pops up and we cannot go wrong. But if South holds Kx he must decide whether or not to cover.

Since South will never cover with Kxx, but might with the equally likely holding Kx, if the Q is not covered, South is at least as likely to hold Kxx as Kx. So if he does not cover on the first round, declarer should run the J on the second round.

If South does cover the Q, declarer's best line depends on how often this particular South would cover holding Kx. Provided South covers less than a third of the time we can do no better than playing the J on the second round, picking up K10 and Kxx but losing to Kx. If, however, we judge that South is weak enough to cover with Kx more than a third of the time, then it is better to finesse the 9. I think in practice, most declarers would adopt this line, but it is slightly insulting. I therefore recommend always ducking with Kx in this position, to exact maximum penalty for this rudeness! The general principle of not covering the first of two touching honours is valid here.

What if South hesitates or fumbles? Now we are on dangerous ground. I think that any reasonable player who hesitates before covering with K10 is being unethical. But no EBU director would rule that it was unreasonable to hesitate holding Kx or Kxx. The best line is to play the J on the second round whether or not South covers.