GAME IN ONE HAND.

This Saturday's not-regular-Saturday bridge column might get into the weeds of combinatorial theory.  The deal itself might not be that interesting, and yes, purists might suggest that bidding Three No-Trump on 25 high-card points is cheeky.

The good news is that all the bots passed.  I can't always depend on that: a few days ago I got into one where I bid a heart and partner bot bid a spade and I raised to Three No-Trump and partner bot decided that was good enough to bid Seven No-Trump.  The East bot doubled, and then partner bot redoubled.  Then the opening lead was an ace.  I forget whether this effort went down four, or five.

In this hand, the Spades might be trouble: the Ace, Jack, and three other cards are outstanding; in Hearts, there is one winner, and finessing the Queen with eight outstanding is trouble.  With proper play, all four Diamonds ought be good, although with five outstanding there is still a three-in-eight chance of a defender being able to promote a Diamond trick later.  There are also three good Clubs, with eight still outstanding the defender has a chance of promoting a few tricks in that suit.

There are two principles in conflict here.  One is that finessing a card is safer early in the hand, the better to be able to use the stopper cards in other suits.  The other is that forcing the defenders to dip into their long suits when they are no longer able to follow suit makes it harder for them to promote a trick once they gain the lead.  Losing the lead, I've discovered, can be risky.  For example, with the lead, Ace-low on the board and King-low in hand are two tricks and two entry opportunities.  Lose the lead, though, and those low cards might have to go because the rules require a card to be played on each trick.  Thus instead of two winners and two reversals, there's maybe one stop.

On to the play.  Opening lead, ♥5, fourth highest, six higher cards out against it.  East played the ten.  If the Three to the Queen brings out the King, there are a lot of Heart tricks for the defense to win.  I chose to work on the Clubs while the lead was in hand:  Three to the Queen, Nine to the King, Ace, and discard the ♠2 from the board.  Four tricks.

In retrospect, it might have been better to have worked on the Diamonds first:  Deuce to the Jack, Six back to the Queen, King pulls out East's ♠7.  The Ace brings out the ♥7 and ♥6.  Eight tricks.

Now the nail-biting begins.  I've used up the dummy entries, thus the ♠Q comes from hand, forcing out the Ace.  (That ♥3 to the Queen is retrospectively even worse, as that could give West the lead.) The King is good, provided the defense leads another Spade.  The lead is in East: first the two promoted Club tricks, the Eight and the Six with everybody else showing out, then the ♥K (had the outstanding honors been reversed, the lead would be with West and the contract broken) and all East had to lead was the ♠J, covered by the King, and nine tricks made.