### by Jonathan Mestel

Last newsletter, I set the problem of finding the worst combined trump holding with which it is still possible to make a given number of tricks. The defence is assumed to be optimal, except that a trump lead was disallowed. `Worst' is defined by total length and then by highest card, so that 432 is better than A2 which is better than KQ.

Here are the solutions and best entries. It may still be possible to improve on these, but I think they're optimal.

(a) To make 1, a singleton 2 suffices, provided the other trumps split 6-6. Declarer makes seven tricks in the side-suits.

(b) To make 2, declarer must make a trump trick. A stiff ace of spades is enough against any lead, but if a trump lead is barred, a stiff K suffices. Declarer cashes seven tricks in the side suits and then leads a suit in which he is void towards the king. If RHO has the A, the king scores en passant.

(c) To make 3, the AK in one hand would suffice, with trumps breaking 6-5. However, the worst possible holding is K8 split between the hands. Consider the following layout:

Contract 3
 K KJ10xxx Kxxx AK
 AQJ109 Qxx xxx xx
 N W E S
 765432 xx xxx xx
 8 Ax AQx QJ109876

After cashing 7 tricks declarer leads a heart from dummy, ruffing with the 8, while LHO has to follow. A club from hand allows the K to score en passant once more.

If a trump lead is permitted, then A8 in one hand will do.

(d) To make 4, declarer needs at least three trumps between the hands. Many entrants missed the optimum holding on this one, as curiously the best layout for declarer involves an uneven trump break!

Contract 4
 8 AKQJ AKxxxx AK
 765432 xxx QJ xx
 N W E S
 AQJ10 xxx xxx xxx
 K9 xxx xx Qxxxxx

After the seven side suit winners declarer ruffs a diamond in hand and a club in dummy, with LHO under-ruffing twice, and then scores the K as before.

If a trump lead is permitted, K102 in one hand suffices, on a 5-5 break.

(e) To make 5 requires 3 trump tricks. AQ9 in hand is optimal, whether a trump lead is permitted or not. Declarer scores eight tricks then leads towards the AQ9, intending to ruff low and exit with a side suit. RHO, holding KJ10xx must ruff with an honour, but declarer discards. RHO must then lead a trump which declarer wins and exits with a side suit.

(f) A small slam requires 4 trumps between the hands, with a 5-4 break. Best seems to be AQ10 in hand and a stiff 7 in dummy. After 8 winners declarer scores the 7 while LHO, with his familiar holding of 65432, wishes he'd led one. Then a ruff in hand and an endplay as before.

A trump lead is very damaging to this plan. Giving declarer AQ108 in one hand is enough on a trump lead, but not if LHO tries a side suit. I think AKJ7 in hand, with Q1098 on the right is the weakest possible holding.

(g) And finally, what do you need for a Grand? Clearly AKQ87 in one hand is enough (fortunately the J109 drop). But the weakest holding, found by only two entrants (and myself!) is A8 opposite KJ7, provided The Q109 are under the jack. For instance:

Contract 7
 KJ7 AKQJ xx AKxxx