My cat recently signed a contract for a rent increase, but now she refuses to pay!
She purrs it’s the “best contract” she ever had, as I have no legal right to collect because it was signed “Mabel” and not “Mabel Jr.” When I told her I saw her sign it, she purr-purred that away by offering to settle it in court… I can say whatever I want, but she’ll just claim my vision has deteriorated.
No problem, kitto! I reminded her there’s more than one way to skin a cat, and she forked over the rent.
Meanwhile, back to bridge. Choosing the best contract on a bridge deal is straightforward: Determine the number of tricks won by each plausible contract with logical play (not double-dummy) then choose the one producing the highest score.
Looking at only two hands, however, this is more difficult. You don’t know the location of the missing high cards, or the distribution of the unseen hands. Therefore, determining the best contract must be done by probability, or the average expected score if the situation were to occur many times. Further, if duplicate, you may need to consider what might happen at the other table or tables.
This will be your task on the following six problems. You are shown the West and East hands, the form of scoring, and the vulnerability. Assume all players* are experts. Contracts must be at the two level or higher and not doubled or redoubled. West or East must be declarer (you do not need to stipulate which) lest any wiseguys claim the best contract for East-West is 7 NT by North down about nine.
*including those at the other table if IMPs or board-a-match, and all tables if matchpoints.
North and South will pass throughout. For the sake of analysis, the beginning of each auction is shown to clarify the opportunities North and South had to bid, as their silence may affect the chance of a contract’s success.
IMPs, None vul, North deals
Best contract? 3 NT 4 6 6 NT 7 7 NT
Cavendish*, N-S vul, East deals
*same as Chicago (four-deal bridge) except dealer is nonvulnerable on Deals 2 and 3
Best contract? 2 2 2 NT 3 NT 4 5
Matchpoints, E-W vul, South deals
Best contract? 2 2 NT 3 3 NT 4
Quit
Board-a-match, Both vul, West deals
Best contract? 6 6 6 NT 7 7 7 NT
Chicago, None vul, North deals
Best contract? 2 2 2 NT 3 3 NT 5 5
Matchpoints, N-S vul, East deals
*Solvers did not have the benefit of multiple choice, which was added for this writeup.
For each of the six problems, contracts are rated on a 0-10 scale (10 being best). The average score of all entries was 48.92, and everyone who scored above average is ranked below. Ties are broken by date-time of entry (earliest wins).
Correction 1-2-25: An oversight on Problem 4 caused runners-up 7 NT and 7 to be underrated, so both are now awarded 9 (too close to differentiate between them). Also, on Problem 1, further consideration shows that awards for also-rans 4 and 6 should be switched, so now 6 = 6 and 4 = 5. All scores and rankings have been updated accordingly.
The notable aspect of these hands is that, with the inevitable spade lead, the chance of making a grand slam is the same as a small slam. Commonly called a “five or seven” deal, if the heart finesse loses you make 5 (losing a heart and a spade); if it wins you make 7 , since the spade loser goes away on the fourth diamond. Hence, if you’re going to be in slam, you might as well bid seven.
The same condition exists in notrump — 7 NT is better than 6 NT — but there is no vigorish for notrump. When it makes, the 10-point gain over hearts wins nothing at IMPs; but when it fails, you are set multiple tricks as the opponents rattle off the spade suit.
Now that we know seven is better than six, and hearts beats notrump, it only remains to compare 7 with 4 .* If the heart finesse loses, 4 gains 11 IMPs (+450 vs -100); if it wins, 4 loses 14 IMPs (+510 vs +1510). Thus we have a winner: Bid for ‘em all in hearts!
*There is no reason to consider 5 , because 4 scores the same at lesser risk.
In my analysis I ignored the chance of a non-spade lead. This would narrow the advantage of 7 but not change its overall superiority.
Grant Peacock: Despite any slight risk of ruffs, 7 is better because 7 NT can go down five or six.
Nicholas Greer: Five or seven on a spade lead, but hard to judge how likely a spade lead is.
Foster Tom: I expect the other table to be in 6 , so 7 only loses big if the K is off and they don’t lead a spade. 7 NT makes any time 7 does, but the undertricks could hurt.
Stan Zhang: I calculate that 7 beats 6 if a spade is led against 6 more than 25 percent of the time (obviously true)…
Cavendish, N-S vul, East deals
Many solvers no doubt thought “Say what?” to Cavendish scoring. The name comes from the Cavendish Bridge Club (New York City) which, circa 1960, changed Chicago scoring to make the dealing side nonvulnerable on Deals 2 and 3. Many consider this to be superior, because first-seat nonvulnerable tends to produce more competitive auctions.
As in any standard form of non-duplicate bridge, honor bonuses exist, so right away notrump stands out with West’s 150 honors (four aces in one hand). Unfortunately, chances of making 3 NT are poor. Clubs cannot be established (lack of entries) so declarer is obliged to rely on 3-3 diamonds, about 36 percent. Therefore, in 100 deals the expectation of 3 NT is 36×550 + 64×100 = 26,200. Note that honors count even when set, so the usual -50 becomes +100. Expectation of 2 NT is better: 36×300 + 64×270 = 28,080.*
*The immediate score for 2 NT is only 70, but the partscore carried forward has value, which by expert consensus is worth about 50 points.
Aha! Clubs may be useless in notrump, but if trumps, only two tricks will be lost with a normal 3-2 split. This makes 5 about 68 percent, so in 100 deals the expectation is 68×400 - 28×50 - 4×100 = 25,400. Alas, those 150 honors are beginning to haunt, as this still falls short of notrump. In practice 5 would fare slightly worse, because it might be doubled when trumps are 4-1 or 5-0.
But wait! There’s another game that’s a big favorite but easy to overlook: Four spades needs only the top tricks to score and one heart ruff with the 6 (a second heart ruff is a lock with the J). Being a complex analysis, I used my Hand Pattern Analyzer to determine about 72 percent. Expectation is therefore 72×420 - 25×50 - 3×100 = 28,690. I guessed at the rare occasions of being set two, but it clearly beats the expectation of 2 NT. Of course, reaching 4 is another story, requiring profound inspiration, or more likely a bidding accident.
Foster Tom: Four spades looks like the best contract at any form of scoring. I’m curious why Cavendish was specified.
Reason was to bring honors into the picture, and Chicago was impossible with the dealer-vul condition. -RP
Andrew Spooner: Four spades will always succeed barring some very bad breaks…
Richard Stein: On a trump lead, there are 10 tricks with normal breaks.
Brad Johnston: I would be unlucky not to be able to manage two heart ruffs.
It is evident that 4 has no play*, and 3 NT is laydown unless the opponents run five spades. At IMPs or any quantitative form of scoring, there is no question that 3 NT is the best contract. With both opponents silent at favorable vulnerability, the chance of a long spade suit is greatly reduced, and even if there is, leader may have the shorter spades and not lead one.
*a slight exaggeration, but constructing a layout to make 4 against best defense is a puzzle in itself
But the scoring is matchpoints, and 3 NT seems unreachable by any logical means. For example, two-over-one bidders will start 1 1 NT (forcing); 2 , then for East to suggest notrump is far-fetched holding worthless spades and diamonds. Raising to 3 is routine, and West has a clear pass. If game is reached, I would expect few if any pairs to be in 3 NT, while overbidding to 4 would be common.
Therefore, I believe the best contract is two notrump. If nine tricks come home, this beats all the heart bidders at any level. The advantage is that if the defense runs five spades, this still beats the 4 bidders, whereas 3 NT only ties them.
Foster Tom: Notrump beats hearts unless the opponents run five spades (less likely with silent opponents); 3 NT makes in such cases, but I’m happy with a confident plus in 2 NT.
With 12 top tricks in spades or notrump, and a club finesse for 13 in either, it is clear that notrump is better than spades at the same level. The problem is whether 6 NT or 7 NT offers the better chance to win the board, and the choice may depend on what you think might occur at the other table.
Hold the phone! There’s another strain to consider, albeit bizarre: With hearts trump, 13 tricks do not require the club finesse but only a 4-3 trump break (about 62 percent) since a diamond can be ruffed in the East hand before drawing trumps. This certainly complicates matters, leaving no obvious solution.
For an overall picture, let’s compare each contract with the four plausible contracts at the other table (reaching 6 or 7 is too unrealistic to consider happening). The following table shows the expected wins in 100 deals for each comparison (50 = break even):
Calculating the heart wins is slightly complicated. For instance, versus 7 with the K onside, 7 ties 62 percent and loses the rest; with the K offside, 7 wins 62 percent, ties about 34 percent, and loses about 4 percent (when 7 goes down two). Hence 7 achieves 31 wins and 48 ties = 55. Actually, all the heart wins should be reduced a notch or two, because a spade void is deadly (in either hand with a trump lead).
The table shows that 6 NT achieves the most wins, particularly against a small slam, which seems more likely to be bid at the other table. (Most expert pairs will discover the K is missing, often with RKC then 5 NT for specific kings, and reluctantly settle for six.)
I decided to make 7 NT and 7 close seconds with equal awards. Even though 7 NT has the edge overall, and cannot be topped when it makes, 7 fares better against a small slam.
This was the toughest problem of the set. While easy to diagnose the chances of 3 NT (4-4 hearts or no heart lead) and 5 (3-2 trumps), the stickler is whether 150 honors in diamonds will offset its inferior chances to produce game.
The logical play in diamonds after a heart lead is to ruff, cash three spades to pitch a club, then ace and another club; ruff the heart return, cash one diamond and lead a club. This brings 11 tricks when North and South both are 3-2 in the minors.
To obtain realistic percentages given the opponents’ silence, I used my Hand Pattern Analyzer and arbitrarily excluded all hands with 6+ hearts (no weak two-bid) or 10+ cards in the majors. Further, I halved the weights of six spades in any hand, or five hearts in North (no overcall after the obvious 1 opening); and slightly reduced weights of other patterns that might inspire action. This showed, rounded to whole percents: 4-4 hearts = 52%; 3-2 clubs = 75%; North-South both 3-2 in the minors = 35%.
Regarding 3 NT and 5 , I will estimate the chance of a non-heart lead as one-third of the failing cases, which is low if anything, because East will respond 1 to West’s 1 opening. This boosts the success rate of 3 NT to 68 percent, and 5 to 57 percent, although the latter will still fail against some distributions (e.g., four clubs in North) so call it 53 percent. From these numbers I can construct a table based on 100 plays of each contract:
Note that partscores include the widely accepted future value of 50 points. Actually, a 70 partscore has greater potential than 40 (a bid of only 1 , 1 , 1 NT, 2 or 2 on the next deal is game) so I ranked 2 NT ahead of 2 despite losing by a whisker in the table.
Analysis of club contracts against 4-1 trumps and a heart lead shows declarer will win 10, 9 or 8 tricks when the four-trump hand has 3, 4 or 5 hearts, respectively. Without a heart lead, 10 tricks are routine. My apportionment for 8-10 tricks is only approximate (not thoroughly calculated) but adjustments would not have a significant impact.
There you have it, folks, staring you in the face! That gorgeous diamond suit wins the game contest, and its lowly partscores finish second. Between 3 and 2 , the former is better (carrying forward 60 instead of 40) but too slight to warrant a higher award.
The last problem is obviously a two-horse race — game in spades or notrump — but analysis is complicated by three finessing chances. Consider the eight possible arrangements of the three critical cards and the likely result of each game:
Spades does better in Cases 1-3 (37 percent); notrump does better in Cases 4-6 (39 percent). The two remaining cases depend on both the heart division and the opening lead:
Case 7: Both games are usually down one, but 3 NT would be down two if North has five hearts or leads a heart from three. If this occurs half the time, 4 would pick up 6.5 percent (the remaining 6.5 would be a tie).
Case 8: 3 NT would usually make while 4 fails, but 3 NT would also fail if North has three hearts and chooses to lead one. This is clearly odds-against, and I’ll estimate one-fourth of the time. Therefore, 3 NT picks up 8.25 percent (the remaining 2.75 would be a tie).
Summing it all up with my ball-park estimates makes 3 NT the winner, 47.25 to 43.5 percent. The notrump edge would actually be greater, as I ignored a bad trump break in spades and the possibility of a ruff.
Nicholas Greer: The defense can cash hearts against 3 NT about half the time, but I think the possibility of a favorable lead is enough to make 3 NT better.
Foster Tom: If the A is onside, 3 NT looks better; and if offside, 4 lets me ruff a heart. However, ruffing a heart might not help if North has K-x-x-x.
Andrew Spooner: With the A onside, 3 NT is better; if offside, 4 is usually better but not always, so 3 NT seems better overall.
Richard Stein: On the surface 4 is our play; but how often will it make while notrump doesn’t do just as well? If the A is onside, I want to be in 3 NT.
Mabel Jr. opted to take me to court after all, and it was not my lucky day. The case was assigned to Judge Felix Grimalkin, who immediately ruled against me. Not only that, but he issued a restraining order despite my admission that I didn’t even know one way to skin a cat. Oh, well. Time to close shop with a few final words of wisdom, or not.
Brad Johnston: Hopefully my answers are bad enough to distract you from purr-suing any further contract negotiations.
Dan Baker: I nearly forgot the honor bonuses in Chicago/Cavendish!
Richard Stein: It’s been a long time since the last puzzle contest, so good to see you back.
Charles Blair: I have a feeling I’m missing a lot of subtleties.
No worries… Joan is sure of it.
© 2024 Richard Pavlicek
by Richard Pavlicek by Richard Pavlicek