Analysis 7A56 Main

The Truth Will Out

by Richard Pavlicek

The following declarer-play problem appeared recently in a bridge magazine. After the auction shown, South is declarer in 6 with the K lead (East follows) won by the ace. The contract is laydown with an even heart division, so declarer starts with the A (both follow) then a heart to the king.

 IMPs A K 7 5 4 West North East South Both vul K 7 3 2 1 Pass 2 J 5 Pass 4 Pass 4 NT K 6 Pass 5 Pass 6 Pass Pass Pass Lead: K Q 10 A 8 6 5 4 A 10 8 4 6 South A 10

No luck, as an opponent shows out on the second heart. The magazine analysis goes on to state:

“If East is the one with the remaining heart winner, you have to hope that East has J-x-x-x as well. If he does, after finessing the 10, you can cash the Q, cross to the K and play the A-K and the last spade to discard three diamonds. East can ruff the fifth spade, but it’s too late to cash a diamond.”

Surely the spade finesse is the best chance to make 6 , and the correct play at total points, but the form of scoring was IMPs. Therefore I wondered if the recommended play was truly correct. The obvious caveat is that your chances of success are poor, and most of the time you will finish down two instead of one. Further, if East has a singleton diamond, you can make 6 against any 4-2 or 3-3 spade division.

When East turns up with three trumps, there are two reasonable lines of play:

A. Finesse the 10. If it wins cash the Q and A* then cross to the K and lead good spades.
*overlooked in magazine but clearly correct in case East has no more diamonds

B. Win the Q, K (unless J fell) and A, then cross to the K and lead spades from the top.

Note that in Line B, if East has a singleton diamond you will always succeed (barring 5-1 spades) because he must yield a ruff-sluff if he ruffs any spade, or if not then, when you exit with a trump.

Problems like this bug me, as I can’t have peace of mind until I find an answer. Alas, this one seemed out of reach to solve intuitively, and certainly a three-piper for Sherlock Holmes. Fortunately, many years ago I created a Hand Pattern Analyzer, so I fed it the data to obtain a listing of all possible distributions (excluding a spade void, 8+ clubs or a singleton K lead) and the percent chance of each. Results are tabulated below:

#WtWestEastTricksFactorsPercentA PctB PctA AvgB AvgA IMPB IMPA MPB MPA TPB TP
110031543325ab3 bb320 4 10 12615.360010.5011.0001.500.501.50-150-100
210031453334ab3 bb320 4 10 12615.360010.5011.0001.500.501.50-150-100
310041442335ab4 bb215 4 10 12611.520010.3311.0002.000.331.67-167-100
48021554324ab2 cb415 4 10 1269.2266.67011.3311.0011.331.001.330.67887-100
510041532326ab4 bb215 4 10 847.680010.3311.0002.000.331.67-167-100
610021464333ab2 cb415 4 10 847.6866.67011.3311.0011.331.001.330.67887-100
710041352344ab4 bb215 4 5 1265.760010.3311.0002.000.331.67-167-100
810031363343ab3 bb320 4 5 845.120010.5011.0001.500.501.50-150-100
99031633316ac3 cc320 4 5 844.6150.0010011.0012.0008.500.501.506151430
108021644315ac2 cc415 4 5 1264.6166.6710011.3312.0005.670.671.338871430
1110051431336ab5 bb16 4 10 843.070010.1711.0002.500.171.83-183-100
1210051341345ab5 bb16 4 5 1262.300010.1711.0002.500.171.83-183-100
1310051521327ab5 bb16 4 10 361.320010.1711.0002.500.171.83-183-100
148041622317ac4 cc215 4 5 361.3233.3310010.6712.00011.330.331.673431430
154011565323ab1 bb56 4 10 841.230010.8311.0000.500.831.17-117-100
164011655314ac1 cb56 4 5 1260.9283.3316.6711.6711.1714.172.831.670.331158155
175021374342ab2 cb415 4 5 360.8266.67011.3311.0011.331.001.330.67887-100
1810041262353ab4 bb215 4 1 840.770010.3311.0002.000.331.67-167-100
195011475332ab1 bb56 4 10 360.660010.8311.0000.500.831.17-117-100
2010051251354ab5 bb16 4 1 1260.460010.1711.0002.500.171.83-183-100
215031273352ab3 bb320 4 1 360.220010.5011.0001.500.501.50-150-100
Totals and Averages656,256100.0018.4010.6910.6611.112.142.250.601.4012963

Factors: Number of combinations for each suit distribution between West and East. For example, Case 1 allows 20 ways (6c3) for a 3-3 spade split, 4 ways (4c1) for a 1-3 heart split, and 126 ways (9c4) for a 4-5 club split. Diamonds require special treatment, since West is presumed to hold K-Q for his lead, so there are only 10 ways (5c3) to compose K-Q-x-x-x opposite x-x. Multiplying the numbers shows how many layouts are represented. Case 1 comprises 20×4×10×126 = 100,800 layouts, and the column total shows that all 21 cases comprise 656,256 layouts.

Tricks: Number of tricks won by each line according to the J location. For each group of three characters, the first shows the tricks won by Line A, and the second by Line B, both in hexadecimal. The third number shows the applicable portion (number of sixths). For example, cb4 means 12 tricks for Line A, 11 tricks for Line B, 4/6 of the time. Note that the portions are identical to the spade divisions.

Weight: An attempt to obtain realistic results by reducing the influence of patterns with which an opponent may have bid. For example, with 6-5 in the minors (Case 15,16) I estimated West would bid 60 percent of the time (probably unusual 2 NT) so the fact that he passed reduces the weight of these patterns to 40 percent. Similarly, with seven clubs (Case 17,19,21) I judged West would bid half the time; and with six diamonds (Case 9,10,14) or 5-5 minors (Case 4) 20 percent of the time (but only 10 percent if 3=1=6=3). While my numbers are arbitrary, failure to apply weights would surely be off track.

Percent: The frequency of each West-East pattern after weight adjustment relative to all, which dictates a column total of 100 percent. Note that patterns are listed in order of frequency.

The last 10 columns compare Line A and Line B in five categories: Make percent (how often 6 makes), average number of tricks won, average IMPs won, average matchpoints won (counting 2 for a win, 1 for a tie) and average total points won.

The crux of the problem is the IMP comparison. Each row shows the average number of IMPs won per layout. Case 1 (100,800 layouts) is obvious, because Line A never wins, and Line B wins 3 IMPs on half the layouts, or 1.50 IMPs per layout. Case 4 favors Line A, winning 17 IMPs 2/3 of the time (11.33 avg) while Line B wins 3 IMPs 1/3 of the time (1.00 avg). Note that averages are for IMPs won, so the net gain or loss would be found by the difference.

My suspicion proved to be right. At IMP scoring, Line B is the better play, winning an average of 0.11 IMPs per deal, albeit not a windfall. At matchpoints, Line B is a huge winner. Only at total points is Line A the superior play, and greatly so at that.

Vulnerability matters

Curiously, if North-South were nonvulnerable, the bottom line becomes:

FactorsPercentA PctB PctA AvgB AvgA IMPB IMPA MPB MPA TPB TP
656,256100.0018.4010.6910.6611.111.761.630.601.4011360

Only the IMP and TP columns are affected by vulnerability. Notice the turnaround at IMPs: Line A is now superior, winning an average of 0.14 IMPs per deal. Evidently, the sacrifice of only 2 IMPs when the spade finesse loses (versus 3 IMPs vulnerable) justifies playing for the best chance to make.

Another consideration

The preceding analysis assumes 6 is reached at the other table, which is an accurate assumption in expert competition. After the 4 raise, South’s cards are golden, so it’s hard to imagine a reputable player not driving to slam. Nonetheless, if you thought the slam would not (or might not) be bid, you should take the best chance to make (Line A) because down one or two will not matter. If only 11 tricks can be made, you will lose 13 IMPs regardless.

 Analysis 7A56 Main Top The Truth Will Out