Study AC01 Main

Point Count Methods

by Richard Pavlicek

This study will compare various hand evaluation methods to determine their effectiveness in suit bidding. More specifically, it will discover the point-count thresholds for which bidding suit games and slams becomes a winning proposition.

How many points do you need for game? That’s a loaded question for sure, as it depends not only the evaluation method used but on other factors, including form of scoring and vulnerability. To narrow the scope, I will assume IMP scoring, because total-point scoring is obsolete, and many (including me) would question whether matchpoints is truly bridge.

Algorithms

Four of a Major

Five of a Minor

Six of a Suit

Seven of a Suit

Algorithms

There may be more ways to count points than to skin a cat — a frightening metaphor, as Mabel Jr. darts under the bed. For starters I will consider the six methods listed here. In the future I may add others, and existing ones may be adjusted for possible oversights in my interpretation.

For each summary table, the top lines apply to both partnership hands at all times, light gold-tinted to each suit, and darker gold to the hand as a whole. After a trump fit is found, blue-tinted lines apply to dummy, and green-tinted lines to declarer. All points are cumulative, hence “+1” means in addition to points awarded by other applicable rules.

For evaluation purposes, “declarer” is the hand with longer trumps regardless of orientation. For example, after a 1 NT opening and a transfer to spades, responder evaluates as declarer despite becoming dummy; reciprocally, opener evaluates as dummy. With equal length, the stronger hand is deemed “declarer.”

1. Goren (short suit) Count

Surely the most well-known evaluation method, widely associated with Standard American bidding, is the short-suit count. While invented by William Anderson of Toronto, the method is attributed to Charles Goren who promoted and popularized it.

Ace = 4 King = 3 Queen = 2 Jack = 1

Void = 3 Singleton = 2 Doubleton = 1

Singleton king or queen -2 Singleton jack -1

Doubleton Q-J, Q-x or J-x -1

All four aces +1 Aceless -1

Dummy with four trumps: Void +2 Singleton +1

Dummy with trump K, Q or J (less than 4 trump HCP) +1

Dummy with 4-3-3-3 shape -1

Declarer with five trumps +1 Each trump over five +2

2. Karpin (long suit) Count

Opposite to the short-suit count is the long-suit count. Origination is unknown (at least to me) but Fred Karpin of Silver Spring, Maryland, was one of its earliest proponents. Many variations have been introduced over the years, particularly in regard to revaluation with a trump fit, but for now I’ll use Karpin’s original version.

Ace = 4 King = 3 Queen = 2 Jack = 1

Each card over four in any suit +1

Singleton K or Q; doubleton K-Q, K-J, Q-J, Q-x or J-x -1

Dummy with three trumps: Void +2 Singleton +1

Dummy with four trumps: Void +3 Singleton +2 Doubleton +1

Declarer with five trumps: Void +2 Singleton +1

Declarer with six trumps: Void +3 Singleton +2 Doubleton +1

3. Pavlicek Point Count

When I began to teach bridge in the early ‘70s, I researched various point-count schemes and found the short-suit method to be superior, most notably because it awarded 5-4-3-1 shape 1 point more than 5-3-3-2, while the long-suit method counted them the same. Therefore, I adopted the Goren method as a model, removing a few things I thought were dubious, and adding some tweaks to improve accuracy. An important consideration was to keep it simple enough for the average player to use.

Ace = 4 King = 3 Queen = 2 Jack = 1

Void = 3 Singleton = 2 Doubleton = 1 With singleton K, Q, J or doubleton KQ, KJ, QJ, Qx or Jx count the greater of the suit’s HCP or shortness but not both. Do not count shortness in partner’s suit unless another fit exists.

Any four aces and/or 10s +1

Dummy with four trumps: Void +2 Singleton +1

Dummy 4-3-3-3, aceless and minimal fit -1

Declarer with five trumps (only if not shown) +1

Declarer with six or more trumps: Each card over five +2

Declarer with a side suit: Each card over three +1

4. Modified Bissell Count

The original Bissell Count was invented by Harold Bissell of New York City, circa 1936, but presented here is a modified version that has attracted some following in recent times.

I am unaware of any trump-fit adjustment for this method, so it will be implemented as defined. Indeed, the exaggerated points for suit length might already account for this, though it could spell trouble on misfit hands.

Ace = 3 King = 2 Queen = 1

Each suit: 4+ cards +1 5+ cards +2 Each card over five +3

Ace with 3+ cards +1 Ace or king with 4+ cards +1

At least two of A-K-Q-J in same suit +1

At least two of Q-J-10-9 in same suit with 3+ cards +1

Singleton K, Q or J; doubleton QJ, Qx or Jx -1

Maximum points allowed in any suit is length times 3.

5. Zar Point Count

Are we still on planet Earth? This next scheme by Zar Petkov of Aurora, Ontario, might leave you wondering. His basic formula for hand evaluation is: HCP + controls + 2*(longest suit length) + (second longest suit length) - (shortest suit length). Say what? This produces an awful lot of points, but the curious sum is exactly twice what we are accustomed to; i.e., an opening bid requires 26 points, and the goal for game is 52 points. Therefore, to put Zar points on an equal scale, the final partnership total will be halved in this study. If this leaves a half point, I will round up if the partnership body-card sum (3*tens + 2*nines + eights) is at least 12; else down.

Ace = 6 King = 4 Queen = 2 Jack = 1

Singleton A, K, Q or J; doubleton KQ KJ QJ Qx Jx -1

Length points: 2*(longest) + (second longest) - (shortest)

11-14 HCP with all HCP in two suits +1

15+ HCP with all HCP in three suits +1

Dummy with 8+ fit: Each trump honor (maximum 2) +1

With 9+ fit and shortest suit singleton: Each trump over eight +1

With 9+ fit and shortest suit void: Each trump over eight +2

Secondary 9-card fit +1 Secondary 10-card fit +2

Red-tinted rules apply to either player with knowledge of their existence.

6. Kaplan-Rubens Count

In the early 1980s Edgar Kaplan devised a point-count scheme with many tweaks, attempting to translate expert judgment into a formula. Jeff Rubens provided the mathematical expertise and oversight to fine-tune the project. While too complicated for practical use, it is often implemented by computer, and even to this day is highly regarded as the definitive gauge in hand evaluation.

Rather than rewrite the elaborate algorithm, refer to Kaplan-Rubens Evaluation Stats, an earlier study explaining it in detail. Or better yet, just take Kaplan’s word for it! To be sure, anyone who tries to use K-R count at the table is on a direct path to the Looney Bin.

The K-R count is fractional in 1/20 point (0.05 decimal) increments, but in fairness to other methods, the final partnership total will be a whole number. Fractions of 12/20 and higher are rounded up; 8/20 and lower are rounded down (dropped). Arbitrarily I chose to round mid values (9/20, 10/20, 11/20) up if the partnership body-card sum (3*tens + 2*nines + eights) is at least 12; else down.

Examples to verify the algorithms: Four Spades Five Diamonds Six Hearts Seven Clubs

Study AC01 Main

Top Point Count Methods

Four of a Major

In order to perform any analyses I needed a data source, for which my database of 10,485,760 (10 MB) random deals was made to order. Each deal is double-dummy solved 20 times for makable tricks by each hand in each strain — a project that itself took almost two years of computer time. (This resource is freely available to anyone interested: Random Solved Deals)

For this first analysis, I selected deals with a partnership fit of 8+ cards*, and to focus on common situations, I rejected any partnership with a 7+ card suit. For uniformity I made the trump suit spades, swapping suits and double-dummy results as needed, which is effectively transparent. This provided a sample of 9,252,096 deals.

*If a partnership had two fits, the longer (more evenly divided if same) was chosen.
If both partnerships of a deal qualified, the side with more HCP was chosen.

Sometimes the number of makable tricks depends on which hand is declarer, as a damaging opening lead may be prevented if played from the right side. It would not be feasible to construct auctions on 9+ million deals, nor fair to right-side every contract, so the stronger* hand (more HCP) is made declarer. This emulates the great majority of actual occurrences, especially those involving transfer bids. (Note that for evaluation purposes, “declarer” is determined differently and may not be the same.)

*With equal HCP, longer trumps (stronger if same) determine declarer.

Declarer’s Advantage

Another factor in these analyses is the relationship between double-dummy and actual play. There is no doubt that declarer in actual play has the advantage (below the slam level) because the defense at double-dummy gets the first shot — and it will always be right — while in actual play, the opening lead is often inferior, and sometimes an outright gift.

In order to quantify this, I turned to Actual Play vs. Double-Dummy, a previous study based on 72 major events from 1996-2014. Results there showed that 4 was made 8044 times out of 12,086 occurrences, but at double-dummy would have been made only 7865 times — an increase of 2.28 percent — and this was against world-class opening leaders. Versus less stellar defense it would be higher, perhaps as much as 10 percent against typical club players.

Arbitrarily I decided that 4 percent would be a good general adjustment, and even if inaccurate would be the same for all methods and comparatively have little if any effect. To implement this I increased the number of successful occurrences by 4 percent, as shown by the gold-tinted cells in the following table. All calculations are then based on the Adjusted column.

Result	Occurred	Percent	Adjusted	Percent
0	250	0.00	250	0.00
1	1271	0.01	1271	0.01
2	5602	0.06	5602	0.06
3	20,926	0.23	20,926	0.22
4	63,358	0.68	63,358	0.67
5	144,909	1.57	144,909	1.53
6	287,110	3.10	287,110	3.04
7	605,004	6.54	605,004	6.41
8	1,291,626	13.96	1,291,626	13.68
9	2,034,958	21.99	2,034,958	21.55
10	2,179,486	23.56	2,266,665	24.00
11	1,604,655	17.34	1,668,841	17.67
12	795,297	8.60	827,109	8.76
13	217,644	2.35	226,350	2.40
Totals	9,252,096	100	9,443,979	100

Point Count Results

The following six tables show the results of 4 contracts for each evaluation method, according to the total points (23-29) obtained from its algorithm. The last four columns are the most significant, showing the average IMP gain (+) or loss (-) versus contracts of 2 and 3 , nonvul (green) and vul (red). A minus means you’re better off in the partscore, and a plus means you’re better off in game.

An important result of each table is the point total that makes 4 a winning bid versus stopping in a partscore. For the Goren count and my own, the generally accepted 26-point threshold is true; for Karpin and Bissell, it’s 25 points; for Zar, 27 points; and for Kaplan-Rubens only 24 points. The last is surprising to me, as I assumed K-R count was scaled to the standard threshold.

A notable aspect of all tables is that 4 usually fails when bid at the threshold; i.e., the make percent — even with my boost for declarer’s advantage — is below 50 percent. Nonetheless, this is sound bridge; IMP odds clearly favor bidding games as low as 38 percent vulnerable (46 percent nonvulnerable) and experts tend to push the envelope with less. Which brings to mind the Wolff game try: “First I bid game. Then I try to make it.” -Bobby Wolff

Goren (short suit) Count

Points	Cases	4 make%	Trick Avg	2	3	2	3
23	564,403	13.10	8.4193	-3.22	-1.99	-4.09	-2.33
24	711,598	22.30	8.7509	-2.34	-1.39	-2.62	-1.27
25	822,924	33.65	9.0646	-1.18	-0.48	-0.77	+0.20
26	890,206	46.11	9.3640	+0.14	+0.63	+1.27	+1.93
27	900,049	58.01	9.6452	+1.43	+1.76	+3.20	+3.66
28	854,420	68.69	9.9252	+2.58	+2.81	+4.94	+5.24
29	765,346	77.94	10.2132	+3.59	+3.73	+6.43	+6.63

Karpin (long suit) Count

Points	Cases	4 make%	Trick Avg	2	3	2	3
23	729,793	16.63	8.6065	-2.94	-1.85	-3.55	-2.02
24	857,907	28.60	8.9557	-1.73	-0.95	-1.60	-0.52
25	929,481	42.40	9.2821	-0.27	+0.25	+0.66	+1.37
26	927,031	56.68	9.6038	+1.27	+1.60	+2.99	+3.43
27	868,786	69.45	9.9181	+2.66	+2.85	+5.06	+5.32
28	775,916	79.57	10.2215	+3.76	+3.88	+6.70	+6.86
29	656,723	86.88	10.5162	+4.56	+4.63	+7.88	+7.97

Pavlicek Point Count

Points	Cases	4 make%	Trick Avg	2	3	2	3
23	589,345	9.60	8.3271	-3.59	-2.23	-4.68	-2.75
24	750,991	18.25	8.6828	-2.80	-1.76	-3.30	-1.85
25	872,782	30.50	9.0260	-1.54	-0.83	-1.29	-0.31
26	929,431	45.21	9.3626	+0.02	+0.47	+1.12	+1.73
27	919,236	60.41	9.6942	+1.67	+1.93	+3.60	+3.95
28	853,735	73.40	10.0119	+3.09	+3.23	+5.70	+5.90
29	752,434	83.42	10.3224	+4.18	+4.26	+7.33	+7.43

Modified Bissell Count

Points	Cases	4 make%	Trick Avg	2	3	2	3
23	629,958	23.08	8.7330	-2.23	-1.29	-2.47	-1.14
24	720,251	33.14	9.0175	-1.21	-0.48	-0.84	+0.17
25	778,074	44.00	9.2922	-0.07	+0.47	+0.93	+1.67
26	799,315	54.69	9.5537	+1.08	+1.46	+2.66	+3.20
27	779,003	64.41	9.8103	+2.13	+2.40	+4.24	+4.62
28	730,171	72.69	10.0592	+3.02	+3.22	+5.58	+5.85
29	652,279	79.38	10.3018	+3.75	+3.89	+6.67	+6.86

Zar Point Count

Points	Cases	4 make%	Trick Avg	2	3	2	3
23	349,737	1.22	7.4447	-3.84	-2.25	-5.65	-3.19
24	604,969	4.68	8.0090	-3.93	-2.39	-5.39	-3.16
25	918,124	13.54	8.5146	-3.25	-2.07	-4.05	-2.39
26	1,178,638	29.16	8.9853	-1.68	-0.94	-1.51	-0.48
27	1,285,150	49.52	9.4448	+0.49	+0.88	+1.82	+2.36
28	1,226,959	69.11	9.8925	+2.62	+2.80	+5.01	+5.26
29	1,043,091	83.56	10.3275	+4.20	+4.28	+7.35	+7.45

Kaplan-Rubens Count

Points	Cases	4 make%	Trick Avg	2	3	2	3
23	871,032	33.00	9.0792	-1.27	-0.61	-0.88	+0.04
24	914,662	47.37	9.4034	+0.26	+0.69	+1.47	+2.06
25	898,026	61.60	9.7197	+1.80	+2.06	+3.79	+4.14
26	828,557	73.75	10.0256	+3.13	+3.28	+5.76	+5.97
27	728,471	82.83	10.3190	+4.12	+4.21	+7.23	+7.35
28	611,695	89.06	10.6027	+4.80	+4.85	+8.24	+8.31
29	494,607	93.18	10.8801	+5.25	+5.28	+8.90	+8.94

Study AC01 Main

Top Point Count Methods

Five of a Minor

The next scenario will be to analyze evaluation methods for contracts of five of a minor, for which I made diamonds trump for uniformity. It is rare to bid a minor-suit game with only eight trumps, so this venture will require a 9+ card fit and allow suit lengths up to seven cards (but not 8+). These conditions reduced the sample size to 5,413,732 deals.

As previously for four of a major, I made the same 4-percent boost of successful cases, which now becomes 11+ tricks, as shown by the gold-tinted cells in the table below. All calculations are then based on the Adjusted column.

Result	Occurred	Percent	Adjusted	Percent
0	0	0	0	0
1	18	0.00	18	0.00
2	242	0.00	242	0.00
3	2883	0.05	2883	0.05
4	18,792	0.35	18,792	0.34
5	73,181	1.35	73,181	1.33
6	197,634	3.65	197,634	3.60
7	400,768	7.40	400,768	7.31
8	693,786	12.82	693,786	12.65
9	1,041,166	19.23	1,041,166	18.98
10	1,224,595	22.62	1,224,595	22.33
11	1,027,330	18.98	1,068,423	19.48
12	566,966	10.47	589,645	10.75
13	166,371	3.07	173,026	3.16
Totals	5,413,732	100	5,484,159	100

Point Count Results

The following six tables show the results of 5 contracts for each evaluation method, according to the total points (26-32) obtained from its algorithm. The last four columns are the most significant, showing the average IMP gain (+) or loss (-) versus contracts of 3 and 4 , nonvul (green) and vul (red). A minus means you’re better off in the partscore, and a plus means you’re better off in game.

It is generally accepted that minor-suit games require 29+ total points, and three algorithms (Goren, Zar and my own) are right on target. Karpin and Bissell are close, with 28 points being a favorite except nonvul vs. 3 . Kaplan-Rubens is further below the norm with 27 points, which drops to only 26 if vulnerable.

Goren (short suit) Count

Points	Cases	5 make%	Trick Avg	3	4	3	4
26	414,090	15.16	9.5244	-3.05	-1.90	-3.77	-2.14
27	434,065	23.61	9.8020	-2.22	-1.31	-2.40	-1.13
28	433,223	33.82	10.0719	-1.16	-0.48	-0.74	+0.21
29	413,129	44.95	10.3325	+0.02	+0.51	+1.08	+1.75
30	375,035	55.68	10.5779	+1.17	+1.51	+2.82	+3.30
31	327,300	65.28	10.8062	+2.21	+2.45	+4.38	+4.72
32	272,042	74.07	11.0428	+3.17	+3.33	+5.81	+6.03

Karpin (long suit) Count

Points	Cases	5 make%	Trick Avg	3	4	3	4
26	443,543	18.37	9.6810	-2.78	-1.75	-3.27	-1.83
27	451,248	28.75	9.9746	-1.72	-0.97	-1.58	-0.53
28	434,493	40.92	10.2600	-0.43	+0.08	+0.42	+1.12
29	393,704	53.96	10.5406	+0.97	+1.30	+2.55	+2.99
30	341,315	66.02	10.8056	+2.28	+2.49	+4.51	+4.79
31	283,564	76.20	11.0608	+3.39	+3.52	+6.16	+6.33
32	226,489	83.70	11.2995	+4.21	+4.29	+7.37	+7.47

Pavlicek Point Count

Points	Cases	5 make%	Trick Avg	3	4	3	4
26	434,150	11.98	9.4900	-3.43	-2.19	-4.33	-2.59
27	458,188	21.12	9.8044	-2.54	-1.62	-2.84	-1.56
28	456,305	32.91	10.1046	-1.30	-0.68	-0.89	-0.03
29	429,594	46.54	10.3995	+0.16	+0.55	+1.34	+1.87
30	380,989	60.34	10.6823	+1.66	+1.89	+3.59	+3.90
31	322,399	72.61	10.9536	+3.00	+3.13	+5.58	+5.76
32	261,400	82.27	11.2101	+4.05	+4.12	+7.14	+7.24

Modified Bissell Count

Points	Cases	5 make%	Trick Avg	3	4	3	4
26	392,804	20.75	9.6847	-2.48	-1.49	-2.85	-1.46
27	396,965	29.82	9.9457	-1.57	-0.79	-1.39	-0.30
28	387,373	39.52	10.1887	-0.55	+0.04	+0.20	+1.02
29	362,515	49.46	10.4206	+0.51	+0.95	+1.82	+2.42
30	328,016	58.65	10.6419	+1.50	+1.82	+3.31	+3.76
31	285,579	67.11	10.8574	+2.42	+2.65	+4.68	+5.01
32	240,693	73.85	11.0514	+3.15	+3.33	+5.77	+6.02

Zar Point Count

Points	Cases	5 make%	Trick Avg	3	4	3	4
26	533,652	4.98	9.1501	-4.03	-2.49	-5.43	-3.24
27	612,429	13.81	9.5960	-3.28	-2.16	-4.03	-2.47
28	632,503	29.23	10.0255	-1.70	-1.02	-1.49	-0.56
29	585,249	48.96	10.4349	+0.42	+0.78	+1.74	+2.22
30	491,549	68.14	10.8286	+2.51	+2.68	+4.86	+5.09
31	374,539	82.07	11.1975	+4.03	+4.11	+7.11	+7.22
32	262,967	90.61	11.5380	+4.97	+5.01	+8.49	+8.54

Kaplan-Rubens Count

Points	Cases	5 make%	Trick Avg	3	4	3	4
26	433,883	40.47	10.2727	-0.50	-0.02	+0.35	+1.00
27	395,102	54.14	10.5487	+0.98	+1.28	+2.58	+2.98
28	342,361	67.11	10.8175	+2.40	+2.57	+4.69	+4.93
29	285,966	77.64	11.0721	+3.55	+3.66	+6.39	+6.54
30	227,834	85.40	11.3131	+4.40	+4.46	+7.65	+7.73
31	176,138	90.38	11.5384	+4.95	+4.99	+8.45	+8.50
32	130,562	93.80	11.7561	+5.32	+5.35	+9.00	+9.04

Study AC01 Main

Top Point Count Methods

Six of a Suit

Next we turn to slam bidding, for which the distinction between major and minor suits is minimal — and should be even less, but no-o-o-o, the bridge world has ignored my Broken Scoring Fix. Wise up, people!

An obvious fact of slam bidding is that point count means nothing if the defense can win the first two tricks. Therefore, to obtain a fair comparison of evaluation methods alone, any partnership off two top tricks will be rejected. In other words, selected partnerships must have first-round control (ace or void) in three suits, and at least second-round control (king or singleton) in the remaining suit.

Since the control requirement reduces the number of eligible deals, partnership selection will be more liberal, requiring only an 8+ card fit and allowing suit lengths up to seven cards (note 7-1 fits are included). Still, this produced a healthy sample of 4,186,420 deals. For variety in my uniformity, this time the trump suit will be hearts.

Say good-bye to declarer’s advantage, which disappears at the slam level. This is because in most cases the opening lead will not matter, which shifts the edge to the double-dummy declarer. Therefore, no boost will be made to the double-dummy results. Arguably, the results should be lowered slightly, but it’s too close to call, so I’ll leave them untouched per the table below.

Result	Occurred	Percent
0	0	0
1	0	0
2	1*	0.00
3	26	0.00
4	177	0.00
5	1805	0.04
6	14,162	0.34
7	71,776	1.71
8	238,971	5.71
9	562,661	13.44
10	962,202	22.98
11	1,151,755	27.51
12	925,387	22.10
13	257,497	6.15
Totals	4,186,420	100

*Curious how a partnership could win only two tricks with an 8+ card fit and controls required for slam.
But I won’t divulge the layout — call it a hidden puzzle for anyone who has time to waste.

Point Count Results

The following six tables show the results of 6 contracts for each evaluation method, according to the total points (29-35) obtained from its algorithm. The last four columns are the most significant, showing the average IMP gain (+) or loss (-) versus contracts of 4 and 5 , nonvul (green) and vul (red). A minus means you’re better off in game only, and a plus means you’re better off in slam.

In previous analyses of 4 and 5 , it was shown that IMP odds favor bidding games that usually fail (46 percent nonvul, 38 percent vul). Not so for slams! Generally, a six-bid requires at least an even-money chance (50 percent) because the IMP gain if made is the same as the IMP loss if set (versus stopping in game) at any vulnerability, save a slight variance whether game is a four- or five-bid.

The widely endorsed goal for a six-bid is 33+ total points, and three algorithms (Goren, Bissell and my own) confirm this to be the case. Karpin and Zar require only 32 points, and Kaplan-Rubens once again is low man on the totem pole with 31 points.

A notable aspect of all methods is that with hands 1 point below the threshold, it is better to bid six than to stop in five. In other words, if you can’t play game at the four level, or exceed that level with slam tries, it pays to take the final plunge.

Goren (short suit) Count

Points	Cases	6 make%	Trick Avg	4	5	4	5
29	395,900	16.90	10.5756	-6.24	-3.25	-7.57	-4.13
30	390,181	24.48	10.8171	-4.92	-2.52	-5.94	-3.19
31	365,173	33.15	11.0383	-3.23	-1.38	-3.91	-1.80
32	323,597	42.79	11.2488	-1.27	+0.13	-1.56	+0.03
33	273,445	52.14	11.4404	+0.69	+1.73	+0.77	+1.96
34	219,428	60.70	11.6204	+2.49	+3.26	+2.92	+3.79
35	168,800	68.34	11.7832	+4.13	+4.70	+4.87	+5.51

Karpin (long suit) Count

Points	Cases	6 make%	Trick Avg	4	5	4	5
29	402,631	22.74	10.8155	-5.39	-2.96	-6.48	-3.72
30	377,205	32.72	11.0595	-3.41	-1.64	-4.11	-2.09
31	332,382	43.83	11.2897	-1.11	+0.13	-1.36	+0.05
32	279,329	54.62	11.4952	+1.18	+2.06	+1.36	+2.36
33	222,364	64.04	11.6794	+3.20	+3.83	+3.77	+4.48
34	169,436	71.45	11.8418	+4.80	+5.25	+5.66	+6.17
35	122,556	77.90	11.9936	+6.19	+6.53	+7.31	+7.69

Pavlicek Point Count

Points	Cases	6 make%	Trick Avg	4	5	4	5
29	419,676	15.96	10.6162	-6.67	-3.57	-8.03	-4.49
30	410,546	24.46	10.8883	-5.16	-2.83	-6.18	-3.54
31	376,363	35.23	11.1442	-2.99	-1.39	-3.58	-1.77
32	325,347	46.90	11.3768	-0.54	+0.51	-0.66	+0.52
33	265,431	58.51	11.5891	+1.96	+2.63	+2.30	+3.06
34	206,777	69.23	11.7890	+4.28	+4.70	+5.05	+5.52
35	150,147	77.60	11.9663	+6.10	+6.37	+7.21	+7.51

Modified Bissell Count

Points	Cases	6 make%	Trick Avg	4	5	4	5
29	365,754	20.28	10.6506	-5.53	-2.88	-6.72	-3.66
30	355,697	27.70	10.8564	-4.16	-1.97	-5.05	-2.53
31	329,484	36.06	11.0544	-2.50	-0.74	-3.06	-1.04
32	293,540	43.81	11.2207	-0.90	+0.53	-1.15	+0.49
33	250,993	51.15	11.3780	+0.62	+1.80	+0.67	+2.02
34	206,771	57.39	11.5146	+1.94	+2.94	+2.23	+3.38
35	164,901	62.75	11.6402	+3.07	+3.92	+3.58	+4.56

Zar Point Count

Points	Cases	6 make%	Trick Avg	4	5	4	5
29	605,454	14.48	10.5528	-6.85	-3.69	-8.27	-4.65
30	583,783	27.40	10.9464	-4.52	-2.44	-5.42	-3.06
31	490,962	44.15	11.2993	-1.07	+0.15	-1.30	+0.08
32	368,667	60.50	11.6098	+2.41	+3.10	+2.83	+3.60
33	246,605	73.69	11.8821	+5.27	+5.66	+6.21	+6.66
34	149,192	82.65	12.1184	+7.21	+7.44	+8.52	+8.78
35	81,320	88.62	12.3172	+8.51	+8.65	+10.06	+10.21

Kaplan-Rubens Count

Points	Cases	6 make%	Trick Avg	4	5	4	5
29	372,137	34.43	11.0994	-3.07	-1.39	-3.70	-1.79
30	322,041	45.72	11.3254	-0.72	+0.46	-0.89	+0.44
31	265,848	56.79	11.5303	+1.65	+2.48	+1.92	+2.86
32	208,239	66.51	11.7192	+3.74	+4.34	+4.40	+5.08
33	155,282	73.57	11.8832	+5.26	+5.72	+6.20	+6.72
34	111,021	78.75	12.0280	+6.38	+6.73	+7.53	+7.93
35	75,391	82.62	12.1556	+7.22	+7.50	+8.53	+8.84

Study AC01 Main

Top Point Count Methods

Seven of a Suit

For the grand finale, my attorneys warn me that angry club players will file a class-action lawsuit unless I give them equal time. Okay, you got me! The grand slam will be in clubs.

As in the previous section, adequate controls must be held for point-count comparisons to be meaningful, so a partnership now requires first-round control (ace or void) in every suit. Further, in this day and age, a grand slam is rarely bid off a key card, so the K will also be required. These stringent conditions, together with an 8+ card fit and suit lengths no longer than seven, reduced the eligible deals to only 813,349. Even so, that’s surely an adequate sample for reliable stats.

Declarer’s advantage not only disappears but is a clear disadvantage at the seven level. The opening lead rarely matters, and the double-dummy declarer always finds the winning line, never misguessing. In my Actual Play vs. Double-Dummy study, seven of a suit was made 435 times out of 698 occurrences but would have made 480 times at double-dummy — a decrease of 9.06 percent for the actual declarer. Average defense would mitigate this with occasional poor leads, so 8 percent seemed like a good general reduction, implemented in the gold-tinted cell below. All calculations are then based on the Adjusted column.

Result	Occurred	Percent	Adjusted	Percent
0	0	0	0	0
1	0	0	0	0
2	0	0	0	0
3	0	0	0	0
4	0	0	0	0
5	1*	0.00	1	0.00
6	56	0.01	56	0.01
7	953	0.12	953	0.12
8	7806	0.96	7806	0.98
9	35,462	4.36	35,462	4.46
10	99,861	12.28	99,861	12.57
11	190,065	23.37	190,065	23.92
12	245,640	30.20	245,640	30.91
13	233,505	28.71	214,825	27.03
Totals	813,349	100	794,669	100

*Another curio… with an 8+ card club fit, A-K and first-round control in every suit!
Unlucky, partner, down eight. Talk about a bad day at Black Rock.

Point Count Results

The following six tables show the results of 7 contracts for each evaluation method, according to the total points (32-38) obtained from its algorithm. The last four columns are the most significant, showing the average IMP gain (+) or loss (-) versus contracts of 5 and 6 , nonvul (green) and vul (red). A minus means you’re better off in five or six, and a plus means you’re better off in the grand.

While it is reasonable to bid six with a 50-50 chance, grand slams are a different story. There is more to lose if defeated than to gain if successful. Assuming a six-bid makes, IMP scoring barely justifies bidding seven with a 56-percent chance (fractionally less vulnerable) but the wise player waits for better odds.

Over the years I’ve read that a grand slam requires 36 or 37 points, with about equal recommendation for either number. Well, which is it? Evidence shows the lower, assuming my prerequisite of the trump A-K and all first-round controls. Three algorithms (Goren, Bissell and my own) required 36 points. Karpin required only 35, while Zar and Kaplan-Rubens lowered the bar another notch to 34.

Goren (short suit) Count

Points	Cases	7 make%	Trick Avg	5	6	5	6
32	80,647	17.68	11.6541	-5.39	-4.65	-6.51	-5.54
33	78,867	24.79	11.8676	-3.85	-4.10	-4.61	-4.80
34	72,393	33.32	12.0646	-1.92	-2.97	-2.25	-3.41
35	62,785	42.35	12.2332	+0.21	-1.48	+0.33	-1.62
36	51,337	51.47	12.3826	+2.37	+0.24	+2.94	+0.41
37	39,597	59.78	12.5041	+4.36	+1.92	+5.34	+2.40
38	28,967	67.24	12.6029	+6.15	+3.52	+7.51	+4.27

Karpin (long suit) Count

Points	Cases	7 make%	Trick Avg	5	6	5	6
32	80,581	24.42	11.8896	-4.00	-4.39	-4.78	-5.12
33	74,254	33.77	12.0890	-1.82	-3.06	-2.13	-3.50
34	63,971	43.98	12.2699	+0.59	-1.28	+0.79	-1.37
35	52,034	53.53	12.4201	+2.87	+0.57	+3.55	+0.81
36	40,026	62.42	12.5410	+5.00	+2.46	+6.12	+3.03
37	29,165	68.46	12.6231	+6.43	+3.76	+7.85	+4.55
38	20,279	73.98	12.6927	+7.76	+4.98	+9.46	+5.98

Pavlicek Point Count

Points	Cases	7 make%	Trick Avg	5	6	5	6
32	85,124	17.72	11.7028	-5.55	-4.84	-6.67	-5.74
33	82,053	25.80	11.9431	-3.77	-4.28	-4.48	-4.98
34	74,327	35.60	12.1586	-1.47	-3.00	-1.69	-3.40
35	61,807	46.40	12.3424	+1.12	-1.09	+1.44	-1.13
36	49,059	57.05	12.4952	+3.68	+1.08	+4.53	+1.43
37	36,372	66.75	12.6191	+6.01	+3.21	+7.35	+3.92
38	25,112	73.98	12.7095	+7.75	+4.82	+9.45	+5.81

Modified Bissell Count

Points	Cases	7 make%	Trick Avg	5	6	5	6
32	74,982	19.64	11.6983	-4.89	-4.49	-5.91	-5.32
33	72,008	27.32	11.8940	-3.19	-3.61	-3.82	-4.22
34	66,161	34.96	12.0540	-1.42	-2.47	-1.67	-2.84
35	57,586	42.89	12.1954	+0.44	-1.07	+0.58	-1.17
36	48,232	50.05	12.3145	+2.13	+0.28	+2.63	+0.42
37	38,668	56.09	12.4043	+3.56	+1.51	+4.37	+1.88
38	29,728	61.02	12.4721	+4.74	+2.55	+5.79	+3.10

Zar Point Count

Points	Cases	7 make%	Trick Avg	5	6	5	6
32	121,359	25.87	11.9382	-3.70	-4.29	-4.41	-4.99
33	100,984	41.81	12.2469	+0.06	-1.79	+0.15	-1.97
34	73,508	57.31	12.4807	+3.76	+1.30	+4.63	+1.67
35	46,694	69.34	12.6380	+6.64	+3.92	+8.11	+4.74
36	25,855	77.72	12.7445	+8.65	+5.78	+10.54	+6.92
37	12,998	82.78	12.8077	+9.86	+6.92	+12.00	+8.25
38	5809	86.92	12.8516	+10.86	+7.90	+13.21	+9.39

Kaplan-Rubens Count

Points	Cases	7 make%	Trick Avg	5	6	5	6
32	76,185	33.60	12.1009	-1.88	-3.21	-2.20	-3.67
33	64,838	44.58	12.2859	+0.74	-1.23	+0.97	-1.32
34	51,793	55.05	12.4401	+3.23	+0.90	+3.98	+1.20
35	38,767	63.86	12.5551	+5.34	+2.82	+6.54	+3.45
36	27,787	70.54	12.6418	+6.94	+4.28	+8.47	+5.16
37	18,928	74.24	12.6866	+7.83	+5.14	+9.54	+6.15
38	12,384	76.74	12.7199	+8.43	+5.67	+10.27	+6.78

Examples to verify the algorithms: Four Spades Five Diamonds Six Hearts Seven Clubs

Study AC01 Main

Top Point Count Methods

© 2022 Richard Pavlicek