Study 8C17   Main

Tripleton Leads

by Richard Pavlicek

Having recently completed a study on Doubleton Leads, and several years ago on leads from Four and Five, it seemed only fitting to include the missing link: Tripletons. In my major-event database the opening lead was from three cards 16,167 times in 60,481 opportunities (leader held at least one tripleton) which is certainly significant, although diminished by some cases being clearly indicated, such as partner’s bid suit. This study will examine the merits and demerits of leading from each three-card holding.

Tripletons headed by Ace

King

Queen

Jack

Ten or Nine

Eight or Lower

There are 286 possible tripletons, from A-K-Q through 4-3-2, and each can occur in 60,291 specific layouts. For this study I will exclude layouts with a void or 6+ card suit, because the bidding would usually reveal the lopsided distribution, giving the leader special knowledge, hence strategy would deviate from the general scenario I try to present here. This reduces the field to 44,730 common layouts per tripleton, or 12,792,780 in all. The following table shows the breakdown by distribution:

Suit Pattern	Combinations	Ways	Specific	Ratio
3 (4-3-3)	10c4 × 6c3 = 4200	3	12600	220
3 (4-4-2)	10c4 × 6c4 = 3150	3	9450	150
3 (5-3-2)	10c5 × 5c3 = 2520	6	15120	108
3 (5-4-1)	10c5 × 5c4 = 1260	6	7560	45
Totals	11130	18	44730	523

The Ratio column shows the relative probability of a specific layout in that distribution group. This is used as a weight factor to obtain true averages and comparisons, i.e., 4-3-3 distributions (in any order) are the most common, so their results get the greatest weight.

The contract is presumed to be notrump, however, all analyses would apply equally at a suit contract with regard to the tricks winnable in a single suit without ruffing ability. That is, it could be a side suit that no one can (or chooses not to) ruff; or it could be the trump suit itself if being led without ruffing.

Trick Average

The basic calculation was to determine the number of tricks won by the defense on each of the 12,792,780 layouts in five cases: (1) West leading high, (2) West leading middle (3) West leading low, (4) Partner leading first to best advantage, and (5) Declarer leading throughout. In Cases 1-4, declarer was required to lead after the first trick, and in all cases declarer could lead from either hand to best advantage. This required almost 64 million intense calculations, which sounds formidable, but the only real task was designing and debugging the software; once done, my computer spewed out the results in five minutes.

For each layout, tricks won by the defense are the difference between the maximum declarer could win and the actual result. For example, if declarer has five cards in dummy and two in hand, the most he could win is five (including two long cards); so if he wins three, the defense gets two. Depending on the distribution, declarer’s maximum varies from 3-5, but the most the defense could ever win is four (partner cannot have five when declarer does).

For each tripleton, the average tricks won is then calculated in each of the five cases. Weight factors (220, 150, 108, 45) are applied for each distribution group to obtain a true average based on frequency of occurrence. Trick average is most relevant in comparing lead choices within each tripleton, rather than one tripleton to another, since it’s generally proportional to the strength of the tripleton.

Double-Dummy Issues

Except for West’s forced lead in Cases 1-3, play by both sides is double-dummy. This means declarer gets to make the first double-dummy lead in Cases 1-3 and 5, which gives him a definite advantage. Consequently, trick averages for tenuous holdings would be higher in practice, though relative differences among lead choices would rarely change. (In Case 4 partner gets the first stroke, so those trick averages would be closer to actual play.) To level the playing field somewhat, in Case 5 declarer is not allowed to start with a backward finesse. For example:

Any contract	K 6 5
Q 8 2		10 7 3
South leads	A J 9 4

If West doesn’t lead this suit, he deserves a trick; but declarer could start with the jack to win all four tricks. This is more like cheating than a correct guess, so I block it. My algorithm is: If North or South’s second-highest card (of three or more) is the eight, nine, 10 or jack, it cannot be led to the first trick unless the N-S side has a touching card.

Versus Partner Leading

For each lead option (high, middle, low) a comparison is made with the results if partner had led the suit first. This is shown as a signed whole number: plus if your first lead does better, or minus if partner’s first lead does better. The higher the number, the greater the advantage or disadvantage. The actual value is the expected net trick gain or loss in 1000 opportunities; hence the theoretical range is +1000 (winning on every layout) to -1000 (losing on every layout) but nothing close to either extreme is possible.

This comparison rating is not just a count of winning and losing cases but includes a weight adjustment for each distribution group; hence a common layout like 4-3-3 gets more weight than a rarer one (say, 5-3-2) in the exact proportion of their ratios.

Versus Declarer Leading

A similar comparison is made for each lead option (high, middle, low) versus the results if declarer had led the suit entirely, however in this case it’s a no-win situation. The best you can do is break even, and unless your lead is 100-percent safe, it must show a loss. This is the main criterion for judging the safety of a lead; the higher the negative number, the more risky the lead.

Even though almost all leads show a loss here, it is nothing to be paranoid about. It’s a fact of card play that the side leading a suit is at a disadvantage (regardless of length) because an opponent plays after each of their plays. The goal is to minimize these losses. You have to lead something.

Study 8C17   Main

Top   Tripleton Leads

Ace Holdings

The first table (of six) shows the results of the 66 tripletons headed by the ace. The best lead (high, middle, low) from each holding is tinted gold, and when three distinct options exist, the second-best lead is tinted blue. An equal sign (=) indicates a touching card to the one above, which of course must have the same value.

Comparisons with partner or declarer leading are always net losses, except for A-K-Q or A-K-J which break even. The worst holding to lead from is A-J-9, losing a trick 377 times (out of 1000) versus declarer leading.

Noteworthy is that A-Q-x is the only tripleton without touching cards where leading high is best (assuming you choose to lead it). This holds for all tripletons, not just those headed by the ace.

Holding	High	Middle	Low	Partner	Declarer	High v P	Mid v P	Low v P	High v D	Mid v D	Low v D
A-K-Q	3.0951	=	=	3.0951	3.0951	0	=	=	0	=	=
A-K-J	2.8224	=	2.4727	2.8224	2.8224	0	=	-350	0	=	-350
A-K-10	2.6101	=	2.4285	2.6946	2.6946	-85	=	-266	-85	=	-266
A-K-9	2.5565	=	2.4176	2.6198	2.6269	-63	=	-202	-70	=	-209
A-K-8	2.5331	=	2.4176	2.5584	2.5947	-25	=	-141	-62	=	-177
A-K-7	2.5317	=	2.4176	2.5501	2.5661	-18	=	-132	-34	=	-148
A-K-6	2.5317	=	2.4176	2.5439	2.5595	-12	=	-126	-28	=	-142
A-K-5	2.5317	=	2.4176	2.5435	2.5541	-12	=	-126	-22	=	-136
A-K-4	2.5317	=	2.4176	2.5435	2.5537	-12	=	-126	-22	=	-136
A-K-3	2.5317	=	2.4176	2.5435	2.5537	-12	=	-126	-22	=	-136
A-K-2	2.5317	=	2.4176	2.5435	2.5537	-12	=	-126	-22	=	-136
A-Q-J	2.4763	2.4727	=	2.7857	2.7857	-309	-313	=	-309	-313	=
A-Q-10	2.3862	2.2843	2.2479	2.5875	2.5875	-201	-303	-340	-201	-303	-340
A-Q-9	2.3125	2.1432	2.2316	2.4812	2.4883	-169	-338	-250	-176	-345	-257
A-Q-8	2.3016	2.1064	2.2307	2.3974	2.4367	-96	-291	-167	-135	-330	-206
A-Q-7	2.2992	2.0910	2.2307	2.3786	2.4048	-79	-288	-148	-106	-314	-174
A-Q-6	2.2992	2.0903	2.2307	2.3713	2.3976	-72	-281	-141	-98	-307	-167
A-Q-5	2.2992	2.0903	2.2307	2.3709	2.3922	-72	-281	-140	-93	-302	-161
A-Q-4	2.2992	2.0903	2.2307	2.3709	2.3919	-72	-281	-140	-93	-302	-161
A-Q-3	2.2992	2.0903	2.2307	2.3709	2.3919	-72	-281	-140	-93	-302	-161
A-Q-2	2.2992	2.0903	2.2307	2.3709	2.3919	-72	-281	-140	-93	-302	-161
A-J-10	1.9494	2.1488	=	2.4736	2.4883	-524	-325	=	-539	-339	=
A-J-9	1.9227	1.9803	2.0038	2.3649	2.3809	-442	-385	-361	-458	-401	-377
A-J-8	1.9024	1.9188	1.9988	2.2571	2.3033	-355	-338	-258	-401	-384	-304
A-J-7	1.9012	1.8956	1.9967	2.2099	2.2684	-309	-314	-213	-367	-373	-272
A-J-6	1.9010	1.8915	1.9967	2.2011	2.2622	-300	-310	-204	-361	-371	-266
A-J-5	1.9010	1.8913	1.9967	2.1985	2.2591	-298	-307	-202	-358	-368	-262
A-J-4	1.9010	1.8913	1.9967	2.1985	2.2587	-298	-307	-202	-358	-367	-262
A-J-3	1.9010	1.8913	1.9967	2.1985	2.2587	-298	-307	-202	-358	-367	-262
A-J-2	1.9010	1.8913	1.9967	2.1985	2.2587	-298	-307	-202	-358	-367	-262
A-10-9	1.7988	1.9272	=	2.0613	2.1660	-262	-134	=	-367	-239	=
A-10-8	1.7966	1.8750	1.9094	2.0435	2.1280	-247	-168	-134	-331	-253	-219
A-10-7	1.7959	1.8363	1.9053	2.0131	2.1053	-217	-177	-108	-309	-269	-200
A-10-6	1.7958	1.8317	1.9039	2.0102	2.1032	-214	-178	-106	-307	-271	-199
A-10-5	1.7958	1.8304	1.9039	2.0090	2.1018	-213	-179	-105	-306	-271	-198
A-10-4	1.7958	1.8303	1.9039	2.0085	2.1018	-213	-178	-105	-306	-271	-198
A-10-3	1.7958	1.8303	1.9039	2.0085	2.1018	-213	-178	-105	-306	-271	-198
A-10-2	1.7958	1.8303	1.9039	2.0085	2.1018	-213	-178	-105	-306	-271	-198
A-9-8	1.7598	1.8662	=	1.8889	2.0037	-129	-23	=	-244	-138	=
A-9-7	1.7598	1.8297	1.8397	1.8826	1.9978	-123	-53	-43	-238	-168	-158
A-9-6	1.7598	1.8232	1.8377	1.8776	1.9950	-118	-54	-40	-235	-172	-157
A-9-5	1.7598	1.8200	1.8376	1.8756	1.9943	-116	-56	-38	-235	-174	-157
A-9-4	1.7598	1.8199	1.8376	1.8754	1.9940	-116	-55	-38	-234	-174	-156
A-9-3	1.7598	1.8199	1.8376	1.8754	1.9940	-116	-55	-38	-234	-174	-156
A-9-2	1.7598	1.8199	1.8376	1.8754	1.9940	-116	-55	-38	-234	-174	-156
A-8-7	1.7549	1.8237	=	1.8354	1.9218	-80	-12	=	-167	-98	=
A-8-6	1.7549	1.8149	1.8212	1.8346	1.9192	-80	-20	-13	-164	-104	-98
A-8-5	1.7549	1.8098	1.8208	1.8338	1.9169	-79	-24	-13	-162	-107	-96
A-8-4	1.7549	1.8097	1.8207	1.8334	1.9168	-78	-24	-13	-162	-107	-96
A-8-3	1.7549	1.8097	1.8207	1.8334	1.9168	-78	-24	-13	-162	-107	-96
A-8-2	1.7549	1.8097	1.8207	1.8334	1.9168	-78	-24	-13	-162	-107	-96
A-7-6	1.7547	1.8144	=	1.8145	1.8922	-60	-0	=	-138	-78	=
A-7-5	1.7547	1.8090	1.8095	1.8145	1.8919	-60	-5	-5	-137	-83	-82
A-7-4	1.7547	1.8088	1.8093	1.8145	1.8918	-60	-6	-5	-137	-83	-82
A-7-3	1.7547	1.8088	1.8093	1.8145	1.8918	-60	-6	-5	-137	-83	-82
A-7-2	1.7547	1.8088	1.8093	1.8145	1.8918	-60	-6	-5	-137	-83	-82
A-6-5	1.7547	1.8088	=	1.8124	1.8820	-58	-4	=	-127	-73	=
A-6-4	1.7547	1.8085	1.8085	1.8124	1.8820	-58	-4	-4	-127	-74	-74
A-6-3	1.7547	1.8085	1.8085	1.8124	1.8820	-58	-4	-4	-127	-74	-74
A-6-2	1.7547	1.8085	1.8085	1.8124	1.8820	-58	-4	-4	-127	-74	-74
A-5-4	1.7547	1.8085	=	1.8123	1.8806	-58	-4	=	-126	-72	=
A-5-3	1.7547	1.8085	1.8085	1.8123	1.8806	-58	-4	-4	-126	-72	-72
A-5-2	1.7547	1.8085	1.8085	1.8123	1.8806	-58	-4	-4	-126	-72	-72
A-4-3	1.7547	1.8085	=	1.8123	1.8806	-58	-4	=	-126	-72	=
A-4-2	1.7547	1.8085	1.8085	1.8123	1.8806	-58	-4	-4	-126	-72	-72
A-3-2	1.7547	1.8085	=	1.8123	1.8806	-58	-4	=	-126	-72	=

Study 8C17   Main

Top   Tripleton Leads

King Holdings

This table shows the results of the 55 tripletons headed by the king. The best lead (high, middle, low) from each holding is tinted gold, and when three distinct options exist, the second-best is tinted blue.

Comparisons with partner or declarer leading are always net losses, except for K-Q-J which breaks even. The worst holding to lead from is K-J-9, losing a trick 279 times (out of 1000) versus declarer leading.

Holding	High	Middle	Low	Partner	Declarer	High v P	Mid v P	Low v P	High v D	Mid v D	Low v D
K-Q-J	2.4727	=	=	2.4727	2.4727	0	=	=	0	=	=
K-Q-10	2.2843	=	2.0351	2.3834	2.3834	-99	=	-348	-99	=	-348
K-Q-9	2.1432	=	2.0086	2.3398	2.3398	-197	=	-331	-197	=	-331
K-Q-8	2.1064	=	2.0028	2.3107	2.3137	-204	=	-308	-207	=	-311
K-Q-7	2.0910	=	2.0028	2.2874	2.3019	-196	=	-285	-211	=	-299
K-Q-6	2.0903	=	2.0028	2.2827	2.2982	-192	=	-280	-208	=	-295
K-Q-5	2.0903	=	2.0028	2.2800	2.2978	-190	=	-277	-208	=	-295
K-Q-4	2.0903	=	2.0028	2.2799	2.2974	-190	=	-277	-207	=	-295
K-Q-3	2.0903	=	2.0028	2.2799	2.2974	-190	=	-277	-207	=	-295
K-Q-2	2.0903	=	2.0028	2.2799	2.2974	-190	=	-277	-207	=	-295
K-J-10	1.9383	2.0342	=	2.2442	2.2588	-306	-210	=	-320	-225	=
K-J-9	1.8570	1.8802	1.8907	2.1611	2.1700	-304	-281	-270	-313	-290	-279
K-J-8	1.7940	1.7950	1.8797	2.1053	2.1218	-311	-310	-226	-328	-327	-242
K-J-7	1.7852	1.7732	1.8788	2.0732	2.0950	-288	-300	-194	-310	-322	-216
K-J-6	1.7833	1.7682	1.8788	2.0641	2.0891	-281	-296	-185	-306	-321	-210
K-J-5	1.7833	1.7680	1.8788	2.0609	2.0884	-278	-293	-182	-305	-320	-210
K-J-4	1.7833	1.7680	1.8788	2.0607	2.0880	-277	-293	-182	-305	-320	-209
K-J-3	1.7833	1.7680	1.8788	2.0607	2.0880	-277	-293	-182	-305	-320	-209
K-J-2	1.7833	1.7680	1.8788	2.0607	2.0880	-277	-293	-182	-305	-320	-209
K-10-9	1.5190	1.7860	=	1.9437	2.0214	-425	-158	=	-502	-235	=
K-10-8	1.4998	1.6818	1.7086	1.9114	1.9711	-412	-230	-203	-471	-289	-263
K-10-7	1.4867	1.6457	1.7050	1.8747	1.9341	-388	-229	-170	-447	-288	-229
K-10-6	1.4860	1.6377	1.7036	1.8636	1.9269	-378	-226	-160	-441	-289	-223
K-10-5	1.4860	1.6362	1.7036	1.8605	1.9257	-375	-224	-157	-440	-289	-222
K-10-4	1.4860	1.6361	1.7036	1.8601	1.9257	-374	-224	-156	-440	-290	-222
K-10-3	1.4860	1.6361	1.7036	1.8601	1.9257	-374	-224	-156	-440	-290	-222
K-10-2	1.4860	1.6361	1.7036	1.8601	1.9257	-374	-224	-156	-440	-290	-222
K-9-8	1.4066	1.6441	=	1.7182	1.8321	-312	-74	=	-426	-188	=
K-9-7	1.4054	1.6203	1.6363	1.7128	1.8199	-307	-92	-76	-414	-200	-184
K-9-6	1.4050	1.6124	1.6342	1.7082	1.8122	-303	-96	-74	-407	-200	-178
K-9-5	1.4050	1.6117	1.6342	1.7067	1.8118	-302	-95	-72	-407	-200	-178
K-9-4	1.4050	1.6116	1.6342	1.7065	1.8116	-302	-95	-72	-407	-200	-177
K-9-3	1.4050	1.6116	1.6342	1.7065	1.8116	-302	-95	-72	-407	-200	-177
K-9-2	1.4050	1.6116	1.6342	1.7065	1.8116	-302	-95	-72	-407	-200	-177
K-8-7	1.3794	1.6134	=	1.6447	1.7443	-265	-31	=	-365	-131	=
K-8-6	1.3794	1.6054	1.6094	1.6434	1.7432	-264	-38	-34	-364	-138	-134
K-8-5	1.3794	1.6051	1.6093	1.6418	1.7430	-262	-37	-32	-364	-138	-134
K-8-4	1.3794	1.6051	1.6092	1.6414	1.7430	-262	-36	-32	-364	-138	-134
K-8-3	1.3794	1.6051	1.6092	1.6414	1.7430	-262	-36	-32	-364	-138	-134
K-8-2	1.3794	1.6051	1.6092	1.6414	1.7430	-262	-36	-32	-364	-138	-134
K-7-6	1.3769	1.6046	=	1.6223	1.7208	-245	-18	=	-344	-116	=
K-7-5	1.3769	1.6043	1.6045	1.6222	1.7208	-245	-18	-18	-344	-117	-116
K-7-4	1.3769	1.6041	1.6043	1.6221	1.7208	-245	-18	-18	-344	-117	-117
K-7-3	1.3769	1.6041	1.6043	1.6221	1.7208	-245	-18	-18	-344	-117	-117
K-7-2	1.3769	1.6041	1.6043	1.6221	1.7208	-245	-18	-18	-344	-117	-117
K-6-5	1.3768	1.6041	=	1.6192	1.7115	-242	-15	=	-335	-107	=
K-6-4	1.3768	1.6038	1.6038	1.6192	1.7115	-242	-15	-15	-335	-108	-108
K-6-3	1.3768	1.6038	1.6038	1.6192	1.7115	-242	-15	-15	-335	-108	-108
K-6-2	1.3768	1.6038	1.6038	1.6192	1.7115	-242	-15	-15	-335	-108	-108
K-5-4	1.3768	1.6038	=	1.6190	1.7104	-242	-15	=	-334	-107	=
K-5-3	1.3768	1.6038	1.6038	1.6190	1.7104	-242	-15	-15	-334	-107	-107
K-5-2	1.3768	1.6038	1.6038	1.6190	1.7104	-242	-15	-15	-334	-107	-107
K-4-3	1.3768	1.6038	=	1.6190	1.7104	-242	-15	=	-334	-107	=
K-4-2	1.3768	1.6038	1.6038	1.6190	1.7104	-242	-15	-15	-334	-107	-107
K-3-2	1.3768	1.6038	=	1.6190	1.7104	-242	-15	=	-334	-107	=

Study 8C17   Main

Top   Tripleton Leads

Queen Holdings

This table shows the results of the 45 tripletons headed by the queen. The best lead (high, middle, low) from each holding is tinted gold, and when three distinct options exist, the second-best is tinted blue.

Comparisons with partner or declarer leading are always net losses, except for Q-J-10 which gains versus partner and breaks even versus declarer. The worst holding to lead from is Q-10-8, losing a trick 259 times (out of 1000) versus declarer leading.

Holding	High	Middle	Low	Partner	Declarer	High v P	Mid v P	Low v P	High v D	Mid v D	Low v D
Q-J-10	1.9227	=	=	1.8333	1.9227	+89	=	=	0	=	=
Q-J-9	1.8069	=	1.5832	1.8095	1.8771	-3	=	-226	-70	=	-294
Q-J-8	1.7188	=	1.5645	1.7979	1.8543	-79	=	-233	-136	=	-290
Q-J-7	1.7014	=	1.5602	1.7891	1.8394	-88	=	-229	-138	=	-279
Q-J-6	1.6960	=	1.5602	1.7846	1.8362	-89	=	-224	-140	=	-276
Q-J-5	1.6958	=	1.5602	1.7834	1.8347	-88	=	-223	-139	=	-274
Q-J-4	1.6958	=	1.5602	1.7831	1.8346	-87	=	-223	-139	=	-274
Q-J-3	1.6958	=	1.5602	1.7831	1.8346	-87	=	-223	-139	=	-274
Q-J-2	1.6958	=	1.5602	1.7831	1.8346	-87	=	-223	-139	=	-274
Q-10-9	1.4634	1.5818	=	1.7092	1.7940	-246	-127	=	-331	-212	=
Q-10-8	1.4262	1.4865	1.4976	1.6862	1.7566	-260	-200	-189	-330	-270	-259
Q-10-7	1.3982	1.4459	1.4896	1.6687	1.7339	-270	-223	-179	-336	-288	-244
Q-10-6	1.3964	1.4386	1.4890	1.6619	1.7244	-265	-223	-173	-328	-286	-235
Q-10-5	1.3961	1.4370	1.4890	1.6597	1.7222	-264	-223	-171	-326	-285	-233
Q-10-4	1.3961	1.4370	1.4890	1.6594	1.7222	-263	-222	-170	-326	-285	-233
Q-10-3	1.3961	1.4370	1.4890	1.6594	1.7222	-263	-222	-170	-326	-285	-233
Q-10-2	1.3961	1.4370	1.4890	1.6594	1.7222	-263	-222	-170	-326	-285	-233
Q-9-8	1.2620	1.4307	=	1.5275	1.6527	-265	-97	=	-391	-222	=
Q-9-7	1.2576	1.3838	1.3945	1.5198	1.6350	-262	-136	-125	-377	-251	-241
Q-9-6	1.2544	1.3692	1.3926	1.5144	1.6225	-260	-145	-122	-368	-253	-230
Q-9-5	1.2541	1.3678	1.3924	1.5122	1.6204	-258	-144	-120	-366	-253	-228
Q-9-4	1.2540	1.3676	1.3924	1.5122	1.6201	-258	-145	-120	-366	-253	-228
Q-9-3	1.2540	1.3676	1.3924	1.5122	1.6201	-258	-145	-120	-366	-253	-228
Q-9-2	1.2540	1.3676	1.3924	1.5122	1.6201	-258	-145	-120	-366	-253	-228
Q-8-7	1.2108	1.3623	=	1.4342	1.5653	-223	-72	=	-354	-203	=
Q-8-6	1.2105	1.3560	1.3597	1.4336	1.5604	-223	-78	-74	-350	-204	-201
Q-8-5	1.2104	1.3554	1.3594	1.4334	1.5570	-223	-78	-74	-347	-202	-198
Q-8-4	1.2103	1.3553	1.3594	1.4334	1.5568	-223	-78	-74	-346	-201	-197
Q-8-3	1.2103	1.3553	1.3594	1.4334	1.5568	-223	-78	-74	-346	-201	-197
Q-8-2	1.2103	1.3553	1.3594	1.4334	1.5568	-223	-78	-74	-346	-201	-197
Q-7-6	1.2022	1.3533	=	1.4091	1.5413	-207	-56	=	-339	-188	=
Q-7-5	1.2022	1.3521	1.3521	1.4091	1.5408	-207	-57	-57	-339	-189	-189
Q-7-4	1.2022	1.3521	1.3521	1.4091	1.5406	-207	-57	-57	-338	-188	-188
Q-7-3	1.2022	1.3521	1.3521	1.4091	1.5406	-207	-57	-57	-338	-188	-188
Q-7-2	1.2022	1.3521	1.3521	1.4091	1.5406	-207	-57	-57	-338	-188	-188
Q-6-5	1.2013	1.3518	=	1.4043	1.5343	-203	-52	=	-333	-183	=
Q-6-4	1.2013	1.3518	1.3518	1.4043	1.5343	-203	-52	-52	-333	-183	-183
Q-6-3	1.2013	1.3518	1.3518	1.4043	1.5343	-203	-52	-52	-333	-183	-183
Q-6-2	1.2013	1.3518	1.3518	1.4043	1.5343	-203	-52	-52	-333	-183	-183
Q-5-4	1.2013	1.3518	=	1.4037	1.5331	-202	-52	=	-332	-181	=
Q-5-3	1.2013	1.3518	1.3518	1.4037	1.5331	-202	-52	-52	-332	-181	-181
Q-5-2	1.2013	1.3518	1.3518	1.4037	1.5331	-202	-52	-52	-332	-181	-181
Q-4-3	1.2013	1.3518	=	1.4037	1.5331	-202	-52	=	-332	-181	=
Q-4-2	1.2013	1.3518	1.3518	1.4037	1.5331	-202	-52	-52	-332	-181	-181
Q-3-2	1.2013	1.3518	=	1.4037	1.5331	-202	-52	=	-332	-181	=

Study 8C17   Main

Top   Tripleton Leads

Jack Holdings

This table shows the results of the 36 tripletons headed by the jack. The best lead (high, middle, low) from each holding is tinted gold, and when three distinct options exist, the second-best is tinted blue.

Note the complete turnaround versus partner leading, as every holding now shows a gain. Versus declarer, of course, everything still loses, except J-10-9 which breaks even. The worst holding to lead from is J-8-6, losing a trick 114 times (out of 1000) versus declarer leading.

A curiosity arises with J-7-x and J-6-x, where leading the middle card is best, albeit by a negligible margin over low. I know the reason but won’t explain it here to avoid compromising a puzzle.

Holding	High	Middle	Low	Partner	Declarer	High v P	Mid v P	Low v P	High v D	Mid v D	Low v D
J-10-9	1.4199	=	=	1.2577	1.4199	+162	=	=	0	=	=
J-10-8	1.3724	=	1.2917	1.2514	1.4042	+121	=	+40	-32	=	-112
J-10-7	1.3371	=	1.2783	1.2484	1.3959	+89	=	+30	-59	=	-118
J-10-6	1.3336	=	1.2754	1.2464	1.3901	+87	=	+29	-57	=	-115
J-10-5	1.3321	=	1.2754	1.2455	1.3896	+87	=	+30	-57	=	-114
J-10-4	1.3321	=	1.2754	1.2455	1.3895	+87	=	+30	-57	=	-114
J-10-3	1.3321	=	1.2754	1.2455	1.3895	+87	=	+30	-57	=	-114
J-10-2	1.3321	=	1.2754	1.2455	1.3895	+87	=	+30	-57	=	-114
J-9-8	1.2003	1.2907	=	1.2199	1.3653	-20	+71	=	-165	-75	=
J-9-7	1.1918	1.2422	1.2453	1.2162	1.3529	-24	+26	+29	-161	-111	-108
J-9-6	1.1845	1.2261	1.2400	1.2139	1.3445	-29	+12	+26	-160	-118	-105
J-9-5	1.1836	1.2241	1.2396	1.2129	1.3418	-29	+11	+27	-158	-118	-102
J-9-4	1.1833	1.2239	1.2396	1.2129	1.3415	-30	+11	+27	-158	-118	-102
J-9-3	1.1833	1.2239	1.2396	1.2129	1.3415	-30	+11	+27	-158	-118	-102
J-9-2	1.1833	1.2239	1.2396	1.2129	1.3415	-30	+11	+27	-158	-118	-102
J-8-7	1.1110	1.2111	=	1.1632	1.3172	-52	+48	=	-206	-106	=
J-8-6	1.1094	1.1969	1.1974	1.1626	1.3114	-53	+34	+35	-202	-114	-114
J-8-5	1.1081	1.1929	1.1963	1.1624	1.3068	-54	+30	+34	-199	-114	-111
J-8-4	1.1079	1.1927	1.1962	1.1624	1.3065	-55	+30	+34	-199	-114	-110
J-8-3	1.1079	1.1927	1.1962	1.1624	1.3065	-55	+30	+34	-199	-114	-110
J-8-2	1.1079	1.1927	1.1962	1.1624	1.3065	-55	+30	+34	-199	-114	-110
J-7-6	1.0920	1.1873	=	1.1519	1.2951	-60	+35	=	-203	-108	=
J-7-5	1.0919	1.1861	1.1859	1.1519	1.2942	-60	+34	+34	-202	-108	-108
J-7-4	1.0918	1.1860	1.1857	1.1519	1.2939	-60	+34	+34	-202	-108	-108
J-7-3	1.0918	1.1860	1.1857	1.1519	1.2939	-60	+34	+34	-202	-108	-108
J-7-2	1.0918	1.1860	1.1857	1.1519	1.2939	-60	+34	+34	-202	-108	-108
J-6-5	1.0889	1.1851	=	1.1491	1.2908	-60	+36	=	-202	-106	=
J-6-4	1.0889	1.1851	1.1850	1.1491	1.2908	-60	+36	+36	-202	-106	-106
J-6-3	1.0889	1.1851	1.1850	1.1491	1.2908	-60	+36	+36	-202	-106	-106
J-6-2	1.0889	1.1851	1.1850	1.1491	1.2908	-60	+36	+36	-202	-106	-106
J-5-4	1.0888	1.1850	=	1.1479	1.2901	-59	+37	=	-201	-105	=
J-5-3	1.0888	1.1850	1.1850	1.1479	1.2901	-59	+37	+37	-201	-105	-105
J-5-2	1.0888	1.1850	1.1850	1.1479	1.2901	-59	+37	+37	-201	-105	-105
J-4-3	1.0888	1.1850	=	1.1479	1.2901	-59	+37	=	-201	-105	=
J-4-2	1.0888	1.1850	1.1850	1.1479	1.2901	-59	+37	+37	-201	-105	-105
J-3-2	1.0888	1.1850	=	1.1479	1.2901	-59	+37	=	-201	-105	=

Study 8C17   Main

Top   Tripleton Leads

Ten or Nine Holdings

This table shows the results of the 49 tripletons headed by the 10 or nine. The best lead (high, middle, low) from each holding is tinted gold, and when three distinct options exist, the second-best is tinted blue.

Comparisons versus partner leading always show a gain, obviously because partner often has a high honor from which he cannot lead safely. As always, any holding except three touching cards loses versus declarer leading, though the losses are small.

The curiosity continues: Advocates of M-U-D (middle-up-down) will be pleased to see that their choice has a technical advantage (versus low) from 10-8-x, 10-7-x, 10-6-x, 9-7-x and 9-6-x.

Holding	High	Middle	Low	Partner	Declarer	High v P	Mid v P	Low v P	High v D	Mid v D	Low v D
10-9-8	1.1687	=	=	0.9943	1.1687	+174	=	=	0	=	=
10-9-7	1.1547	=	1.0994	0.9935	1.1636	+161	=	+106	-9	=	-64
10-9-6	1.1434	=	1.0916	0.9933	1.1606	+150	=	+98	-17	=	-69
10-9-5	1.1424	=	1.0902	0.9932	1.1584	+149	=	+97	-16	=	-68
10-9-4	1.1421	=	1.0902	0.9932	1.1582	+149	=	+97	-16	=	-68
10-9-3	1.1421	=	1.0902	0.9932	1.1582	+149	=	+97	-16	=	-68
10-9-2	1.1421	=	1.0902	0.9932	1.1582	+149	=	+97	-16	=	-68
10-8-7	1.0703	1.0994	=	0.9817	1.1566	+89	+118	=	-86	-57	=
10-8-6	1.0673	1.0844	1.0803	0.9814	1.1533	+86	+103	+99	-86	-69	-73
10-8-5	1.0649	1.0796	1.0776	0.9813	1.1508	+84	+98	+96	-86	-71	-73
10-8-4	1.0646	1.0793	1.0774	0.9813	1.1506	+83	+98	+96	-86	-71	-73
10-8-3	1.0646	1.0793	1.0774	0.9813	1.1506	+83	+98	+96	-86	-71	-73
10-8-2	1.0646	1.0793	1.0774	0.9813	1.1506	+83	+98	+96	-86	-71	-73
10-7-6	1.0429	1.0707	=	0.9721	1.1448	+71	+99	=	-102	-74	=
10-7-5	1.0427	1.0696	1.0675	0.9721	1.1440	+71	+98	+95	-101	-74	-77
10-7-4	1.0426	1.0694	1.0672	0.9721	1.1438	+71	+97	+95	-101	-74	-77
10-7-3	1.0426	1.0694	1.0672	0.9721	1.1438	+71	+97	+95	-101	-74	-77
10-7-2	1.0426	1.0694	1.0672	0.9721	1.1438	+71	+97	+95	-101	-74	-77
10-6-5	1.0370	1.0669	=	0.9693	1.1422	+68	+98	=	-105	-75	=
10-6-4	1.0370	1.0669	1.0666	0.9693	1.1422	+68	+98	+97	-105	-75	-76
10-6-3	1.0370	1.0669	1.0666	0.9693	1.1422	+68	+98	+97	-105	-75	-76
10-6-2	1.0370	1.0669	1.0666	0.9693	1.1422	+68	+98	+97	-105	-75	-76
10-5-4	1.0366	1.0666	=	0.9686	1.1415	+68	+98	=	-105	-75	=
10-5-3	1.0366	1.0666	1.0666	0.9686	1.1415	+68	+98	+98	-105	-75	-75
10-5-2	1.0366	1.0666	1.0666	0.9686	1.1415	+68	+98	+98	-105	-75	-75
10-4-3	1.0366	1.0666	=	0.9686	1.1415	+68	+98	=	-105	-75	=
10-4-2	1.0366	1.0666	1.0666	0.9686	1.1415	+68	+98	+98	-105	-75	-75
10-3-2	1.0366	1.0666	=	0.9686	1.1415	+68	+98	=	-105	-75	=
9-8-7	1.0578	=	=	0.9079	1.0578	+150	=	=	0	=	=
9-8-6	1.0524	=	1.0342	0.9079	1.0570	+144	=	+126	-5	=	-23
9-8-5	1.0479	=	1.0306	0.9079	1.0567	+140	=	+123	-9	=	-26
9-8-4	1.0476	=	1.0300	0.9079	1.0566	+140	=	+122	-9	=	-27
9-8-3	1.0476	=	1.0300	0.9079	1.0566	+140	=	+122	-9	=	-27
9-8-2	1.0476	=	1.0300	0.9079	1.0566	+140	=	+122	-9	=	-27
9-7-6	1.0259	1.0342	=	0.9069	1.0540	+119	+127	=	-28	-20	=
9-7-5	1.0254	1.0333	1.0300	0.9069	1.0537	+118	+126	+123	-28	-20	-24
9-7-4	1.0252	1.0331	1.0293	0.9069	1.0535	+118	+126	+122	-28	-20	-24
9-7-3	1.0252	1.0331	1.0293	0.9069	1.0535	+118	+126	+122	-28	-20	-24
9-7-2	1.0252	1.0331	1.0293	0.9069	1.0535	+118	+126	+122	-28	-20	-24
9-6-5	1.0172	1.0297	=	0.9058	1.0508	+111	+124	=	-34	-21	=
9-6-4	1.0172	1.0297	1.0291	0.9058	1.0508	+111	+124	+123	-34	-21	-22
9-6-3	1.0172	1.0297	1.0291	0.9058	1.0508	+111	+124	+123	-34	-21	-22
9-6-2	1.0172	1.0297	1.0291	0.9058	1.0508	+111	+124	+123	-34	-21	-22
9-5-4	1.0164	1.0291	=	0.9052	1.0504	+111	+124	=	-34	-21	=
9-5-3	1.0164	1.0291	1.0291	0.9052	1.0504	+111	+124	+124	-34	-21	-21
9-5-2	1.0164	1.0291	1.0291	0.9052	1.0504	+111	+124	+124	-34	-21	-21
9-4-3	1.0164	1.0291	=	0.9052	1.0504	+111	+124	=	-34	-21	=
9-4-2	1.0164	1.0291	1.0291	0.9052	1.0504	+111	+124	+124	-34	-21	-21
9-3-2	1.0164	1.0291	=	0.9052	1.0504	+111	+124	=	-34	-21	=

Study 8C17   Main

Top   Tripleton Leads

Eight or Lower Holdings

This table shows the results of the 35 tripletons headed by the eight or lower. The best lead (high, middle, low) from each holding is tinted gold, and when three distinct options exist, the second-best is tinted blue.

Comparisons versus partner leading always show a gain, for the same reason as jack, 10 and nine holdings. Losses (if any) versus declarer leading are very small, and 15 of the holdings are completely safe (cannot lose a trick).

Holding	High	Middle	Low	Partner	Declarer	High v P	Mid v P	Low v P	High v D	Mid v D	Low v D
8-7-6	1.0215	=	=	0.8792	1.0215	+142	=	=	0	=	=
8-7-5	1.0208	=	1.0124	0.8792	1.0215	+142	=	+133	-1	=	-9
8-7-4	1.0206	=	1.0116	0.8792	1.0215	+141	=	+132	-1	=	-10
8-7-3	1.0206	=	1.0116	0.8792	1.0215	+141	=	+132	-1	=	-10
8-7-2	1.0206	=	1.0116	0.8792	1.0215	+141	=	+132	-1	=	-10
8-6-5	1.0118	1.0124	=	0.8792	1.0211	+133	+133	=	-9	-9	=
8-6-4	1.0118	1.0124	1.0116	0.8792	1.0211	+133	+133	+132	-9	-9	-10
8-6-3	1.0118	1.0124	1.0116	0.8792	1.0211	+133	+133	+132	-9	-9	-10
8-6-2	1.0118	1.0124	1.0116	0.8792	1.0211	+133	+133	+132	-9	-9	-10
8-5-4	1.0108	1.0116	=	0.8792	1.0209	+132	+132	=	-10	-9	=
8-5-3	1.0108	1.0116	1.0116	0.8792	1.0209	+132	+132	+132	-10	-9	-9
8-5-2	1.0108	1.0116	1.0116	0.8792	1.0209	+132	+132	+132	-10	-9	-9
8-4-3	1.0108	1.0116	=	0.8792	1.0209	+132	+132	=	-10	-9	=
8-4-2	1.0108	1.0116	1.0116	0.8792	1.0209	+132	+132	+132	-10	-9	-9
8-3-2	1.0108	1.0116	=	0.8792	1.0209	+132	+132	=	-10	-9	=
7-6-5	1.0102	=	=	0.8740	1.0102	+136	=	=	0	=	=
7-6-4	1.0102	=	1.0091	0.8740	1.0102	+136	=	+135	0	=	-1
7-6-3	1.0102	=	1.0091	0.8740	1.0102	+136	=	+135	0	=	-1
7-6-2	1.0102	=	1.0091	0.8740	1.0102	+136	=	+135	0	=	-1
7-5-4	1.0088	1.0091	=	0.8740	1.0102	+135	+135	=	-1	-1	=
7-5-3	1.0088	1.0091	1.0091	0.8740	1.0102	+135	+135	+135	-1	-1	-1
7-5-2	1.0088	1.0091	1.0091	0.8740	1.0102	+135	+135	+135	-1	-1	-1
7-4-3	1.0088	1.0091	=	0.8740	1.0102	+135	+135	=	-1	-1	=
7-4-2	1.0088	1.0091	1.0091	0.8740	1.0102	+135	+135	+135	-1	-1	-1
7-3-2	1.0088	1.0091	=	0.8740	1.0102	+135	+135	=	-1	-1	=
6-5-4	1.0088	=	=	0.8739	1.0088	+135	=	=	0	=	=
6-5-3	1.0088	=	1.0088	0.8739	1.0088	+135	=	+135	0	=	0
6-5-2	1.0088	=	1.0088	0.8739	1.0088	+135	=	+135	0	=	0
6-4-3	1.0088	1.0088	=	0.8739	1.0088	+135	+135	=	0	0	=
6-4-2	1.0088	1.0088	1.0088	0.8739	1.0088	+135	+135	+135	0	0	0
6-3-2	1.0088	1.0088	=	0.8739	1.0088	+135	+135	=	0	0	=
5-4-3	1.0088	=	=	0.8739	1.0088	+135	=	=	0	=	=
5-4-2	1.0088	=	1.0088	0.8739	1.0088	+135	=	+135	0	=	0
5-3-2	1.0088	1.0088	=	0.8739	1.0088	+135	+135	=	0	0	=
4-3-2	1.0088	=	=	0.8739	1.0088	+135	=	=	0	=	=

Study 8C17   Main

Top   Tripleton Leads

© 2018 Richard Pavlicek