Study 7A23 Main

Par for the Course

by Richard Pavlicek

What is the “par score” for a bridge deal? How is it calculated? You won’t find the answers in the classic, “Bubba Watson on the Play of the Hand,” but you might find them here. Most experienced players have a general idea of the logic, but few understand it completely. This study describes the algorithm I use to calculate par scores, the special case of par zero deals, and statistics of actual par scores.

Par Score Calculation Par Zero Deals Par Score Stats

Par Score Calculation

The first step to calculate the par score is to determine the highest makable contract by rank (not score) for each side. The side that makes the higher contract must be the board winner, because a plus score can be assured by bidding and making that contract or by defeating the opponents if they bid any further. In most cases this determines par immediately.

1. None vul Q
4
K 7 5 3 2
K J 10 6 5 3 Makes
North
South
West
East
Deal NT
10
=
3
=
26
9
=
4
=
26
8
=
5
=
26
9
=
3
=
24
12
=
1
=
26
K 9 6 3
J 5 2
J 10 9 6
7 2 Table J 8 5
Q 8 7 6 3
Q 8 4
A 9
A 10 7 4 2
A K 10 9
A
Q 8 4 Par: NS +920, 6 N or S

North or South can make 6 , either by ruffing two diamonds, or by establishing the long spade if the defense leads two rounds of trumps. No higher contract is makable by anyone, so this appears to be par.

Profitable sacrifice?

The next step is to determine if the losing side has a profitable sacrifice at the vulnerability, and if so, that sacrifice becomes the par score for the winning side. On the previous deal the most East-West could win is five tricks in hearts, so a sacrifice in 6 would be down seven doubled (minus 1700) which is hardly profitable to put it mildly. Now consider this deal:

2. Both vul A 9 7 5
9 5 4
6 3
K Q 7 3 Makes
North
South
West
East
Deal NT
3
=
7
=
20
2
=
10
=
24
9
=
4
=
26
2
=
10
=
24
9
=
4
=
26
Q 10 6 4 2
K 8
A 9 8 2
A 2 Table K J 8
7 2
K Q J 10 7
J 8 6
3
A Q J 10 6 3
5 4
10 9 5 4 Par: EW +500, 5 × S or N

East-West own the highest makable contract at 4 , but N-S can win nine tricks in hearts. Therefore, N-S do better to bid 5 , which when doubled surrenders only 500 instead of 620. Curiously, N-S could also sacrifice in clubs, but the score would be the same, which is all that matters.

Higher scoring contract?

If the losing side has no profitable sacrifice against the highest makable contract, it is necessary to check the winning side for alternate contracts that score higher. Three situations must be considered:

Highest make 5 or 5 → check for 4 NT, 4 or 4
Highest make 4 or 4 → check for 3 NT, 3 or 3
Highest make 3 or 3 → check for 2 NT

If a higher scoring contract exists, it does not necessarily become the par contract; a further check for sacrifices is required. The losing side may have a profitable save against the higher scoring contract but not against the higher ranking contract, in which case this new sacrifice becomes par, unless the winning side does better to take its score for the higher ranking contract. This is easier than it sounds, applying little more than common sense. Consider this deal:

3. E-W vul K 7 2
J 10 5
A K 8 2
A 4 2 Makes
North
South
West
East
Deal NT
9
=
4
=
26
9
=
4
=
26
5
=
6
=
22
10
=
2
=
24
4
=
9
=
26
10 8
A K 9 7 4
4
K Q J 9 3 Table J 9 5 3
8 2
7 6 5
10 8 7 5
A Q 6 4
Q 6 3
Q J 10 9 3
6 Par: NS +200, 4 × W or E

North-South own the highest make at 4 , against which E-W have no profitable sacrifice; but N-S also make 3 NT. Should the par score then be 400 for N-S? No, because if N-S bid 3 NT, E-W would have a profitable sacrifice in 4 , which becomes the par contract (doubled, down one). If E-W were nonvulnerable, 4 × would net only 100, so N-S do better to ignore it and take their 4 ; then par would be 130.

Study 7A23 Main Top Par for the Course

Par Zero Deals

Almost all practical cases of determining the par score are resolved by the preceding calculation, but a few loose ends still exist. What if neither side can make any contract? On the following deal no one can win more than six tricks in any strain. This is rare, but the obvious consequence is a par score of zero. A perfect passout!

4. None vul 7 6 5 4
8 3
9 5
A K Q J 10 Makes
North
South
West
East
Deal NT
5
=
3
=
16
5
=
5
=
20
4
=
6
=
20
6
=
3
=
18
6
=
3
=
18
A K Q 8 2
J
10 8 3
9 5 3 2 Table 10
A K Q 9 7 2
7 4 2
8 6 4
J 9 3
10 6 5 4
A K Q J 6
7 Par: Zero, Passout

Another possible situation, even more unlikely, is both sides making the same contract. Consider the following symmetric deal in which everybody makes 1 NT. Who deserves the 90 points? One possibility is to assign plus par to the dealing side, because they have the first opportunity to bid 1 NT, but this seems unfair.

5. None vul 10 8
A 5 4
Q J 3 2
K 9 7 6 Makes
North
South
West
East
Deal NT
7
=
7
=
28
6
=
6
=
24
6
=
6
=
24
6
=
6
=
24
6
=
6
=
24
A 5 4
Q J 3 2
K 9 7 6
10 8 Table K 9 7 6
10 8
A 5 4
Q J 3 2
Q J 3 2
K 9 7 6
10 8
A 5 4 Par: Zero, Passout

My decision is to assign “par zero” to any deal in which the highest make for both sides is the same contract. Vulnerability does not matter. The deal is like a jump ball in basketball; whoever grabs the opportunity beats par.

Enter the bizarre

Consider the following deal I composed for my Optimum Contract puzzle, where the object was to find the best contracts for N-S and E-W. Remarkably, 3 is the best contract for both sides. South and East both win nine tricks in spades against any defense. Hard to believe, but proof is in the play.

6. None vul 9
J 10 9
A Q 4 3 2
A K Q 2 Makes
North
South
West
East
Deal NT
5
=
8
=
26
4
9
4
9
26
5
=
8
=
26
4
5
8
9
26
5
=
8
=
26
—
—
J 10 9 8 7 6 5
8 7 6 5 4 3 Table A K Q 8
A K Q 8 7
K
J 10 9
J 10 7 6 5 4 3 2
6 5 4 3 2
—
— Par: Zero, Passout

For the ultimate fantasy, witness this deal from my Fewest HCP To Make Notrump record book. With only 11 HCP, South is cold for 7 NT with any lead. Strange enough, but the eerie fact is that so is West. Therefore, to keep in line with my arbitrary rule, par remains at zero. Bid 7 NT first to claim your prize!

7. None vul —
—
8 7 6 5 4 3 2
7 6 5 4 3 2 Makes
North
South
West
East
Deal NT
0
13
13
0
26
12
13
1
0
26
1
2
12
11
26
1
2
12
11
26
0
=
13
=
26
K
K Q J 10 9 8 7 6 5 4 3 2
—
— Table —
—
A K Q J 10 9
A K Q J 10 9 8
A Q J 10 9 8 7 6 5 4 3 2
A
—
— Par: Zero, Passout

Somehow, I wouldn’t predict a passout. East will surely bid 7 , and South will bid 7 not realizing he has left West a chance to be a hero. But hey, maybe East will try to be the hero. In any case, both sides making 7 NT may be the short route to the looney bin.

Study 7A23 Main Top Par for the Course

Par Score Stats

Source for this par-score analysis was my database of 10,485,760 random deals, each of which is solved 20 times (4 hands x 5 strains) with double-dummy play. Separate analyses were made for each of the four vulnerability conditions, because this affects the number and value of par scores. Results are summarized below.

Par scores are considered as absolute values, regardless of which side is plus, or neither side if par zero. Note that the first four rows of the table are the same for each column, since vulnerability has no effect on which side is plus to par. The number of deals with N-S plus or E-W plus (Rows 2 and 3) are equal in theory, but only bear close proximity in this small sample. Yes, 10+ million is small, indeed infinitesimal in relation to 53+ octillion possible bridge deals.

Summary None Vul Both Vul N-S Vul E-W Vul
Deals analyzed 10,485,760 10,485,760 10,485,760 10,485,760
Par plus N-S 5,241,511 5,241,511 5,241,511 5,241,511
Par plus E-W 5,243,995 5,243,995 5,243,995 5,243,995
Par zero 254 254 254 254
Unique scores 25 26 38 38
Minimum 0 0 0 0
Maximum 1520 2220 2220 2220
Mode 100 140 140 140
Median 400 600 400 400
Mean 372 530 441 441
Standard deviation 317 475 403 403

Summary	None Vul	Both Vul	N-S Vul	E-W Vul
Deals analyzed	10,485,760	10,485,760	10,485,760	10,485,760
Par plus N-S	5,241,511	5,241,511	5,241,511	5,241,511
Par plus E-W	5,243,995	5,243,995	5,243,995	5,243,995
Par zero	254	254	254	254
Unique scores	25	26	38	38
Minimum	0	0	0	0
Maximum	1520	2220	2220	2220
Mode	100	140	140	140
Median	400	600	400	400
Mean	372	530	441	441
Standard deviation	317	475	403	403

In duplicate bridge there are 205 possible absolute scores, but only 25 to 38 of them (depending on vulnerability) are possible as par scores. For instance, common bridge scores of 50, 150 and 170 can never be par scores.

The following four tables show the breakdown of absolute par scores by vulnerability condition. Percents of par score occurrence are shown in three ways: for the specific score, at least that score, and at most that score. To find the chance of multiple scores (e.g., 110 to 140) add the specific percents for each score.

None Vul
Score Deals Specific At Least At Most
0 254 0.0024 100 0.0024
70 544 0.0052 99.9976 0.0076
80 25,534 0.2435 99.9924 0.2511
90 311,654 2.9722 99.7489 3.2233
100 1,703,194 16.2429 96.7767 19.4662
110 471,717 4.4986 80.5338 23.9649
120 426,000 4.0627 76.0351 28.0275
130 396,295 3.7794 71.9725 31.8069
140 1,126,959 10.7475 68.1931 42.5544
300 574,095 5.4750 57.4456 48.0294
400 857,017 8.1732 51.9706 56.2025
420 1,172,303 11.1800 43.7975 67.3825
430 635,759 6.0631 32.6175 73.4456
450 940,380 8.9682 26.5544 82.4137
460 440,033 4.1965 17.5863 86.6102
500 42,423 0.4046 13.3898 87.0148
800 92,577 0.8829 12.9852 87.8977
920 231,696 2.2096 12.1023 90.1073
980 379,389 3.6181 9.8927 93.7254
990 354,495 3.3807 6.2746 97.1062
1100 7832 0.0747 2.8938 97.1808
1400 11,700 0.1116 2.8192 97.2924
1440 50,858 0.4850 2.7076 97.7774
1510 87,483 0.8343 2.2226 98.6117
1520 145,569 1.3883 1.3883 100
U = 25 10,485,760 100 2600

None Vul
Score	Deals	Specific	At Least	At Most
0	254	0.0024	100	0.0024
70	544	0.0052	99.9976	0.0076
80	25,534	0.2435	99.9924	0.2511
90	311,654	2.9722	99.7489	3.2233
100	1,703,194	16.2429	96.7767	19.4662
110	471,717	4.4986	80.5338	23.9649
120	426,000	4.0627	76.0351	28.0275
130	396,295	3.7794	71.9725	31.8069
140	1,126,959	10.7475	68.1931	42.5544
300	574,095	5.4750	57.4456	48.0294
400	857,017	8.1732	51.9706	56.2025
420	1,172,303	11.1800	43.7975	67.3825
430	635,759	6.0631	32.6175	73.4456
450	940,380	8.9682	26.5544	82.4137
460	440,033	4.1965	17.5863	86.6102
500	42,423	0.4046	13.3898	87.0148
800	92,577	0.8829	12.9852	87.8977
920	231,696	2.2096	12.1023	90.1073
980	379,389	3.6181	9.8927	93.7254
990	354,495	3.3807	6.2746	97.1062
1100	7832	0.0747	2.8938	97.1808
1400	11,700	0.1116	2.8192	97.2924
1440	50,858	0.4850	2.7076	97.7774
1510	87,483	0.8343	2.2226	98.6117
1520	145,569	1.3883	1.3883	100
U = 25	10,485,760	100	2600

Both Vul
Score Deals Specific At Least At Most
0 254 0.0024 100 0.0024
70 544 0.0052 99.9976 0.0076
80 25,534 0.2435 99.9924 0.2511
90 311,654 2.9722 99.7489 3.2233
110 1,125,546 10.7340 96.7767 13.9573
120 596,794 5.6915 86.0427 19.6488
130 494,735 4.7182 80.3512 24.3670
140 1,626,205 15.5087 75.6330 39.8757
200 280,305 2.6732 60.1243 42.5489
500 574,095 5.4750 57.4511 48.0239
600 857,597 8.1787 51.9761 56.2025
620 1,172,303 11.1800 43.7975 67.3825
630 635,759 6.0631 32.6175 73.4456
650 940,380 8.9682 26.5544 82.4137
660 440,033 4.1965 17.5863 86.6102
800 42,423 0.4046 13.3898 87.0148
1100 92,577 0.8829 12.9852 87.8977
1370 231,696 2.2096 12.1023 90.1073
1400 96,494 0.9202 9.8927 91.0275
1430 305,448 2.9130 8.9725 93.9405
1440 339,774 3.2403 6.0595 97.1808
1700 11,700 0.1116 2.8192 97.2924
2000 14,757 0.1407 2.7076 97.4332
2140 39,650 0.3781 2.5668 97.8113
2210 83,934 0.8005 2.1887 98.6117
2220 145,569 1.3883 1.3883 100
U = 26 10,485,760 100 2700

Both Vul
Score	Deals	Specific	At Least	At Most
0	254	0.0024	100	0.0024
70	544	0.0052	99.9976	0.0076
80	25,534	0.2435	99.9924	0.2511
90	311,654	2.9722	99.7489	3.2233
110	1,125,546	10.7340	96.7767	13.9573
120	596,794	5.6915	86.0427	19.6488
130	494,735	4.7182	80.3512	24.3670
140	1,626,205	15.5087	75.6330	39.8757
200	280,305	2.6732	60.1243	42.5489
500	574,095	5.4750	57.4511	48.0239
600	857,597	8.1787	51.9761	56.2025
620	1,172,303	11.1800	43.7975	67.3825
630	635,759	6.0631	32.6175	73.4456
650	940,380	8.9682	26.5544	82.4137
660	440,033	4.1965	17.5863	86.6102
800	42,423	0.4046	13.3898	87.0148
1100	92,577	0.8829	12.9852	87.8977
1370	231,696	2.2096	12.1023	90.1073
1400	96,494	0.9202	9.8927	91.0275
1430	305,448	2.9130	8.9725	93.9405
1440	339,774	3.2403	6.0595	97.1808
1700	11,700	0.1116	2.8192	97.2924
2000	14,757	0.1407	2.7076	97.4332
2140	39,650	0.3781	2.5668	97.8113
2210	83,934	0.8005	2.1887	98.6117
2220	145,569	1.3883	1.3883	100
U = 26	10,485,760	100	2700

N-S Vul
Score Deals Specific At Least At Most
0 254 0.0024 100 0.0024
70 544 0.0052 99.9976 0.0076
80 25,534 0.2435 99.9924 0.2511
90 311,654 2.9722 99.7489 3.2233
100 850,820 8.1141 96.7767 11.3373
110 799,092 7.6207 88.6627 18.9581
120 511,183 4.8750 81.0419 23.8331
130 445,768 4.2512 76.1669 28.0843
140 1,376,763 13.1298 71.9157 41.2141
200 140,241 1.3374 58.7859 42.5515
300 287,045 2.7375 57.4485 45.2890
400 506,497 4.8303 54.7110 50.1194
420 741,276 7.0694 49.8806 57.1887
430 329,155 3.1391 42.8113 60.3278
450 503,739 4.8040 39.6722 65.1318
460 221,220 2.1097 34.8682 67.2415
500 464,681 4.4315 32.7585 71.6731
600 295,134 2.8146 28.3269 74.4877
620 388,047 3.7007 25.5123 78.1884
630 286,214 2.7295 21.8116 80.9179
650 402,711 3.8406 19.0821 84.7585
660 215,268 2.0530 15.2415 86.8114
800 67,398 0.6428 13.1886 87.4542
920 136,161 1.2985 12.5458 88.7527
980 211,293 2.0150 11.2473 90.7678
990 179,206 1.7090 9.2322 92.4768
1100 80,221 0.7650 7.5232 93.2419
1370 86,306 0.8231 6.7581 94.0650
1400 75,703 0.7220 5.9350 94.7869
1430 104,401 0.9956 5.2131 95.7826
1440 182,287 1.7384 4.2174 97.5210
1510 45,358 0.4326 2.4790 97.9535
1520 72,677 0.6931 2.0465 98.6467
1700 7313 0.0697 1.3533 98.7164
2000 7979 0.0761 1.2836 98.7925
2140 13,688 0.1305 1.2075 98.9230
2210 40,037 0.3818 1.0770 99.3048
2220 72,892 0.6952 0.6952 100
U = 38 10,485,760 100 3900

N-S Vul
Score	Deals	Specific	At Least	At Most
0	254	0.0024	100	0.0024
70	544	0.0052	99.9976	0.0076
80	25,534	0.2435	99.9924	0.2511
90	311,654	2.9722	99.7489	3.2233
100	850,820	8.1141	96.7767	11.3373
110	799,092	7.6207	88.6627	18.9581
120	511,183	4.8750	81.0419	23.8331
130	445,768	4.2512	76.1669	28.0843
140	1,376,763	13.1298	71.9157	41.2141
200	140,241	1.3374	58.7859	42.5515
300	287,045	2.7375	57.4485	45.2890
400	506,497	4.8303	54.7110	50.1194
420	741,276	7.0694	49.8806	57.1887
430	329,155	3.1391	42.8113	60.3278
450	503,739	4.8040	39.6722	65.1318
460	221,220	2.1097	34.8682	67.2415
500	464,681	4.4315	32.7585	71.6731
600	295,134	2.8146	28.3269	74.4877
620	388,047	3.7007	25.5123	78.1884
630	286,214	2.7295	21.8116	80.9179
650	402,711	3.8406	19.0821	84.7585
660	215,268	2.0530	15.2415	86.8114
800	67,398	0.6428	13.1886	87.4542
920	136,161	1.2985	12.5458	88.7527
980	211,293	2.0150	11.2473	90.7678
990	179,206	1.7090	9.2322	92.4768
1100	80,221	0.7650	7.5232	93.2419
1370	86,306	0.8231	6.7581	94.0650
1400	75,703	0.7220	5.9350	94.7869
1430	104,401	0.9956	5.2131	95.7826
1440	182,287	1.7384	4.2174	97.5210
1510	45,358	0.4326	2.4790	97.9535
1520	72,677	0.6931	2.0465	98.6467
1700	7313	0.0697	1.3533	98.7164
2000	7979	0.0761	1.2836	98.7925
2140	13,688	0.1305	1.2075	98.9230
2210	40,037	0.3818	1.0770	99.3048
2220	72,892	0.6952	0.6952	100
U = 38	10,485,760	100	3900

E-W Vul
Score Deals Specific At Least At Most
0 254 0.0024 100 0.0024
70 544 0.0052 99.9976 0.0076
80 25,534 0.2435 99.9924 0.2511
90 311,654 2.9722 99.7489 3.2233
100 852,374 8.1289 96.7767 11.3522
110 798,171 7.6120 88.6478 18.9641
120 511,611 4.8791 81.0359 23.8432
130 445,262 4.2463 76.1568 28.0896
140 1,376,401 13.1264 71.9104 41.2159
200 140,064 1.3358 58.7841 42.5517
300 287,050 2.7375 57.4483 45.2892
400 507,378 4.8387 54.7108 50.1280
420 742,218 7.0783 49.8720 57.2063
430 329,950 3.1466 42.7937 60.3529
450 502,760 4.7947 39.6471 65.1476
460 221,243 2.1099 34.8524 67.2576
500 465,264 4.4371 32.7424 71.6947
600 293,876 2.8026 28.3053 74.4973
620 386,854 3.6893 25.5027 78.1866
630 285,765 2.7253 21.8134 80.9119
650 403,453 3.8476 19.0881 84.7595
660 215,379 2.0540 15.2405 86.8135
800 67,602 0.6447 13.1865 87.4582
920 136,400 1.3008 12.5418 88.7591
980 211,021 2.0125 11.2409 90.7715
990 180,089 1.7175 9.2285 92.4890
1100 79,335 0.7566 7.5110 93.2456
1370 86,315 0.8232 6.7544 94.0687
1400 75,452 0.7196 5.9313 94.7883
1430 104,952 1.0009 5.2117 95.7892
1440 181,405 1.7300 4.2108 97.5192
1510 45,237 0.4314 2.4808 97.9506
1520 72,892 0.6952 2.0494 98.6458
1700 7444 0.0710 1.3542 98.7168
2000 8077 0.0770 1.2832 98.7938
2140 13,686 0.1305 1.2062 98.9243
2210 40,117 0.3826 1.0757 99.3069
2220 72,677 0.6931 0.6931 100
U = 38 10,485,760 100 3900

E-W Vul
Score	Deals	Specific	At Least	At Most
0	254	0.0024	100	0.0024
70	544	0.0052	99.9976	0.0076
80	25,534	0.2435	99.9924	0.2511
90	311,654	2.9722	99.7489	3.2233
100	852,374	8.1289	96.7767	11.3522
110	798,171	7.6120	88.6478	18.9641
120	511,611	4.8791	81.0359	23.8432
130	445,262	4.2463	76.1568	28.0896
140	1,376,401	13.1264	71.9104	41.2159
200	140,064	1.3358	58.7841	42.5517
300	287,050	2.7375	57.4483	45.2892
400	507,378	4.8387	54.7108	50.1280
420	742,218	7.0783	49.8720	57.2063
430	329,950	3.1466	42.7937	60.3529
450	502,760	4.7947	39.6471	65.1476
460	221,243	2.1099	34.8524	67.2576
500	465,264	4.4371	32.7424	71.6947
600	293,876	2.8026	28.3053	74.4973
620	386,854	3.6893	25.5027	78.1866
630	285,765	2.7253	21.8134	80.9119
650	403,453	3.8476	19.0881	84.7595
660	215,379	2.0540	15.2405	86.8135
800	67,602	0.6447	13.1865	87.4582
920	136,400	1.3008	12.5418	88.7591
980	211,021	2.0125	11.2409	90.7715
990	180,089	1.7175	9.2285	92.4890
1100	79,335	0.7566	7.5110	93.2456
1370	86,315	0.8232	6.7544	94.0687
1400	75,452	0.7196	5.9313	94.7883
1430	104,952	1.0009	5.2117	95.7892
1440	181,405	1.7300	4.2108	97.5192
1510	45,237	0.4314	2.4808	97.9506
1520	72,892	0.6952	2.0494	98.6458
1700	7444	0.0710	1.3542	98.7168
2000	8077	0.0770	1.2832	98.7938
2140	13,686	0.1305	1.2062	98.9243
2210	40,117	0.3826	1.0757	99.3069
2220	72,677	0.6931	0.6931	100
U = 38	10,485,760	100	3900

Mathematical check total of At Least + At Most columns = 100(U+1).

Study 7A23 Main Top Par for the Course

1. None vul	Q 4 K 7 5 3 2 K J 10 6 5 3		Makes `North` `South` `West` `East` Deal	NT 10 = 3 = 26	9 = 4 = 26	8 = 5 = 26	9 = 3 = 24	12 = 1 = 26
K 9 6 3 J 5 2 J 10 9 6 7 2		J 8 5 Q 8 7 6 3 Q 8 4 A 9
	A 10 7 4 2 A K 10 9 A Q 8 4		Par: NS +920, 6 N or S

2. Both vul	A 9 7 5 9 5 4 6 3 K Q 7 3		Makes North South West East Deal	NT 3 = 7 = 20	2 = 10 = 24	9 = 4 = 26	2 = 10 = 24	9 = 4 = 26
Q 10 6 4 2 K 8 A 9 8 2 A 2		K J 8 7 2 K Q J 10 7 J 8 6
	3 A Q J 10 6 3 5 4 10 9 5 4		Par: EW +500, 5 × S or N

3. E-W vul	K 7 2 J 10 5 A K 8 2 A 4 2		Makes `North` `South` West East Deal	NT 9 = 4 = 26	9 = 4 = 26	5 = 6 = 22	10 = 2 = 24	4 = 9 = 26
10 8 A K 9 7 4 4 K Q J 9 3		J 9 5 3 8 2 7 6 5 10 8 7 5
	A Q 6 4 Q 6 3 Q J 10 9 3 6		Par: NS +200, 4 × W or E

4. None vul	7 6 5 4 8 3 9 5 A K Q J 10		Makes `North` `South` `West` `East` Deal	NT 5 = 3 = 16	5 = 5 = 20	4 = 6 = 20	6 = 3 = 18	6 = 3 = 18
A K Q 8 2 J 10 8 3 9 5 3 2		10 A K Q 9 7 2 7 4 2 8 6 4
	J 9 3 10 6 5 4 A K Q J 6 7		Par: Zero, Passout

5. None vul	10 8 A 5 4 Q J 3 2 K 9 7 6		Makes `North` `South` `West` `East` Deal	NT 7 = 7 = 28	6 = 6 = 24	6 = 6 = 24	6 = 6 = 24	6 = 6 = 24
A 5 4 Q J 3 2 K 9 7 6 10 8		K 9 7 6 10 8 A 5 4 Q J 3 2
	Q J 3 2 K 9 7 6 10 8 A 5 4		Par: Zero, Passout

6. None vul	9 J 10 9 A Q 4 3 2 A K Q 2		Makes `North` `South` `West` `East` Deal	NT 5 = 8 = 26	4 9 4 9 26	5 = 8 = 26	4 5 8 9 26	5 = 8 = 26
— — J 10 9 8 7 6 5 8 7 6 5 4 3		A K Q 8 A K Q 8 7 K J 10 9
	J 10 7 6 5 4 3 2 6 5 4 3 2 — —		Par: Zero, Passout

7. None vul	— — 8 7 6 5 4 3 2 7 6 5 4 3 2		Makes `North` `South` `West` `East` Deal	NT 0 13 13 0 26	12 13 1 0 26	1 2 12 11 26	1 2 12 11 26	0 = 13 = 26
K K Q J 10 9 8 7 6 5 4 3 2 — —		— — A K Q J 10 9 A K Q J 10 9 8
	A Q J 10 9 8 7 6 5 4 3 2 A — —		Par: Zero, Passout