Study 8J37 Main

# Hits and Misses

by Richard Pavlicek

Bridge players are a strange breed. Besides their addiction to this great card game, many enjoy looking for bizarre characteristics in a bridge hand. This study — a wasted effort in productivity if there ever was one — will analyze some of these quirks.

I may occasionally add to this stupidity, I mean study, so if you have a favorite diversion, pass it along.

## Magnus Hits

Magnus Olafsson of New York City (originally Iceland) enjoys looking for suits in which the lowest card describes the suit length. For example, K-J-7-4 qualifies because the lowest card is a four in a four-card suit. A bridge hand can have as many as four hits, e.g., A-Q-9-7-5 8-6-3 K-10-3 J-2, where the lowest cards in each suit identify the 5=3=3=2 shape.

Of the 8192 possible suit holdings, only 377 qualify as Magnus hits. The longest suit to qualify is seven cards, i.e., A-K-Q-J-10-9-7 is one of seven possible holdings. Obviously an eight-card suit could not have the eight as its lowest card.

A singleton qualifies only if it’s the ace, which is arbitrarily considered “one” to meet the rule. A void also qualifies, because having “no lowest card” matches having “no cards” — or at least we’ll take Magnus’s word for it. Reminds me of being told by my math teacher for the first time that “zero factorial equals 1.” I thought he had lost his marbles.

The following table shows the percent chance of each number of Magnus hits for each generic hand pattern. Hit percents are relative to the generic pattern in that row. Total percents (last column) are relative to all bridge hands. The bottom row shows the percents for all generic patterns combined. Percents shown as 0.0000 are not zero but round that way to the nearest 10,000th.

Percents for Magnus Hits
PatternNo Hit1 Hit2 Hits3 Hits4 HitsTotal
4-4-3-247.327938.963911.99011.63480.083321.5512
5-3-3-249.796138.170510.69031.28750.055715.5168
5-4-3-155.969135.35217.91620.73820.024312.9307
5-4-2-253.748136.21799.01620.97900.038910.5797
4-3-3-343.847940.163213.77372.09580.119410.5361
6-3-2-255.941735.54917.82830.66610.01495.6425
6-4-2-162.876631.47405.30530.33770.00654.7021
6-3-3-158.253334.55936.71350.46460.00933.4482
5-5-2-163.561630.64985.36980.40750.01133.1739
4-4-4-153.195036.61819.17320.97730.03642.9932
7-3-2-162.828731.87575.04790.24680.00091.8808
6-4-3-0065.020130.78774.08680.10531.3262
5-4-4-0062.470631.97805.27570.27581.2433
5-5-3-0065.728429.91634.17100.18430.8952
6-5-1-174.356722.97092.55160.11880.00190.7053
6-5-2-0073.840423.93022.18030.04920.6511
7-2-2-260.335533.15746.11850.38720.00150.5130
7-4-1-170.617426.30102.97060.11060.00040.3918
7-4-2-0070.127027.18082.68170.01050.3617
7-3-3-0064.970631.20453.80990.01510.2652
8-2-2-166.090129.54034.18750.182100.1924
8-3-1-168.821127.85623.20890.113800.1176
7-5-1-0082.930916.25080.81520.00310.1085
8-3-2-0068.343228.69822.958600.1085
6-6-1-0086.381313.02660.58400.00820.0723
8-4-1-0076.815521.89351.291000.0452
9-2-1-172.098325.12522.68550.091000.0178
9-3-1-0074.556223.96451.479300.0100
9-2-2-0071.597626.03552.366900.0082
7-6-0-00096.34203.64470.01330.0056
8-5-0-00090.20989.790200.0031
10-2-1-0078.106520.71011.183400.0011
9-4-0-00083.216816.783200.0010
10-1-1-178.652719.66321.63860.045500.0004
10-3-0-00080.769219.230800.0002
11-1-1-0085.207114.20120.591700.0000
11-2-0-00084.615415.384600.0000
12-1-0-00092.30777.692300.0000
13-0-0-000010000.0000
All 3949.714838.239610.70011.28930.0563100

Exact Hand Counts for Magnus Hits
PatternNo Hit1 Hit2 Hits3 Hits4 HitsTotal
4-4-3-264,769,581,80053,323,162,20016,408,854,0002,237,241,600114,048,000136,852,887,600
5-3-3-249,066,079,83237,610,906,25610,533,565,9441,268,641,44054,885,60098,534,079,072
5-4-3-145,957,209,76029,028,245,4006,500,130,120606,188,88019,958,40082,111,732,560
5-4-2-236,109,236,24024,331,998,2406,057,270,720657,694,08026,127,36067,182,326,640
4-3-3-329,336,810,58026,871,532,3809,215,444,7001,402,208,50079,860,00066,905,856,160
6-3-2-220,044,221,12012,737,431,8722,804,934,528238,664,4485,322,24035,830,574,208
6-4-2-118,774,201,6009,397,748,1601,584,092,160100,834,5601,935,36029,858,811,840
6-3-3-112,755,413,4407,567,260,6241,470,028,032101,727,1202,032,80021,896,462,016
5-5-2-112,810,641,1846,177,365,2081,082,267,56882,137,8882,286,14420,154,697,992
4-4-4-110,110,954,0006,960,131,5001,743,588,000185,760,0006,912,00019,007,345,500
7-3-2-17,503,959,2323,807,079,056602,898,91229,476,656110,88011,943,524,736
6-4-3-005,475,808,8002,592,854,880344,182,0808,870,4008,421,716,160
5-4-4-004,932,276,3002,524,775,400416,534,40021,772,8007,895,358,900
5-5-3-003,736,437,0121,700,637,444237,105,79210,478,1605,684,658,408
6-5-1-13,330,305,2801,028,827,008114,282,5765,322,24084,6724,478,821,776
6-5-2-003,052,779,840989,345,66490,139,3922,032,1284,134,297,024
7-2-2-21,965,322,6561,080,044,064199,298,88012,610,94448,3843,257,324,928
7-4-1-11,757,125,440654,429,60073,916,3402,752,86010,0802,488,234,320
7-4-2-001,610,698,320624,298,32061,593,120241,9202,296,831,680
7-3-3-001,094,327,388525,590,60464,171,140254,1001,684,343,232
8-2-2-1807,288,768360,833,61651,150,5282,223,93601,221,496,848
8-3-1-1513,729,216207,938,01623,953,644849,4200746,470,296
7-5-1-00571,434,912111,976,3445,617,08021,168689,049,504
8-3-2-00470,918,448197,744,97620,386,0800689,049,504
6-6-1-00396,806,40059,839,6802,682,62437,632459,366,336
8-4-1-00220,540,32062,857,0803,706,5600287,103,960
9-2-1-181,544,32028,416,9603,037,320102,9600113,101,560
9-3-1-0047,567,52015,289,560943,800063,800,880
9-2-2-0037,374,48013,590,7201,235,520052,200,720
7-6-0-00034,043,2801,287,888470435,335,872
8-5-0-00017,930,4841,945,944019,876,428
10-2-1-005,436,2881,441,44082,36806,960,096
9-4-0-0005,105,1001,029,60006,134,700
10-1-1-11,976,832494,20841,184114402,513,368
10-3-0-000792,792188,7600981,552
11-1-1-00134,78422,4649360158,184
11-2-0-00061,77611,232073,008
12-1-0-000187215602028
13-0-0-0000404
All 39315,695,601,300242,826,385,18067,946,955,0368,187,283,152357,334,932635,013,559,600

 Study 8J37 Main Top Hits and Misses

## Symmetric Suits

Peter Boyd of Darnestown, Maryland — one of the truly great players and a former world champion — enjoys looking for suits that are symmetric. That is, the rank differentials remain the same if the suit is turned upside-down. For example, A-K-3-2 has the two highest and two lowest cards, so flipping would not change it. Other examples are Q-4, J-8-5, K-J-5-3 and A-Q-8-4-2.

Of the 8192 possible suit holdings, only 128 are symmetric. Any odd-length symmetric suit must contain the eight, the middle card. Hence the only singleton to qualify is a singleton eight. A void suit certainly qualifies: Turning a void upside-down wouldn’t change it — or if it did, Penn and Teller might have a new act.

The following table shows the percent chance of each number of symmetric hits for each generic hand pattern. Hit percents are relative to the generic pattern in that row. Total percents (last column) are relative to all bridge hands. The bottom row shows the percents for all generic patterns combined. Percents shown as 0.0000 are not zero but round that way to the nearest 10,000th.

Percents for Symmetric Suits
PatternNo Hit1 Hit2 Hits3 Hits4 HitsTotal
4-4-3-286.619112.78660.58340.01080.000121.5512
5-3-3-287.444112.06580.48260.00750.000015.5168
5-4-3-187.444112.06580.48260.00750.000012.9307
5-4-2-282.447316.48021.04990.02250.000110.5797
4-3-3-391.86887.87450.25310.00360.000010.5361
6-3-2-282.447316.48021.04990.02250.00015.6425
6-4-2-182.447316.48021.04990.02250.00014.7021
6-3-3-187.444112.06580.48260.00750.00003.4482
5-5-2-183.232515.83510.91670.01560.00013.1739
4-4-4-186.619112.78660.58340.01080.00012.9932
7-3-2-182.447316.48021.04990.02250.00011.8808
6-4-3-0094.73115.17700.09140.00051.3262
5-4-4-0094.73115.17700.09140.00051.2433
5-5-3-0095.63334.30480.06160.00030.8952
6-5-1-183.232515.83510.91670.01560.00010.7053
6-5-2-0090.16859.64070.18980.00100.6511
7-2-2-277.736020.35071.84870.06410.00050.5130
7-4-1-182.447316.48021.04990.02250.00010.3918
7-4-2-0089.317910.41040.26980.00190.3617
7-3-3-0094.73115.17700.09140.00050.2652
8-2-2-177.736020.35071.84870.06410.00050.1924
8-3-1-182.447316.48021.04990.02250.00010.1176
7-5-1-0090.16859.64070.18980.00100.1085
8-3-2-0089.317910.41040.26980.00190.1085
6-6-1-0090.16859.64070.18980.00100.0723
8-4-1-0089.317910.41040.26980.00190.0452
9-2-1-177.002720.90072.01670.07890.00100.0178
9-3-1-0088.475311.16470.35660.00340.0100
9-2-2-0083.419515.69080.87720.01240.0082
7-6-0-00097.68262.30380.01360.0056
8-5-0-00097.68262.30380.01360.0031
A-2-1-0083.419515.69080.87720.01240.0011
9-4-0-00095.84824.10780.04400.0010
10-1-1-177.002720.90072.01670.07890.00100.0004
10-3-0-00095.84824.10780.04400.0002
11-1-1-0078.652719.66321.63860.04550.0000
11-2-0-00085.207114.20120.59170.0000
12-1-0-00085.207114.20120.59170.0000
13-0-0-000001000.0000
All 3981.843717.18910.94880.01820.0001100

Exact Hand Counts for Symmetric Suits
PatternNo Hit1 Hit2 Hits3 Hits4 HitsTotal
4-4-3-2118,540,800,00017,498,880,000798,336,00014,774,40097,200136,852,887,600
5-3-3-286,162,227,20011,888,916,480475,504,1287,392,38438,88098,534,079,072
5-4-3-171,801,856,0009,907,430,400396,253,4406,160,32032,40082,111,732,560
5-4-2-255,390,003,20011,071,779,840705,335,04015,111,36097,20067,182,326,640
4-3-3-361,465,600,0005,268,480,000169,344,0002,419,20012,96066,905,856,160
6-3-2-229,541,335,0405,904,949,248376,178,6888,059,39251,84035,830,574,208
6-4-2-124,617,779,2004,920,791,040313,482,2406,716,16043,20029,858,811,840
6-3-3-119,147,161,6002,641,981,440105,667,5841,642,752864021,896,462,016
5-5-2-116,775,258,1123,191,519,232184,768,1283,136,32016,20020,154,697,992
4-4-4-116,464,000,0002,430,400,000110,880,0002,052,00013,50019,007,345,500
7-3-2-19,847,111,6801,968,316,416125,392,8962,686,46417,28011,943,524,736
6-4-3-007,977,984,000435,993,6007,695,36043,2008,421,716,160
5-4-4-007,479,360,000408,744,0007,214,40040,5007,895,358,900
5-5-3-005,436,426,240244,712,4483,503,52016,2005,684,658,408
6-5-1-13,727,835,136709,226,49641,059,584696,96036004,478,821,776
6-5-2-003,727,835,136398,573,5687,845,12043,2004,134,297,024
7-2-2-22,532,114,432662,888,44860,217,3442,087,42417,2803,257,324,928
7-4-1-12,051,481,600410,065,92026,123,520559,68036002,488,234,320
7-4-2-002,051,481,600239,109,1206,197,76043,2002,296,831,680
7-3-3-001,595,596,80087,198,7201,539,07286401,684,343,232
8-2-2-1949,542,912248,583,16822,581,504782,78464801,221,496,848
8-3-1-1615,444,480123,019,7767,837,056167,9041080746,470,296
7-5-1-00621,305,85666,428,9281,307,5207200689,049,504
8-3-2-00615,444,48071,732,7361,859,32812,960689,049,504
6-6-1-00414,203,90444,285,952871,6804800459,366,336
8-4-1-00256,435,20029,888,640774,7205400287,103,960
9-2-1-187,091,20023,639,0402,280,96089,2801080113,101,560
9-3-1-0056,448,0007,123,200227,520216063,800,880
9-2-2-0043,545,6008,190,720457,920648052,200,720
7-6-0-00034,516,992814,080480035,335,872
8-5-0-00019,415,808457,920270019,876,428
10-2-1-005,806,0801,092,09661,0568646,960,096
9-4-0-0005,880,000252,00027006,134,700
10-1-1-11,935,360525,31250,6881984242,513,368
10-3-0-000940,80040,320432981,552
11-1-1-00124,41631,104259272158,184
11-2-0-00062,20810,36843273,008
12-1-0-0001728288122028
13-0-0-0000044
All 39519,718,577,152109,153,389,5686,025,215,168115,669,312708,400635,013,559,600

 Study 8J37 Main Top Hits and Misses

## Straight Flushes

Jeff Bayone of New York City, owner-manager of Honors Bridge Club and author of “A Taste of Bridge,” enjoys a taste of poker instead, looking for straight flushes. I’m not sure I like this, because every time you find one, it means you’re playing the wrong game, wasting at bridge what could have won big-time at the poker table. Oh well, to each his own.

Of the 8192 possible suit holdings, 1378 contain at least one straight flush, which of course requires at least a five-card suit. Multiple occurrences are possible in the same suit, or in either of two suits. I will assume a liberal view allowing cards to be used more than once, so K-Q-J-10-9-8-7 contains three straight flushes; and A-6-5-4-3-2 contains two, since the ace can be low in poker. Indeed, a 13-card suit has 10 straight flushes!

The following table shows the percent chance of each number of straight-flush hits for each generic hand pattern. Hit percents are relative to the generic pattern in that row. Total percents (last column) are relative to all bridge hands. The bottom row shows the percents for all generic patterns combined. Percents shown as 0.0000 are not zero but round that way to the nearest 10,000th.

Percents for Straight Flushes
PatternNo Hit1 Hit2 Hits3 Hits4+ HitsTotal
4-4-3-2100000021.5512
5-3-3-299.22300.777000015.5168
5-4-3-199.22300.777000012.9307
5-4-2-299.22300.777000010.5797
4-3-3-3100000010.5361
6-3-2-295.86253.61310.5245005.6425
6-4-2-195.86253.61310.5245004.7021
6-3-3-195.86253.61310.5245003.4482
5-5-2-198.45201.54190.0060003.1739
4-4-4-110000002.9932
7-3-2-187.35439.44062.73890.466201.8808
6-4-3-095.86253.61310.5245001.3262
5-4-4-099.22300.77700001.2433
5-5-3-098.45201.54190.0060000.8952
6-5-1-195.11764.32980.54850.004100.7053
6-5-2-095.11764.32980.54850.004100.6511
7-2-2-287.35439.44062.73890.466200.5130
7-4-1-187.35439.44062.73890.466200.3918
7-4-2-087.35439.44062.73890.466200.3617
7-3-3-087.35439.44062.73890.466200.2652
8-2-2-171.173317.87107.77002.64180.54390.1924
8-3-1-171.173317.87107.77002.64180.54390.1176
7-5-1-086.675610.04592.79100.48390.00360.1085
8-3-2-071.173317.87107.77002.64180.54390.1085
6-6-1-091.89616.92711.13610.03790.00280.0723
8-4-1-071.173317.87107.77002.64180.54390.0452
9-2-1-146.293726.293715.52457.83224.05590.0178
9-3-1-046.293726.293715.52457.83224.05590.0100
9-2-2-046.293726.293715.52457.83224.05590.0082
7-6-0-083.740012.20613.42480.59540.03370.0056
8-5-0-070.620318.28527.84852.68160.56440.0031
10-2-1-018.181824.475525.174816.083916.08390.0011
9-4-0-046.293726.293715.52457.83224.05590.0010
10-1-1-118.181824.475525.174816.083916.08390.0004
10-3-0-018.181824.475525.174816.083916.08390.0002
11-1-1-01.28217.692315.384628.205147.43590.0000
11-2-0-01.28217.692315.384628.205147.43590.0000
12-1-0-000001000.0000
13-0-0-000001000.0000
All 3998.32371.41300.22690.03210.0043100

Exact Hand Counts for Straight Flushes
PatternNo Hit1 Hit2 Hits3 Hits4+ HitsTotal
4-4-3-2136,852,887,6000000136,852,887,600
5-3-3-297,768,468,512765,610,56000098,534,079,072
5-4-3-181,473,723,760638,008,80000082,111,732,560
5-4-2-266,660,319,440522,007,20000067,182,326,640
4-3-3-366,905,856,160000066,905,856,160
6-3-2-234,348,073,7601,294,577,856187,922,5920035,830,574,208
6-4-2-128,623,394,8001,078,814,880156,602,1600029,858,811,840
6-3-3-120,990,489,520791,130,912114,841,5840021,896,462,016
5-5-2-119,842,710,472310,770,7201,216,8000020,154,697,992
4-4-4-119,007,345,500000019,007,345,500
7-3-2-110,433,183,9041,127,535,552327,124,51255,680,768011,943,524,736
6-4-3-08,073,265,200304,281,12044,169,840008,421,716,160
5-4-4-07,834,011,90061,347,0000007,895,358,900
5-5-3-05,596,661,92887,653,280343,200005,684,658,408
6-5-1-14,260,148,620193,925,47224,565,164182,52004,478,821,776
6-5-2-03,932,444,880179,008,12822,675,536168,48004,134,297,024
7-2-2-22,845,413,792307,509,69689,215,77615,185,66403,257,324,928
7-4-1-12,173,579,980234,903,24068,150,94011,600,16002,488,234,320
7-4-2-02,006,381,520216,833,76062,908,56010,707,84002,296,831,680
7-3-3-01,471,346,448159,011,42446,132,9447,852,41601,684,343,232
8-2-2-1869,379,264218,293,92094,910,40032,269,5366,643,7281,221,496,848
8-3-1-1531,287,328133,401,84058,000,80019,720,2724,060,056746,470,296
7-5-1-0597,237,57669,221,56819,231,3683,334,03224,960689,049,504
8-3-2-0490,419,072123,140,16053,539,20018,203,3283,747,744689,049,504
6-6-1-0422,139,90031,820,8805,218,824174,09612,636459,366,336
8-4-1-0204,341,28051,308,40022,308,0007,584,7201,561,560287,103,960
9-2-1-152,358,90429,738,59217,558,4248,858,3044,587,336113,101,560
9-3-1-029,535,79216,775,6169,904,7524,996,9922,587,72863,800,880
9-2-2-024,165,64813,725,5048,103,8884,088,4482,117,23252,200,720
7-6-0-029,590,2604,313,1361,210,200210,38411,89235,335,872
8-5-0-014,036,7843,634,4401,560,000533,016112,18819,876,428
10-2-1-01,265,4721,703,5201,752,1921,119,4561,119,4566,960,096
9-4-0-02,839,9801,613,040952,380480,480248,8206,134,700
10-1-1-1456,976615,160632,736404,248404,2482,513,368
10-3-0-0178,464240,240247,104157,872157,872981,552
11-1-1-0202812,16824,33644,61675,036158,184
11-2-0-0936561611,23220,59234,63273,008
12-1-0-0000020282028
13-0-0-0000044
All 39624,368,943,3608,972,493,4001,441,035,444203,578,24027,509,156635,013,559,600

 Study 8J37 Main Top Hits and Misses

## Unique Ranks

The late Margie Gwozdzinsky of New York City was a first-rate player with an eccentric personality, which I guess might be expected from the way she spelled her last name. Too many Z’s there, girl! Margie’s penchant for the extraordinary was to look for hands that had 13 unique card ranks. For example, A-Q-J-8-6 5-3-2 K-9-7 10-4 would bring immediate joy, having one card of each rank.

Toward the other extreme, a bridge hand may contain as few as four different ranks, e.g., A-Q-9-8 A-Q-9 A-Q-8 Q-9-8. The great majority of hands fall in between in a high-biased bell curve, with nine different ranks being the most common.

The following table shows the percent chance of each number of different ranks for each generic hand pattern. Percents are relative to the generic pattern in that row, except Total percents (last column) are relative to all bridge hands. The bottom row shows the percents for all generic patterns combined. Percents shown as 0.0000 are not zero but round that way to the nearest 10,000th.

Percents for Different Ranks
Pattern45678910111213Total
4-4-3-20.00020.02690.79547.215525.187037.088523.31965.86940.48980.007921.5512
5-3-3-200.01570.62706.462324.254337.450624.35706.28890.53540.008815.5168
5-4-3-100.00940.46655.503422.719237.696825.95277.02010.62140.010512.9307
5-4-2-200.01150.52875.912123.429637.632125.22546.67160.57930.009710.5797
4-3-3-30.00030.03600.93327.831725.950136.792322.47075.52610.45240.007210.5361
6-3-2-2000.25864.399121.161438.131427.61347.72400.70000.01215.6425
6-4-2-1000.18623.664119.522537.987829.22588.58820.81100.01454.7021
6-3-3-1000.22574.065020.416538.066128.34638.11680.75050.01323.4482
5-5-2-100.00380.30044.413420.824837.869227.89237.94930.73380.01293.1739
4-4-4-10.00010.01640.59806.188223.706337.453524.89956.55980.56880.00952.9932
7-3-2-10001.774115.308137.618433.372010.81021.09650.02071.8808
6-4-3-0000.14673.166418.176937.637530.53459.39650.92430.01711.3262
5-4-4-000.00490.31304.381620.538937.668328.15208.16080.76680.01371.2433
5-5-3-000.00270.23913.841519.489137.654729.21098.71000.83680.01520.8952
6-5-1-1000.09932.616516.819137.413331.861610.14741.02340.01930.7053
6-5-2-0000.08972.447616.233337.131732.424810.56421.08780.02090.6511
7-2-2-20001.951516.045937.971632.670810.31711.02410.01900.5130
7-4-1-10001.419313.729536.756734.874011.92951.26620.02480.3918
7-4-2-00001.317913.179136.336735.395412.39911.34480.02690.3617
7-3-3-00001.497614.017836.841734.598311.77321.24700.02450.2652
8-2-2-100007.930034.086940.383415.74121.82070.03790.1924
8-3-1-100007.415133.309940.882216.40671.94480.04140.1176
7-5-1-00000.878610.794334.516837.654714.43431.68550.03590.1085
8-3-2-000007.028932.634041.240816.98942.06200.04480.1085
6-6-1-0000.02691.349312.426035.301236.085713.27331.50620.03140.0723
8-4-1-000006.024730.876842.173218.50462.36690.05380.0452
9-2-1-10000022.121148.065525.99003.73240.09100.0178
9-3-1-00000020.333547.767627.59554.19580.10760.0100
9-2-2-00000021.301847.929026.72583.94480.09860.0082
7-6-0-00000.40797.342730.594440.792518.35662.44760.05830.0056
8-5-0-000004.351227.195043.512021.75603.10800.07770.0031
10-2-1-000000044.378745.85809.46750.29590.0011
9-4-0-00000017.622446.993030.20985.03500.13990.0010
10-1-1-100000045.516645.19809.01230.27310.0004
10-3-0-000000041.958047.202810.48950.34970.0002
11-1-1-0000000071.597627.21891.18340.0000
11-2-0-0000000070.512828.20511.28210.0000
12-1-0-00000000092.30777.69230.0000
13-0-0-00000000001000.0000
All 390.00010.01510.55115.830922.988537.423125.62026.94230.61820.0106100

Exact Hand Counts for Different Ranks
Pattern45678910111213Total
4-4-3-2205,92036,756,7201,088,493,1209,874,584,72034,469,154,72050,756,706,00031,913,481,6008,032,424,400670,269,60010,810,800136,852,887,600
5-3-3-2015,444,000617,760,0006,367,561,20023,898,786,91236,901,584,72023,999,976,0006,196,750,560527,567,0408,648,64098,534,079,072
5-4-3-107,722,000383,011,2004,518,914,40018,655,116,48030,953,482,56021,310,248,9605,764,318,560510,269,7608,648,64082,111,732,560
5-4-2-207,722,000355,212,0003,971,887,92015,740,524,80025,282,136,88016,947,010,0804,482,157,680389,188,8006,486,48067,182,326,640
4-3-3-3183,04024,092,640624,349,4405,239,874,64017,362,144,80024,616,191,60015,034,219,2003,697,293,600302,702,4004,804,80066,905,856,160
6-3-2-20092,664,0001,576,214,6407,582,262,68813,662,689,0409,894,044,1602,767,564,800250,810,5604,324,32035,830,574,208
6-4-2-10055,598,4001,094,052,9605,829,183,36011,342,691,3608,726,477,7602,564,321,760242,161,9204,324,32029,858,811,840
6-3-3-10049,420,800890,089,2004,470,482,0168,335,126,8006,206,840,6401,777,295,520164,324,1602,882,88021,896,462,016
5-5-2-10772,20060,540,480889,512,6244,197,184,9927,632,424,8005,621,616,0001,602,160,560147,891,7442,594,59220,154,697,992
4-4-4-111,4403,114,540113,667,8401,176,215,0404,505,941,4407,118,911,8004,732,728,0001,246,845,600108,108,0001,801,80019,007,345,500
7-3-2-1000211,891,6801,828,322,4964,492,968,4803,985,787,5201,291,118,400130,965,1202,471,04011,943,524,736
6-4-3-00012,355,200266,666,4001,530,809,2803,169,726,5602,571,528,960791,350,56077,837,7601,441,4408,421,716,160
5-4-4-00386,10024,710,400345,945,6001,621,620,0002,974,051,0802,222,700,480644,323,68060,540,4801,081,0807,895,358,900
5-5-3-00154,44013,590,720218,378,1601,107,890,7842,140,538,4001,660,538,880495,134,64047,567,520864,8645,684,658,408
6-5-1-1004,447,872117,189,072753,296,5441,675,674,0001,427,025,600454,486,03245,837,792864,8644,478,821,776
6-5-2-0003,706,560101,189,088671,134,4641,535,133,6001,340,539,200436,756,32044,972,928864,8644,134,297,024
7-2-2-200063,567,504522,666,1441,236,858,4801,064,194,560336,061,44033,359,040617,7603,257,324,928
7-4-1-100035,315,280341,621,280914,593,680867,746,880296,833,68031,505,760617,7602,488,234,320
7-4-2-000030,270,240302,702,400834,593,760812,972,160284,787,36030,888,000617,7602,296,831,680
7-3-3-000025,225,200236,107,872620,539,920582,753,600198,300,96021,003,840411,8401,684,343,232
8-2-2-1000096,864,768416,370,240493,281,360192,277,80022,239,360463,3201,221,496,848
8-3-1-1000055,351,296248,648,400305,173,440122,470,92014,517,360308,880746,470,296
7-5-1-00006,054,04874,378,304237,837,600259,459,20099,459,36011,613,888247,104689,049,504
8-3-2-0000048,432,384224,864,640284,169,600117,065,52014,208,480308,880689,049,504
6-6-1-000123,5526,198,19257,081,024162,162,000165,765,60060,972,9126,918,912144,144459,366,336
8-4-1-0000017,297,28088,648,560121,080,96053,127,3606,795,360154,440287,103,960
9-2-1-10000025,019,28054,362,88029,395,0804,221,360102,960113,101,560
9-3-1-00000012,972,96030,476,16017,606,1602,676,96068,64063,800,880
9-2-2-00000011,119,68025,019,28013,951,0802,059,20051,48052,200,720
7-6-0-0000144,1442,594,59210,810,80014,414,4006,486,480864,86420,59235,335,872
8-5-0-00000864,8645,405,4008,648,6404,324,320617,76015,44419,876,428
10-2-1-00000003,088,8003,191,760658,94420,5926,960,096
9-4-0-0000001,081,0802,882,8801,853,280308,88085806,134,700
10-1-1-10000001,144,0001,135,992226,51268642,513,368
10-3-0-0000000411,840463,320102,9603432981,552
11-1-1-00000000113,25643,0561872158,184
11-2-0-0000000051,48020,59293673,008
12-1-0-00000000018721562028
13-0-0-000000000044
All 39400,40096,164,6403,499,651,58437,026,941,952145,979,817,984237,641,564,160162,691,809,28044,084,232,1923,925,868,54467,108,864635,013,559,600

### Margie meets Magnus?

Another interesting pursuit is a hand that meets both Margie’s 13 unique card ranks and Magnus’s lowest-card length identifier in each suit. A little thought reveals there can be only three hand types:

Hand TypeChoose xCombinationsWaysHands
x-x-x-x-x-6 x-x-5-4 3-2 AK Q J 10 9 8 72124504
K-Q-J-10-9-8-7 x-x-3 x-2 A6 5 432472
x-x-x-x-x-x-7 x-6-5-4 3-2 A K Q J 10 9 8724168

Combinations shows the number of distinct ‘x’ choices. Ways shows the number of suit permutations. Totaling the last column shows there are only 744 such hands. Hank Aaron hit more home runs than that (755 if I recall). Considering there are 635+ billion bridge hands, don’t hold your breath for one of these.

 Study 8J37 Main Top Hits and Misses

## Alphabetic Suits

David Burn of London, England, renowned UK expert and journalist, asked what is remarkable about this hand: A-J-7-6 8-5-4 K-3-2 Q-10-3. Well, it looks pretty ordinary aside from almost being a “Margie” — achieved simply by changing either of the threes to a nine. In fact it’s an oddity of a different sort. The card ranks in each suit are alphabetic, i.e., ace-jack-seven-six, eight-five-four, king-three-two and queen-ten-three are each in correct order. Note that holdings such as A-J-5-4 or Q-9-3 would not be acceptable.

Of the 8192 possible suit holdings, only 376 qualify as alphabetic. The longest alphabetic suit is seven cards, of which only three possibilities exist: A-K-Q-7-6-3-2, A-K-9-7-6-3-2 and A-J-9-7-6-3-2. Obviously any singleton must qualify, and so will a void suit to be consistent with other oddities. Now there’s a job for you: Alphabetize a void. My kind of work!

The following table shows the percent chance of each number of alphabetic suits for each generic hand pattern. Hit percents are relative to the generic pattern in that row. Total percents (last column) are relative to all bridge hands. The bottom row shows the percents for all generic patterns combined. Percents shown as 0.0000 are not zero but round that way to the nearest 10,000th.

Percents for Alphabetic Suits
PatternNo Hit1 Hit2 Hits3 Hits4 HitsTotal
4-4-3-214.506944.433132.57857.87530.606321.5512
5-3-3-212.293740.798236.037710.39570.474715.5168
5-4-3-1050.832541.32437.54220.301012.9307
5-4-2-28.205936.775045.36909.26800.382110.5797
4-3-3-321.733441.293928.20998.00950.753210.5361
6-3-2-26.453531.134145.043517.15370.21525.6425
6-4-2-1026.684161.873111.30630.13654.7021
6-3-3-1039.976746.206813.64700.16953.4482
5-5-2-1028.753864.16616.89040.18973.1739
4-4-4-1059.983833.42386.20810.38442.9932
7-3-2-1020.360654.834824.76140.04321.8808
6-4-3-0052.980140.80286.14410.07301.3262
5-4-4-0067.367128.78323.72010.12961.2433
5-5-3-0057.089438.99183.81720.10150.8952
6-5-1-10093.50216.43020.06770.7053
6-5-2-0029.968665.59444.39090.04600.6511
7-2-2-23.286820.909944.353531.39490.05480.5130
7-4-1-10084.188215.78440.02740.3918
7-4-2-0026.983462.263910.73410.01860.3617
7-3-3-0040.425146.270913.28090.02310.2652
8-2-2-1010.272843.556946.170300.1924
8-3-1-10063.636436.363600.1176
7-5-1-00094.55085.44000.00920.1085
8-3-2-0020.396354.895124.708600.1085
6-6-1-00097.45232.53120.01640.0723
8-4-1-00084.335715.664300.0452
9-2-1-10032.051367.948700.0178
9-3-1-00063.636436.363600.0100
9-2-2-0010.272843.556946.170300.0082
7-6-0-00098.54541.45240.00220.0056
8-5-0-00094.71645.283600.0031
10-2-1-00032.051367.948700.0011
9-4-0-00084.335715.664300.0010
10-1-1-100010000.0004
10-3-0-00063.636436.363600.0002
11-1-1-000010000.0000
11-2-0-00032.051367.948700.0000
12-1-0-000010000.0000
13-0-0-000010000.0000
All 398.573040.803140.53599.67750.4106100

Exact Hand Counts for Alphabetic Suits
PatternNo Hit1 Hit2 Hits3 Hits4 HitsTotal
4-4-3-219,853,051,40060,808,000,96844,584,556,64010,777,568,256829,710,336136,852,887,600
5-3-3-212,113,446,80040,200,176,01635,509,371,45610,243,314,432467,770,36898,534,079,072
5-4-3-1041,739,476,68833,932,106,6246,193,025,280247,123,96882,111,732,560
5-4-2-25,512,927,50024,706,302,60030,479,938,9566,226,437,456256,720,12867,182,326,640
4-3-3-314,540,906,01627,628,065,92018,874,093,0565,358,852,096503,939,07266,905,856,160
6-3-2-22,312,310,00011,155,544,40016,139,330,0646,146,265,84077,123,90435,830,574,208
6-4-2-107,967,559,60018,474,579,5523,375,927,98440,744,70429,858,811,840
6-3-3-108,753,480,73610,117,659,5522,988,201,21637,120,51221,896,462,016
5-5-2-105,795,247,90012,932,483,1481,388,735,71238,231,23220,154,697,992
4-4-4-1011,401,323,8046,352,976,4481,179,988,99273,056,25619,007,345,500
7-3-2-102,431,774,8006,549,206,9762,957,383,7285,159,23211,943,524,736
6-4-3-004,461,833,3763,436,296,864517,435,7766,150,1448,421,716,160
5-4-4-005,318,872,4522,272,538,160293,712,38410,235,9047,895,358,900
5-5-3-003,245,338,8242,216,551,584216,997,2485,770,7525,684,658,408
6-5-1-1004,187,791,608287,996,2803,033,8884,478,821,776
6-5-2-001,238,991,6002,711,868,192181,534,3201,902,9124,134,297,024
7-2-2-2107,062,500681,105,0001,444,737,6001,022,633,3041,786,5243,257,324,928
7-4-1-1002,094,800,292392,752,620681,4082,488,234,320
7-4-2-00619,763,4001,430,097,408246,543,480427,3922,296,831,680
7-3-3-00680,896,944779,360,400223,696,512389,3761,684,343,232
8-2-2-10125,482,500532,045,800563,968,54801,221,496,848
8-3-1-100475,026,552271,443,7440746,470,296
7-5-1-000651,501,86437,483,99263,648689,049,504
8-3-2-00140,540,400378,254,448170,254,6560689,049,504
6-6-1-000447,663,21611,627,61675,504459,366,336
8-4-1-000242,131,03244,972,9280287,103,960
9-2-1-10036,250,50076,851,0600113,101,560
9-3-1-00040,600,56023,200,320063,800,880
9-2-2-005,362,50022,737,00024,101,220052,200,720
7-6-0-00034,821,864513,21679235,335,872
8-5-0-00018,826,2361,050,192019,876,428
10-2-1-0002,230,8004,729,29606,960,096
9-4-0-0005,173,740960,96006,134,700
10-1-1-10002,513,36802,513,368
10-3-0-000624,624356,9280981,552
11-1-1-0000158,1840158,184
11-2-0-00023,40049,608073,008
12-1-0-0000202802028
13-0-0-0000404
All 3954,439,704,216259,105,140,428257,408,256,21661,453,240,7842,607,217,956635,013,559,600

### Alphabetic Magnus hit?

Only 45 of 8192 possible suit holdings qualify as alphabetic Magnus hits. Besides a void and singleton ace, these include every doubleton with the two, and alphabetized tripletons with the lowest card three. Curiously, only one holding of 4+ cards qualifies, specifically A-8-5-4, so that must appear at least once in any complete hand that qualifies. For example, A-8-5-4 9-6-3 J-7-3 K-10-3.

### Margie knows the ABC’s

Hands that qualify as a “Margie” (13 unique ranks) with each suit alphabetic are more common than one might think, 58,368 by my count, but still rare in the grand scheme of 635+ billion bridge hands. For example, 8-5-4-3 A-K-10 Q-7-2 J-9-6, or A-J-9-7-6-3-2 K-Q-10 8-5-4 --, the latter having the longest possible suit.

### Alphabetic and symmetric?

Only 10 of 8192 possible suit holdings are both alphabetic and symmetric: Void, singleton 8, A-2, K-3, 9-7, A-8-2, A-K-3-2, A-9-7-2, K-9-7-3 and A-K-9-7-3-2. These can produce 412 hands (counting all combinations and suit permutations), e.g., A-K-9-7-3-2 A-8-2 K-3 9-7.

Crack! There’s a deep line drive into the left-center gap. This will be extra bases!
He’s rounding second… rounding third… Here comes the relay… Oh no!
Thrown out at the plate! (Just like this study should have been.)

 Study 8J37 Main Top Hits and Misses

© 2019 Richard Pavlicek