Main   Study 8J37 by Richard Pavlicek  

Hits and Misses

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.

Magnus HitsSymmetric SuitsStraight FlushesUnique RanksAlphabetic Suits

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., S A-Q-9-7-5 H 8-6-3 D K-10-3 C 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
4-4-3-26476958180053323162200164088540002237241600114048000136852887600
5-3-3-249066079832376109062561053356594412686414405488560098534079072
5-4-3-1459572097602902824540065001301206061888801995840082111732560
5-4-2-2361092362402433199824060572707206576940802612736067182326640
4-3-3-32933681058026871532380921544470014022085007986000066905856160
6-3-2-220044221120127374318722804934528238664448532224035830574208
6-4-2-11877420160093977481601584092160100834560193536029858811840
6-3-3-11275541344075672606241470028032101727120203280021896462016
5-5-2-1128106411846177365208108226756882137888228614420154697992
4-4-4-11011095400069601315001743588000185760000691200019007345500
7-3-2-1750395923238070790566028989122947665611088011943524736
6-4-3-005475808800259285488034418208088704008421716160
5-4-4-0049322763002524775400416534400217728007895358900
5-5-3-0037364370121700637444237105792104781605684658408
6-5-1-1333030528010288270081142825765322240846724478821776
6-5-2-0030527798409893456649013939220321284134297024
7-2-2-21965322656108004406419929888012610944483843257324928
7-4-1-11757125440654429600739163402752860100802488234320
7-4-2-001610698320624298320615931202419202296831680
7-3-3-001094327388525590604641711402541001684343232
8-2-2-180728876836083361651150528222393601221496848
8-3-1-1513729216207938016239536448494200746470296
7-5-1-00571434912111976344561708021168689049504
8-3-2-00470918448197744976203860800689049504
6-6-1-0039680640059839680268262437632459366336
8-4-1-002205403206285708037065600287103960
9-2-1-1815443202841696030373201029600113101560
9-3-1-004756752015289560943800063800880
9-2-2-0037374480135907201235520052200720
7-6-0-000340432801287888470435335872
8-5-0-000179304841945944019876428
10-2-1-00543628814414408236806960096
9-4-0-0005105100102960006134700
10-1-1-1197683249420841184114402513368
10-3-0-0007927921887600981552
11-1-1-00134784224649360158184
11-2-0-0006177611232073008
12-1-0-000187215602028
13-0-0-0000404
All 39315695601300242826385180679469550368187283152357334932635013559600

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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
4-4-3-2118540800000174988800007983360001477440097200136852887600
5-3-3-2861622272001188891648047550412873923843888098534079072
5-4-3-171801856000990743040039625344061603203240082111732560
5-4-2-25539000320011071779840705335040151113609720067182326640
4-3-3-361465600000526848000016934400024192001296066905856160
6-3-2-229541335040590494924837617868880593925184035830574208
6-4-2-124617779200492079104031348224067161604320029858811840
6-3-3-11914716160026419814401056675841642752864021896462016
5-5-2-116775258112319151923218476812831363201620020154697992
4-4-4-116464000000243040000011088000020520001350019007345500
7-3-2-19847111680196831641612539289626864641728011943524736
6-4-3-0079779840004359936007695360432008421716160
5-4-4-0074793600004087440007214400405007895358900
5-5-3-0054364262402447124483503520162005684658408
6-5-1-137278351367092264964105958469696036004478821776
6-5-2-0037278351363985735687845120432004134297024
7-2-2-22532114432662888448602173442087424172803257324928
7-4-1-120514816004100659202612352055968036002488234320
7-4-2-0020514816002391091206197760432002296831680
7-3-3-00159559680087198720153907286401684343232
8-2-2-19495429122485831682258150478278464801221496848
8-3-1-161544448012301977678370561679041080746470296
7-5-1-006213058566642892813075207200689049504
8-3-2-0061544448071732736185932812960689049504
6-6-1-00414203904442859528716804800459366336
8-4-1-00256435200298886407747205400287103960
9-2-1-187091200236390402280960892801080113101560
9-3-1-00564480007123200227520216063800880
9-2-2-00435456008190720457920648052200720
7-6-0-00034516992814080480035335872
8-5-0-00019415808457920270019876428
10-2-1-0058060801092096610568646960096
9-4-0-000588000025200027006134700
10-1-1-11935360525312506881984242513368
10-3-0-00094080040320432981552
11-1-1-0012441631104259272158184
11-2-0-000622081036843273008
12-1-0-0001728288122028
13-0-0-0000044
All 395197185771521091533895686025215168115669312708400635013559600

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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
4-4-3-21368528876000000136852887600
5-3-3-29776846851276561056000098534079072
5-4-3-18147372376063800880000082111732560
5-4-2-26666031944052200720000067182326640
4-3-3-366905856160000066905856160
6-3-2-23434807376012945778561879225920035830574208
6-4-2-12862339480010788148801566021600029858811840
6-3-3-1209904895207911309121148415840021896462016
5-5-2-11984271047231077072012168000020154697992
4-4-4-119007345500000019007345500
7-3-2-110433183904112753555232712451255680768011943524736
6-4-3-0807326520030428112044169840008421716160
5-4-4-07834011900613470000007895358900
5-5-3-0559666192887653280343200005684658408
6-5-1-142601486201939254722456516418252004478821776
6-5-2-039324448801790081282267553616848004134297024
7-2-2-22845413792307509696892157761518566403257324928
7-4-1-12173579980234903240681509401160016002488234320
7-4-2-02006381520216833760629085601070784002296831680
7-3-3-0147134644815901142446132944785241601684343232
8-2-2-1869379264218293920949104003226953666437281221496848
8-3-1-153128732813340184058000800197202724060056746470296
7-5-1-05972375766922156819231368333403224960689049504
8-3-2-049041907212314016053539200182033283747744689049504
6-6-1-042213990031820880521882417409612636459366336
8-4-1-0204341280513084002230800075847201561560287103960
9-2-1-152358904297385921755842488583044587336113101560
9-3-1-0295357921677561699047524996992258772863800880
9-2-2-0241656481372550481038884088448211723252200720
7-6-0-029590260431313612102002103841189235335872
8-5-0-0140367843634440156000053301611218819876428
10-2-1-0126547217035201752192111945611194566960096
9-4-0-0283998016130409523804804802488206134700
10-1-1-14569766151606327364042484042482513368
10-3-0-0178464240240247104157872157872981552
11-1-1-0202812168243364461675036158184
11-2-0-0936561611232205923463273008
12-1-0-0000020282028
13-0-0-0000044
All 396243689433608972493400144103544420357824027509156635013559600

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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, S A-Q-J-8-6 H 5-3-2 D K-9-7 C 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 unique ranks, e.g., S A-Q-9-8 H A-Q-9 D A-Q-8 C Q-9-8. The great majority of hands fall in between in a high-biased bell curve, with nine unique ranks being the most common.

The following table shows the percent chance of each number of unique 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 Unique 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 Unique Ranks
Pattern45678910111213Total
4-4-3-22059203675672010884931209874584720344691547205075670600031913481600803242440067026960010810800136852887600
5-3-3-201544400061776000063675612002389878691236901584720239999760006196750560527567040864864098534079072
5-4-3-10772200038301120045189144001865511648030953482560213102489605764318560510269760864864082111732560
5-4-2-20772200035521200039718879201574052480025282136880169470100804482157680389188800648648067182326640
4-3-3-31830402409264062434944052398746401736214480024616191600150342192003697293600302702400480480066905856160
6-3-2-20092664000157621464075822626881366268904098940441602767564800250810560432432035830574208
6-4-2-10055598400109405296058291833601134269136087264777602564321760242161920432432029858811840
6-3-3-100494208008900892004470482016833512680062068406401777295520164324160288288021896462016
5-5-2-10772200605404808895126244197184992763242480056216160001602160560147891744259459220154697992
4-4-4-111440311454011366784011762150404505941440711891180047327280001246845600108108000180180019007345500
7-3-2-10002118916801828322496449296848039857875201291118400130965120247104011943524736
6-4-3-000123552002666664001530809280316972656025715289607913505607783776014414408421716160
5-4-4-00386100247104003459456001621620000297405108022227004806443236806054048010810807895358900
5-5-3-0015444013590720218378160110789078421405384001660538880495134640475675208648645684658408
6-5-1-100444787211718907275329654416756740001427025600454486032458377928648644478821776
6-5-2-000370656010118908867113446415351336001340539200436756320449729288648644134297024
7-2-2-20006356750452266614412368584801064194560336061440333590406177603257324928
7-4-1-100035315280341621280914593680867746880296833680315057606177602488234320
7-4-2-000030270240302702400834593760812972160284787360308880006177602296831680
7-3-3-000025225200236107872620539920582753600198300960210038404118401684343232
8-2-2-1000096864768416370240493281360192277800222393604633201221496848
8-3-1-100005535129624864840030517344012247092014517360308880746470296
7-5-1-00006054048743783042378376002594592009945936011613888247104689049504
8-3-2-000004843238422486464028416960011706552014208480308880689049504
6-6-1-000123552619819257081024162162000165765600609729126918912144144459366336
8-4-1-000001729728088648560121080960531273606795360154440287103960
9-2-1-1000002501928054362880293950804221360102960113101560
9-3-1-00000012972960304761601760616026769606864063800880
9-2-2-00000011119680250192801395108020592005148052200720
7-6-0-00001441442594592108108001441440064864808648642059235335872
8-5-0-000008648645405400864864043243206177601544419876428
10-2-1-000000030888003191760658944205926960096
9-4-0-00000010810802882880185328030888085806134700
10-1-1-10000001144000113599222651268642513368
10-3-0-00000004118404633201029603432981552
11-1-1-00000000113256430561872158184
11-2-0-00000000514802059293673008
12-1-0-00000000018721562028
13-0-0-000000000044
All 394004009616464034996515843702694195214597981798423764156416016269180928044084232192392586854467108864635013559600


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
S x-x-x-x-x-6 H x-x-5-4 D 3-2 C AK Q J 10 9 8 72124504
S K-Q-J-10-9-8-7 H x-x-3 D x-2 C A6 5 432472
S x-x-x-x-x-x-7 H x-6-5-4 D 3-2 CA 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.

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Alphabetic Suits

David Burn of London, England, renowned UK expert and journalist, asked what is remarkable about this hand: S A-J-7-6 H 8-5-4 D K-3-2 C 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
4-4-3-219853051400608080009684458455664010777568256829710336136852887600
5-3-3-21211344680040200176016355093714561024331443246777036898534079072
5-4-3-104173947668833932106624619302528024712396882111732560
5-4-2-255129275002470630260030479938956622643745625672012867182326640
4-3-3-3145409060162762806592018874093056535885209650393907266905856160
6-3-2-22312310000111555444001613933006461462658407712390435830574208
6-4-2-1079675596001847457955233759279844074470429858811840
6-3-3-1087534807361011765955229882012163712051221896462016
5-5-2-1057952479001293248314813887357123823123220154697992
4-4-4-1011401323804635297644811799889927305625619007345500
7-3-2-10243177480065492069762957383728515923211943524736
6-4-3-004461833376343629686451743577661501448421716160
5-4-4-0053188724522272538160293712384102359047895358900
5-5-3-003245338824221655158421699724857707525684658408
6-5-1-100418779160828799628030338884478821776
6-5-2-001238991600271186819218153432019029124134297024
7-2-2-21070625006811050001444737600102263330417865243257324928
7-4-1-10020948002923927526206814082488234320
7-4-2-0061976340014300974082465434804273922296831680
7-3-3-006808969447793604002236965123893761684343232
8-2-2-1012548250053204580056396854801221496848
8-3-1-1004750265522714437440746470296
7-5-1-0006515018643748399263648689049504
8-3-2-001405404003782544481702546560689049504
6-6-1-0004476632161162761675504459366336
8-4-1-000242131032449729280287103960
9-2-1-10036250500768510600113101560
9-3-1-0004060056023200320063800880
9-2-2-0053625002273700024101220052200720
7-6-0-0003482186451321679235335872
8-5-0-000188262361050192019876428
10-2-1-0002230800472929606960096
9-4-0-000517374096096006134700
10-1-1-1000251336802513368
10-3-0-0006246243569280981552
11-1-1-00001581840158184
11-2-0-0002340049608073008
12-1-0-0000202802028
13-0-0-0000404
All 3954439704216259105140428257408256216614532407842607217956635013559600


Alphabetic Magnus hit?

Only 45 of the 8192 possible suit holdings qualify as alphabetic Magnus hits. Besides a void and a 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, S A-8-5-4 H 9-6-3 D J-7-3 C 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, S 8-5-4-3 H A-K-10 D Q-7-2 C J-9-6, or S A-J-9-7-6-3-2 H K-Q-10 D 8-5-4 C, the latter having the longest possible suit.

Alphabetic and symmetric?

Only 10 of the 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., S A-K-9-7-3-2 H A-8-2 D K-3 C 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! (Like this study should have been.)

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© 2019 Richard Pavlicek