Study 8J65   Main

# Frozen Suits

by Richard Pavlicek

A suit is considered “frozen” to a player if leading any card in that suit allows the opposing side to win an extra trick over what they could win by leading the suit themselves. Suit layouts can be frozen to any number of players (one to four) though the great majority are frozen to no one, i.e., any player can lead the suit without affecting the trick outcome. This study will enter the deep freeze, so get out your winter clothes!

About five years ago I created a database with each of the 67,108,864* suit layouts — well not exactly, as the size could be reduce considerably by equalizing rotations. Any specific suit layout must remain identical in all technical aspects if it is merely rotated around the table. Since every layout has four rotations, this reduced the size to 16,777,216 distinct layouts. Hey, three sevens in the middle! How bad can that be?

*This is easily calculated as 413. Each of the 13 cards can be dealt to any hand, so there are four ways to place the ace, then four ways to place the king (no matter where the ace went), etc., for every card.

Each suit layout was analyzed for winnable tricks by each side if only they lead the suit, which hand(s) must lead to which tricks to achieve this, and which hands if any are frozen or doubly frozen.

I have incorporated this database into an online utility that will analyze any entered suit layout, or multiple layouts, or even random layouts if you just click “Display.” Check it out — it’s fun! — and a handy reference tool: Suit Layout Analyzer

Are you dressed warm enough? Then put on your snow shoes and we’ll head over to the ice rink.

### Skating on thin ice

Which of these four hands are frozen?

 1. K J 5 Q 4 3 2 10 6 A 9 8 7

In order to determine which hands (if any) are frozen, it is first necessary to analyze the trick outcomes when the suit is played solely by each side. North-South clearly win three tricks — not four since West’s queen can only be finessed once. West-East cannot win a single trick on their own.

Next determine if the default productions are increased by a lead from the opposite side. If West leads the suit, can N-S win more than three tricks? No, so West is not frozen. Ah, but if East leads (either the 6 or 10) N-S can win four tricks (play it out to see) so the suit is frozen to East.

What about the other side? West-East are not entitled to a trick, but if North starts the suit, West’s queen will eventually establish; therefore the suit is frozen to North. Observe that South can start the suit without loss (low to the jack) so South is not frozen. That certainly makes sense to me, as in South Florida nothing is frozen.

It often occurs that a suit is frozen all-around, such that any lead by any player allows the other side to win an extra trick. In bridge parlance, the phrase “frozen suit” generally implies this, i.e., omitting any qualifier such as “frozen to West” presumes frozen to all. The most common occurrence is when one side holds the king-queen, and the other the ace-jack, as in this layout:

 2. Q 8 7 3 A J 6 10 4 2 K 9 5

If N-S lead this suit, the default outcome is two tricks (counting North’s long card); or if W-E lead it, they can win only one trick (the ace). In other words, the side that leads the suit first — no matter which player or which card — will lose two of the first three tricks with routine play.

Smart players are well aware of this and avoid leading these suits for as long as possible. Sometimes the opponents will unwittingly help you, and sometimes you can force it upon them — or to paraphrase The Maltese Falcon ending: It’s the stuff that endplays are made of.

### Watch out for the icebergs

As the Titanic crew found out the hard way, frozen objects are not always in plain view.
Which hand (or hands) are frozen in this layout?

 3. A Q 5 4 K 8 6 2 J 3 10 9 7

It appears that N-S can win only two tricks, as West will duck the 10 lead then cover the next; but if West or East leads the suit first, N-S win three tricks. Hence it seems frozen to West and East. Alas, that’s an illusion. North-South actually can win three tricks by force if North starts the suit (low toward 10-9-7), so West or East leading doesn’t cost anything in theory, so it’s not frozen to either.

Curiously, the suit is frozen only to South, because any lead by South gives W-E two tricks when entitled to only one by their own play. So much for my South Florida analogy!

### Frozen twice can raise the price

In the great majority of situations, breaking a frozen suit is a done deal; a trick is lost by the perpetrator, and the suit can be led freely thereafter by anyone — but not always. Consider this layout:

 4. 4 3 A Q 2 9 8 7 5 K J 10 6

North-South can win only one trick on their own (East splits 9-8-7 to prevent finessing the six), and an initial lead by East wouldn’t change this, so East is not frozen. But oh, look at West, who is not only frozen but frozen twice. Any lead by West lets N-S win two tricks, but the unusual aspect is that a second lead costs another, letting N-S win three tricks. Of course nobody in their right mind leads from A-Q-2 — but among my claims, having a right mind is not one of them.

Now consider the other side; W-E can win two tricks (simple finesse) but no more with South covering every lead by East, although South is frozen. If South starts the suit with any card, W-E can win three tricks (try it). Note however that South is frozen only once, unlike poor West.

### The record for ice follies

It is possible for a suit to be quintuply frozen for two opposing players:

 5. 2 A Q 10 8 6 4 — K J 9 7 5 3

If only South (or North first) leads this suit, the N-S rake is zero, as West has every card covered. Any lead from West will increase this, and South would win five tricks if West led the suit five times. Similar analysis applies to West, who is entitled to only one trick (ace) if led entirely but comes to six if South leads five times. In practice of course, once the lie is discovered (imagine being the trump suit) only a fool leads the suit unless he has to, so the outcome will be somewhere in the middle, probably 4:2 in favor of West.

### Doubly frozen all-around

Is it possible for a suit to be doubly frozen to all four players?
Yes, but aside from irrelevant spot cards and rotations, the layout is unique:

 6. Q 8 6 3 J 7 2 K 9 4 A 10 5

North-South can win only two tricks on their own (counting North’s long card) as two tricks must be lost with the futility of any finessing attempt — believe me now or believe me later. West-East are also in dire straights, unable to win a single trick on their own merits. But what tidings some gifts will bring! If West or East leads the suit, it costs one trick, and if either leads again it costs another, giving N-S four tricks. Similarly, if North or South leads the suit, it costs one trick, and if either leads again it costs another, giving W-E two tricks.

This unique layout requires a 4-3-3-3 pattern, with J-7, Q-8, K-9 and A-10 sequentially around the table. Further, the six-spot must be with the Q-8 or the A-10. Lower spots can be anywhere necessary to fill three cards, and the fourth card can go to any hand.

While we’re on the topic of two-trick losses, here’s an interesting poser: Is it possible in a single-suit layout, for one lead to lose more than one trick? Cheer up! The answer is no, so despite your partner’s latest opinion, you’re not such an idiot after all.

### A few more oddities…

Can a suit be frozen to a player with singleton? Yes, but only if LHO is void, and the singleton is a queen or lower. Essentially this could finesse partner, which RHO could not do on his own because of the void.

Can a suit be frozen to a player with a solid sequence? Yes, but again only if LHO is void, and the sequence is headed by the queen or lower. For example, Q-J-10-9-8 looks pretty safe, but it could allow RHO to win the king, which he couldn’t do on his own.

Can a suit be frozen to a player when his RHO has a singleton? No, because the effect of any lead could be duplicated by RHO leading the singleton; hence nothing could ever be lost.

Can a suit be frozen to a player when his RHO has all touching cards? No, for the same reason. The effect of any lead would be identical to RHO leading. The same is true when RHO has only insignificant low cards, whether touching or not.

Can a suit be frozen to a player with a void? Okay, time out! Either I’m suffering from mental frostbite, or Rod Serling will step in to explain how we’ve just entered The Twilight Zone.

 Study 8J65   Main Top   Frozen Suits

## Frozen States

Let’s see… that would be Alaska for sure, and I suppose North Dakota, Minnesota and a few others. Oops! Never mind.

The following table summarizes the frozen states of all suit layouts by generic pattern, of which there are 140 distinct possibilities. Generic suit patterns differ from generic hand patterns because order is significant. For example, a 4=4=3=2 suit layout (around the table) is not generically equivalent to 4=3=4=2, because the latter has both 4-card suits on the same side. Nor is it equivalent to 4=2=3=4, because the side with a 4-3 fit now has four cards behind the enemy four. In suit patterns this matters, but in hand patterns it doesn’t.

The first column lists the 140 generic suit patterns in order of frequency, although patterns with the same four numbers have equal frequency. The designated hand order (W-N-E-S) always assigns West the greatest length, but this is just an arbitrary choice for uniformity; rotation of a suit layout doesn’t change it, so it makes no difference.

The second column shows the number of specific layouts that comprise the generic pattern. Hence the column total should match the 67,108,864 possible suit layouts, which it does. Note that the number of layouts generally decreases according to frequency, but not always. This is because a rarer layout sometimes has more possible arrangements.

The next seven columns show the percent occurrence of each frozen state beneficial to West-East (total across seven columns will be 100 percent), and the last seven do likewise as beneficial to North-South. A plus sign (+) in the column header means doubly frozen (or greater).

Frozen States by Generic Suit Pattern
W-N-E-SLayoutsFrozen To (beneficial to West-East)Frozen To (beneficial to North-South)
NoneNSNSN+S+NS+NoneWEWEW+E+WE+
4-3-3-3480480083.015.844.705.920.340.180.0181.057.614.935.820.400.170.01
4-4-3-2360360080.746.566.765.460.480083.045.078.013.7300.160
4-2-3-4360360084.405.925.214.2300.24081.745.118.564.230.3600
4-3-4-2360360085.555.325.703.050.370084.556.735.413.0500.260
5-3-3-2288288085.856.883.962.700.600084.928.633.642.7000.110
5-2-3-3288288085.387.633.773.0300.19084.177.454.883.030.4700
5-3-2-3288288086.905.564.712.410.430082.2810.024.562.410.7300
5-4-2-2216216082.226.626.693.720.750083.386.457.292.88000
5-2-2-4216216082.979.703.653.6900080.218.266.374.370.7900
5-2-4-2216216088.535.814.720.9300087.466.545.070.93000
4-4-4-1180180091.108.08000.830089.7609.27000.970
5-4-3-1144144089.179.74001.090094.1605.63000.210
5-1-3-4144144094.6505.08000.27089.079.77001.1600
5-4-1-3144144090.218.86000.930089.2910.05000.6600
5-3-1-4144144090.578.93000.490086.5912.25001.1600
5-3-4-1144144091.767.56000.680092.5307.13000.350
5-1-4-3144144093.8805.71000.41090.608.82000.5900
6-3-2-2144144085.987.443.991.790.800084.548.555.131.79000
6-2-2-3144144085.519.882.042.5700083.089.863.732.570.7700
6-2-3-2144144088.696.883.590.8400087.398.093.680.84000
6-3-3-196096089.579.43001.000094.4405.39000.170
6-1-3-396096095.8503.94000.21088.6510.46000.9000
6-3-1-396096088.4610.75000.790084.7314.08001.1900
5-5-2-186486489.859.14001.010096.5303.470000
5-1-2-586486493.0806.92000089.768.56001.6800
5-2-5-186486493.796.210000094.4405.560000
W-N-E-SLayoutsNoneNSNSN+S+NS+NoneWEWEW+E+WE+
6-4-2-172072088.909.59001.520098.0301.970000
6-1-2-472072096.1103.89000086.6711.36001.9700
6-4-1-272072090.497.97001.540089.9310.0700000
6-2-1-472072092.757.250000086.4012.18001.4200
6-2-4-172072092.637.370000095.6004.400000
6-1-4-272072094.9805.02000093.256.7500000
7-2-2-261776089.027.902.430.6500087.259.722.380.65000
5-4-4-036036085.3512.85001.800069.93011.8815.1101.211.86
5-0-4-436036089.6509.38000.97069.9314.45012.522.0101.10
5-4-0-436036064.5714.87016.422.1002.0481.5216.33002.1500
7-3-2-141184088.6210.11001.270098.3201.680000
7-1-2-341184097.9802.02000087.4411.29001.2700
7-3-1-241184090.188.68001.140088.7611.2400000
7-2-1-341184091.128.880000086.3812.49001.1400
7-2-3-141184091.498.510000096.7403.260000
7-1-3-241184096.2003.80000092.037.9700000
5-5-3-028828882.2816.00001.720073.72010.2113.2100.562.30
5-0-3-528828886.79012.63000.58073.7213.25010.222.1000.70
5-3-5-028828890.668.59000.750077.62010.3010.6800.650.75
6-5-1-128828886.4510.38003.1700100000000
6-1-1-528828810000000083.9612.49003.5500
6-1-5-1288288100000000100000000
6-4-3-024024082.8315.04002.140075.1706.9315.2100.252.44
6-0-3-424024093.9305.76000.31075.1715.0507.172.3100.31
6-4-0-324024067.7213.33015.941.6501.3582.0016.65001.3500
6-3-0-424024067.7214.73014.450.9702.1378.5319.21002.2500
6-3-4-024024088.0910.80001.110077.6208.3212.5600.381.11
6-0-4-324024093.1006.44000.45077.6212.2108.720.9900.45
W-N-E-SLayoutsNoneNSNSN+S+NS+NoneWEWEW+E+WE+
7-4-1-120592086.0611.13002.8100100000000
7-1-1-420592010000000082.9014.02003.0700
7-1-4-1205920100000000100000000
6-5-2-014414483.1414.32002.530075.1003.7416.85004.31
6-0-2-514414490.5809.42000075.1016.0304.564.3100
6-5-0-214414474.8212.1809.663.340086.3113.6900000
6-2-0-514414474.827.75014.08003.3482.6015.21002.2000
6-2-5-014414491.968.040000085.9006.068.04000
6-0-5-214414493.2406.76000085.907.3406.76000
7-3-3-013728085.4513.02001.530077.6206.3014.3600.191.53
7-0-3-313728095.3404.43000.23077.6214.0706.671.4100.23
7-3-0-313728065.7314.44016.891.1601.7779.6818.55001.7700
8-2-2-115444090.389.620000097.8502.150000
8-1-2-215444097.4402.56000090.799.2100000
8-2-1-215444091.478.530000088.5711.4300000
7-4-2-010296082.7714.39002.840076.8602.2617.33003.54
7-0-2-410296095.1304.87000076.8616.8802.713.5400
7-4-0-210296073.0011.27013.112.620085.5414.4600000
7-2-0-410296073.0010.45013.93002.6281.8015.90002.3000
7-2-4-010296090.709.300000085.9004.809.30000
7-0-4-210296094.5505.45000085.908.6505.45000
8-3-1-110296086.5811.68001.7400100000000
8-1-1-310296010000000085.0913.17001.7400
8-1-3-1102960100000000100000000
6-6-1-04804881.1412.87005.990073.410018.73007.86
6-0-1-64804810000000073.4118.73007.8600
6-1-6-048048100000000100000000
W-N-E-SLayoutsNoneNSNSN+S+NS+NoneWEWEW+E+WE+
7-5-1-04118478.5915.17006.240071.380021.11007.51
7-0-1-54118410000000071.3821.11007.5100
7-5-0-14118477.1916.61006.2100100000000
7-1-0-54118477.190016.61006.2178.3215.71005.9600
7-1-5-041184100000000100000000
7-0-5-141184100000000100000000
8-3-2-05148084.3713.73001.900080.8401.9115.35001.90
8-0-2-35148097.7302.27000080.8415.0002.271.9000
8-3-0-25148073.1911.11014.121.590085.8814.1200000
8-2-0-35148073.1911.04014.19001.5983.4414.97001.5900
8-2-3-05148089.4610.540000085.9003.5710.54000
8-0-3-25148095.8704.13000085.909.9804.13000
8-4-1-02574078.9716.32004.710074.810020.08005.11
8-0-1-42574010000000074.8120.08005.1100
8-4-0-12574079.7215.91004.3700100000000
8-1-0-42574079.720015.91004.3779.1616.44004.4000
8-1-4-025740100000000100000000
8-0-4-125740100000000100000000
9-2-1-13432089.5110.4900000100000000
9-1-1-23432010000000089.5110.4900000
9-1-2-134320100000000100000000
9-2-2-01716088.2511.750000085.9002.3511.75000
9-0-2-21716097.2302.77000085.9011.3302.77000
9-2-0-21716076.929.86013.2200086.7813.2200000
W-N-E-SLayoutsNoneNSNSN+S+NS+NoneWEWEW+E+WE+
7-6-0-0686456.2926.920016.780056.2926.920016.7800
7-0-0-6686456.29026.920016.78056.2926.920016.7800
7-0-6-06864100000000100000000
9-3-1-01144081.8915.63002.480080.380017.13002.48
9-0-1-31144010000000080.3817.13002.4800
9-3-0-11144083.1114.62002.2700100000000
9-1-0-31144083.110014.62002.2782.7315.00002.2700
9-1-3-011440100000000100000000
9-0-3-111440100000000100000000
8-5-0-0514862.6325.640011.730062.6325.640011.7300
8-0-0-5514862.63025.640011.73062.6325.640011.7300
8-0-5-05148100000000100000000
10-1-1-16864100000000100000000
9-4-0-0286069.9323.08006.990069.9323.08006.9900
9-0-0-4286069.93023.08006.99069.9323.08006.9900
9-0-4-02860100000000100000000
10-2-1-0343287.3012.700000087.300012.70000
10-0-1-2343210000000087.3012.7000000
10-2-0-1343288.2311.7700000100000000
10-1-0-2343288.230011.7700088.2311.7700000
10-1-2-03432100000000100000000
10-0-2-13432100000000100000000
10-3-0-0114477.6219.23003.150077.6219.23003.1500
10-0-0-3114477.62019.23003.15077.6219.23003.1500
10-0-3-01144100000000100000000
11-1-1-0624100000000100000000
11-0-1-1624100000000100000000
11-1-0-1624100000000100000000
11-2-0-031285.9014.100000085.9014.1000000
11-0-0-231285.90014.10000085.9014.1000000
11-0-2-0312100000000100000000
12-1-0-052100000000100000000
12-0-0-152100000000100000000
12-0-1-052100000000100000000
13-0-0-04100000000100000000
W-N-E-SLayoutsNoneNSNSN+S+NS+NoneWEWEW+E+WE+
Totals6710886487.486.473.272.210.460.070.0585.916.954.072.430.460.090.08

The bottom line shows the percent occurrence of each frozen state over all suit patterns combined. Keep in mind that any differences among directions (WNES) are due to my arbitrary choice to assign West the greatest length — perhaps from a recollection of Horace Greeley, “Go West, young man!” or maybe I’m just dreaming about the “young man” part. Obviously there are no differences in theory.

I hope you enjoyed this trek through a frozen wasteland!