Study 8J65   Main

Frozen Suits

by Richard Pavlicek

This study will enter the deep freeze, so get out your winter clothes!

A suit is considered “frozen” to a player if leading that suit allows the opposing side to win an additional trick over what they could win by leading the suit themselves. Suit layouts can be frozen to any number of players (one to four) but the great majority are frozen to no one, i.e., any player can lead the suit without affecting the trick outcome.

Jump to Frozen States table

Years ago (circa 2012) I created a database with each of the 67,108,864* suit layouts — well not exactly. Any specific suit layout must be identical in all technical aspects if merely rotated, so I only considered layouts where West held the ace, thus requiring one-fourth the size, or 16,777,216. 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 reference tool: Suit Layout Analyzer will analyze any layout, or multiple layouts with all distributions between two hands. Check it out. It’s fun!

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.

Freezing your butt off

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 that 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 tables summarize 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.

Column 1 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.

Column 2 shows the number of specific layouts that comprise the generic pattern. 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.

Columns 3-9 show the percent occurrence of each frozen state to North-South (beneficial to West-East). Columns 10-16 do likewise for West-East (beneficial to North-South). A plus sign (+) in the column header means doubly frozen (or greater).

W-N-E-S	Layouts	None	N	S	NS	N+	S+	NS+	None	W	E	WE	W+	E+	WE+
4-3-3-3	4804800	83.01	5.84	4.70	5.92	0.34	0.18	0.01	81.05	7.61	4.93	5.82	0.40	0.17	0.01
4-4-3-2	3603600	80.74	6.56	6.76	5.46	0.48	0	0	83.04	5.07	8.01	3.73	0	0.16	0
4-2-3-4	3603600	84.40	5.92	5.21	4.23	0	0.24	0	81.74	5.11	8.56	4.23	0.36	0	0
4-3-4-2	3603600	85.55	5.32	5.70	3.05	0.37	0	0	84.55	6.73	5.41	3.05	0	0.26	0
5-3-3-2	2882880	85.85	6.88	3.96	2.70	0.60	0	0	84.92	8.63	3.64	2.70	0	0.11	0
5-2-3-3	2882880	85.38	7.63	3.77	3.03	0	0.19	0	84.17	7.45	4.88	3.03	0.47	0	0
5-3-2-3	2882880	86.90	5.56	4.71	2.41	0.43	0	0	82.28	10.02	4.56	2.41	0.73	0	0
5-4-2-2	2162160	82.22	6.62	6.69	3.72	0.75	0	0	83.38	6.45	7.29	2.88	0	0	0
5-2-2-4	2162160	82.97	9.70	3.65	3.69	0	0	0	80.21	8.26	6.37	4.37	0.79	0	0
5-2-4-2	2162160	88.53	5.81	4.72	0.93	0	0	0	87.46	6.54	5.07	0.93	0	0	0
4-4-4-1	1801800	91.10	8.08	0	0	0.83	0	0	89.76	0	9.27	0	0	0.97	0
5-4-3-1	1441440	89.17	9.74	0	0	1.09	0	0	94.16	0	5.63	0	0	0.21	0
5-1-3-4	1441440	94.65	0	5.08	0	0	0.27	0	89.07	9.77	0	0	1.16	0	0
5-4-1-3	1441440	90.21	8.86	0	0	0.93	0	0	89.29	10.05	0	0	0.66	0	0
5-3-1-4	1441440	90.57	8.93	0	0	0.49	0	0	86.59	12.25	0	0	1.16	0	0
5-3-4-1	1441440	91.76	7.56	0	0	0.68	0	0	92.53	0	7.13	0	0	0.35	0
5-1-4-3	1441440	93.88	0	5.71	0	0	0.41	0	90.60	8.82	0	0	0.59	0	0
6-3-2-2	1441440	85.98	7.44	3.99	1.79	0.80	0	0	84.54	8.55	5.13	1.79	0	0	0
6-2-2-3	1441440	85.51	9.88	2.04	2.57	0	0	0	83.08	9.86	3.73	2.57	0.77	0	0
6-2-3-2	1441440	88.69	6.88	3.59	0.84	0	0	0	87.39	8.09	3.68	0.84	0	0	0
6-3-3-1	960960	89.57	9.43	0	0	1.00	0	0	94.44	0	5.39	0	0	0.17	0
6-1-3-3	960960	95.85	0	3.94	0	0	0.21	0	88.65	10.46	0	0	0.90	0	0
6-3-1-3	960960	88.46	10.75	0	0	0.79	0	0	84.73	14.08	0	0	1.19	0	0
5-5-2-1	864864	89.85	9.14	0	0	1.01	0	0	96.53	0	3.47	0	0	0	0
5-1-2-5	864864	93.08	0	6.92	0	0	0	0	89.76	8.56	0	0	1.68	0	0
5-2-5-1	864864	93.79	6.21	0	0	0	0	0	94.44	0	5.56	0	0	0	0

W-N-E-S	Layouts	None	N	S	NS	N+	S+	NS+	None	W	E	WE	W+	E+	WE+
6-4-2-1	720720	88.90	9.59	0	0	1.52	0	0	98.03	0	1.97	0	0	0	0
6-1-2-4	720720	96.11	0	3.89	0	0	0	0	86.67	11.36	0	0	1.97	0	0
6-4-1-2	720720	90.49	7.97	0	0	1.54	0	0	89.93	10.07	0	0	0	0	0
6-2-1-4	720720	92.75	7.25	0	0	0	0	0	86.40	12.18	0	0	1.42	0	0
6-2-4-1	720720	92.63	7.37	0	0	0	0	0	95.60	0	4.40	0	0	0	0
6-1-4-2	720720	94.98	0	5.02	0	0	0	0	93.25	6.75	0	0	0	0	0
7-2-2-2	617760	89.02	7.90	2.43	0.65	0	0	0	87.25	9.72	2.38	0.65	0	0	0
5-4-4-0	360360	85.35	12.85	0	0	1.80	0	0	69.93	0	11.88	15.11	0	1.21	1.86
5-0-4-4	360360	89.65	0	9.38	0	0	0.97	0	69.93	14.45	0	12.52	2.01	0	1.10
5-4-0-4	360360	64.57	14.87	0	16.42	2.10	0	2.04	81.52	16.33	0	0	2.15	0	0
7-3-2-1	411840	88.62	10.11	0	0	1.27	0	0	98.32	0	1.68	0	0	0	0
7-1-2-3	411840	97.98	0	2.02	0	0	0	0	87.44	11.29	0	0	1.27	0	0
7-3-1-2	411840	90.18	8.68	0	0	1.14	0	0	88.76	11.24	0	0	0	0	0
7-2-1-3	411840	91.12	8.88	0	0	0	0	0	86.38	12.49	0	0	1.14	0	0
7-2-3-1	411840	91.49	8.51	0	0	0	0	0	96.74	0	3.26	0	0	0	0
7-1-3-2	411840	96.20	0	3.80	0	0	0	0	92.03	7.97	0	0	0	0	0
5-5-3-0	288288	82.28	16.00	0	0	1.72	0	0	73.72	0	10.21	13.21	0	0.56	2.30
5-0-3-5	288288	86.79	0	12.63	0	0	0.58	0	73.72	13.25	0	10.22	2.10	0	0.70
5-3-5-0	288288	90.66	8.59	0	0	0.75	0	0	77.62	0	10.30	10.68	0	0.65	0.75
6-5-1-1	288288	86.45	10.38	0	0	3.17	0	0	100	0	0	0	0	0	0
6-1-1-5	288288	100	0	0	0	0	0	0	83.96	12.49	0	0	3.55	0	0
6-1-5-1	288288	100	0	0	0	0	0	0	100	0	0	0	0	0	0
6-4-3-0	240240	82.83	15.04	0	0	2.14	0	0	75.17	0	6.93	15.21	0	0.25	2.44
6-0-3-4	240240	93.93	0	5.76	0	0	0.31	0	75.17	15.05	0	7.17	2.31	0	0.31
6-4-0-3	240240	67.72	13.33	0	15.94	1.65	0	1.35	82.00	16.65	0	0	1.35	0	0
6-3-0-4	240240	67.72	14.73	0	14.45	0.97	0	2.13	78.53	19.21	0	0	2.25	0	0
6-3-4-0	240240	88.09	10.80	0	0	1.11	0	0	77.62	0	8.32	12.56	0	0.38	1.11
6-0-4-3	240240	93.10	0	6.44	0	0	0.45	0	77.62	12.21	0	8.72	0.99	0	0.45

W-N-E-S	Layouts	None	N	S	NS	N+	S+	NS+	None	W	E	WE	W+	E+	WE+
7-4-1-1	205920	86.06	11.13	0	0	2.81	0	0	100	0	0	0	0	0	0
7-1-1-4	205920	100	0	0	0	0	0	0	82.90	14.02	0	0	3.07	0	0
7-1-4-1	205920	100	0	0	0	0	0	0	100	0	0	0	0	0	0
6-5-2-0	144144	83.14	14.32	0	0	2.53	0	0	75.10	0	3.74	16.85	0	0	4.31
6-0-2-5	144144	90.58	0	9.42	0	0	0	0	75.10	16.03	0	4.56	4.31	0	0
6-5-0-2	144144	74.82	12.18	0	9.66	3.34	0	0	86.31	13.69	0	0	0	0	0
6-2-0-5	144144	74.82	7.75	0	14.08	0	0	3.34	82.60	15.21	0	0	2.20	0	0
6-2-5-0	144144	91.96	8.04	0	0	0	0	0	85.90	0	6.06	8.04	0	0	0
6-0-5-2	144144	93.24	0	6.76	0	0	0	0	85.90	7.34	0	6.76	0	0	0
7-3-3-0	137280	85.45	13.02	0	0	1.53	0	0	77.62	0	6.30	14.36	0	0.19	1.53
7-0-3-3	137280	95.34	0	4.43	0	0	0.23	0	77.62	14.07	0	6.67	1.41	0	0.23
7-3-0-3	137280	65.73	14.44	0	16.89	1.16	0	1.77	79.68	18.55	0	0	1.77	0	0
8-2-2-1	154440	90.38	9.62	0	0	0	0	0	97.85	0	2.15	0	0	0	0
8-1-2-2	154440	97.44	0	2.56	0	0	0	0	90.79	9.21	0	0	0	0	0
8-2-1-2	154440	91.47	8.53	0	0	0	0	0	88.57	11.43	0	0	0	0	0
7-4-2-0	102960	82.77	14.39	0	0	2.84	0	0	76.86	0	2.26	17.33	0	0	3.54
7-0-2-4	102960	95.13	0	4.87	0	0	0	0	76.86	16.88	0	2.71	3.54	0	0
7-4-0-2	102960	73.00	11.27	0	13.11	2.62	0	0	85.54	14.46	0	0	0	0	0
7-2-0-4	102960	73.00	10.45	0	13.93	0	0	2.62	81.80	15.90	0	0	2.30	0	0
7-2-4-0	102960	90.70	9.30	0	0	0	0	0	85.90	0	4.80	9.30	0	0	0
7-0-4-2	102960	94.55	0	5.45	0	0	0	0	85.90	8.65	0	5.45	0	0	0
8-3-1-1	102960	86.58	11.68	0	0	1.74	0	0	100	0	0	0	0	0	0
8-1-1-3	102960	100	0	0	0	0	0	0	85.09	13.17	0	0	1.74	0	0
8-1-3-1	102960	100	0	0	0	0	0	0	100	0	0	0	0	0	0
6-6-1-0	48048	81.14	12.87	0	0	5.99	0	0	73.41	0	0	18.73	0	0	7.86
6-0-1-6	48048	100	0	0	0	0	0	0	73.41	18.73	0	0	7.86	0	0
6-1-6-0	48048	100	0	0	0	0	0	0	100	0	0	0	0	0	0

W-N-E-S	Layouts	None	N	S	NS	N+	S+	NS+	None	W	E	WE	W+	E+	WE+
7-5-1-0	41184	78.59	15.17	0	0	6.24	0	0	71.38	0	0	21.11	0	0	7.51
7-0-1-5	41184	100	0	0	0	0	0	0	71.38	21.11	0	0	7.51	0	0
7-5-0-1	41184	77.19	16.61	0	0	6.21	0	0	100	0	0	0	0	0	0
7-1-0-5	41184	77.19	0	0	16.61	0	0	6.21	78.32	15.71	0	0	5.96	0	0
7-1-5-0	41184	100	0	0	0	0	0	0	100	0	0	0	0	0	0
7-0-5-1	41184	100	0	0	0	0	0	0	100	0	0	0	0	0	0
8-3-2-0	51480	84.37	13.73	0	0	1.90	0	0	80.84	0	1.91	15.35	0	0	1.90
8-0-2-3	51480	97.73	0	2.27	0	0	0	0	80.84	15.00	0	2.27	1.90	0	0
8-3-0-2	51480	73.19	11.11	0	14.12	1.59	0	0	85.88	14.12	0	0	0	0	0
8-2-0-3	51480	73.19	11.04	0	14.19	0	0	1.59	83.44	14.97	0	0	1.59	0	0
8-2-3-0	51480	89.46	10.54	0	0	0	0	0	85.90	0	3.57	10.54	0	0	0
8-0-3-2	51480	95.87	0	4.13	0	0	0	0	85.90	9.98	0	4.13	0	0	0
8-4-1-0	25740	78.97	16.32	0	0	4.71	0	0	74.81	0	0	20.08	0	0	5.11
8-0-1-4	25740	100	0	0	0	0	0	0	74.81	20.08	0	0	5.11	0	0
8-4-0-1	25740	79.72	15.91	0	0	4.37	0	0	100	0	0	0	0	0	0
8-1-0-4	25740	79.72	0	0	15.91	0	0	4.37	79.16	16.44	0	0	4.40	0	0
8-1-4-0	25740	100	0	0	0	0	0	0	100	0	0	0	0	0	0
8-0-4-1	25740	100	0	0	0	0	0	0	100	0	0	0	0	0	0
9-2-1-1	34320	89.51	10.49	0	0	0	0	0	100	0	0	0	0	0	0
9-1-1-2	34320	100	0	0	0	0	0	0	89.51	10.49	0	0	0	0	0
9-1-2-1	34320	100	0	0	0	0	0	0	100	0	0	0	0	0	0
9-2-2-0	17160	88.25	11.75	0	0	0	0	0	85.90	0	2.35	11.75	0	0	0
9-0-2-2	17160	97.23	0	2.77	0	0	0	0	85.90	11.33	0	2.77	0	0	0
9-2-0-2	17160	76.92	9.86	0	13.22	0	0	0	86.78	13.22	0	0	0	0	0

W-N-E-S	Layouts	None	N	S	NS	N+	S+	NS+	None	W	E	WE	W+	E+	WE+
7-6-0-0	6864	56.29	26.92	0	0	16.78	0	0	56.29	26.92	0	0	16.78	0	0
7-0-0-6	6864	56.29	0	26.92	0	0	16.78	0	56.29	26.92	0	0	16.78	0	0
7-0-6-0	6864	100	0	0	0	0	0	0	100	0	0	0	0	0	0
9-3-1-0	11440	81.89	15.63	0	0	2.48	0	0	80.38	0	0	17.13	0	0	2.48
9-0-1-3	11440	100	0	0	0	0	0	0	80.38	17.13	0	0	2.48	0	0
9-3-0-1	11440	83.11	14.62	0	0	2.27	0	0	100	0	0	0	0	0	0
9-1-0-3	11440	83.11	0	0	14.62	0	0	2.27	82.73	15.00	0	0	2.27	0	0
9-1-3-0	11440	100	0	0	0	0	0	0	100	0	0	0	0	0	0
9-0-3-1	11440	100	0	0	0	0	0	0	100	0	0	0	0	0	0
8-5-0-0	5148	62.63	25.64	0	0	11.73	0	0	62.63	25.64	0	0	11.73	0	0
8-0-0-5	5148	62.63	0	25.64	0	0	11.73	0	62.63	25.64	0	0	11.73	0	0
8-0-5-0	5148	100	0	0	0	0	0	0	100	0	0	0	0	0	0
10-1-1-1	6864	100	0	0	0	0	0	0	100	0	0	0	0	0	0
9-4-0-0	2860	69.93	23.08	0	0	6.99	0	0	69.93	23.08	0	0	6.99	0	0
9-0-0-4	2860	69.93	0	23.08	0	0	6.99	0	69.93	23.08	0	0	6.99	0	0
9-0-4-0	2860	100	0	0	0	0	0	0	100	0	0	0	0	0	0
10-2-1-0	3432	87.30	12.70	0	0	0	0	0	87.30	0	0	12.70	0	0	0
10-0-1-2	3432	100	0	0	0	0	0	0	87.30	12.70	0	0	0	0	0
10-2-0-1	3432	88.23	11.77	0	0	0	0	0	100	0	0	0	0	0	0
10-1-0-2	3432	88.23	0	0	11.77	0	0	0	88.23	11.77	0	0	0	0	0
10-1-2-0	3432	100	0	0	0	0	0	0	100	0	0	0	0	0	0
10-0-2-1	3432	100	0	0	0	0	0	0	100	0	0	0	0	0	0
10-3-0-0	1144	77.62	19.23	0	0	3.15	0	0	77.62	19.23	0	0	3.15	0	0
10-0-0-3	1144	77.62	0	19.23	0	0	3.15	0	77.62	19.23	0	0	3.15	0	0
10-0-3-0	1144	100	0	0	0	0	0	0	100	0	0	0	0	0	0
11-1-1-0	624	100	0	0	0	0	0	0	100	0	0	0	0	0	0
11-0-1-1	624	100	0	0	0	0	0	0	100	0	0	0	0	0	0
11-1-0-1	624	100	0	0	0	0	0	0	100	0	0	0	0	0	0
11-2-0-0	312	85.90	14.10	0	0	0	0	0	85.90	14.10	0	0	0	0	0
11-0-0-2	312	85.90	0	14.10	0	0	0	0	85.90	14.10	0	0	0	0	0
11-0-2-0	312	100	0	0	0	0	0	0	100	0	0	0	0	0	0
12-1-0-0	52	100	0	0	0	0	0	0	100	0	0	0	0	0	0
12-0-0-1	52	100	0	0	0	0	0	0	100	0	0	0	0	0	0
12-0-1-0	52	100	0	0	0	0	0	0	100	0	0	0	0	0	0
13-0-0-0	4	100	0	0	0	0	0	0	100	0	0	0	0	0	0

Overall Summary

Layouts	None	N	S	NS	N+	S+	NS+	None	W	E	WE	W+	E+	WE+
67108864	87.48	6.47	3.27	2.21	0.46	0.07	0.05	85.91	6.95	4.07	2.43	0.46	0.09	0.08

The overall summary shows the percent occurrence of each frozen state in all suit patterns. Note that the number of layouts is equal to 4¹³ (all possible distributions of 13 cards among four hands). In theory there is no difference among directions (WNES) but evident here because of my arbitrary choice to give West the greatest length. Perhaps Horace Greeley’s “Go West, young man!” was an influence, though I had to be dreaming about the “young man” part.

I hope you enjoyed this trek through a frozen wasteland.

Please remove your snow shoes before leaving!

Study 8J65   Main

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© 2019 Richard Pavlicek