This riddle was posted on 538 today. Would love to know what the braintrust here has to say:

You are playing a standard game of Texas Hold ’em, in which you start with two cards. Five community cards will be dealt, and the object is to make the best five-card hand out of the seven cards available to you. However, you’re a perfectionist, and you’re only interested in making the best possible hand once the board has been dealt — such a hand is called the “nuts.” Which two-card starting hand is the most likely to make the nuts once the board has been dealt?

Clearly, preflop, the hand most likely to win hot-and-cold is AA. But how does one approach the problem as stated; i.e., making the best possible hand once the board has been dealt?

Thoughts?

I've only thought about it for a few seconds, but I think the answer has to be ATs. That way, most of the time you make a flush, it's the nuts; plus you can make the nut straight sometimes; plus the T in your hand blocks straight flushes more often than a higher Broadway card would.

EDIT: Actually now I think it's AKs. It blocks fewer straight flushes, but aces full of kings is the nuts significantly more often than aces full of tens.

Off the top of my head I'd guess JTs. You can make the maximum amount of straights (which adds up to a ton of boards when you include different suit combinations), all of them are the nut straight (which isn't true for lower connectors), and being suited helps in that you can make straight flushes on flush boards where a regular straight wouldn't be the nuts.

Start with figuring out how many of each type of board is possible and determining what those types are.

ATs.




This riddle was also posted here in 2010:

https://forumserver.twoplustwo.com/1...?highlight=ats


I said ATs back then too:



Someone ran a simulation to get the answer and listed hands in the order that they are most likely to make the nuts. That post is here:

https://forumserver.twoplustwo.com/s...6&postcount=46

My thought was JTs. The linked thread proves however that CallMeVernon nailed it, ATs.

I had discounted most full houses that are also possible ties with the exact same hand, and forgot those are the nuts as well.

I thought there was another more recent thread in the Probability Forum on this topic. I remember running many cases on a computer.

Also, I thought that this was NOT a problem in which simulation is useful/needed since a computer can whip through all possible boards fairly quickly to determine how many of them give a given specific two-card hand the absolute nuts (for that board).

Maybe I am remembering some other thread/question.

Edit to add: after skimming the earlier thread, I seem to be very confused what is being asked and how it was answered.

I'm not too surprised to see ATs, AJs, AQs and AKs are the top four, but I'm intrigued to learn that the next best are apparently A9s and A6s. A6s???

The 9 and the 6 block a ton of straight flushes and give more true absolute nut flushes for the ace, apparently.

Today I wrote a program to determine the absolute nuts for any 5-card board in NLHE (i.e., the set of hole cards that are guaranteed to win the hand with that board).

I then looped over all possible boards for a few hole cards (one pair of hole cards at a time) and tallied on how many of the total boards possible, given that one player holds those hole cards, do those hole cards make the absolute nuts. Of course, given that a player is dealt a specific pair of hole cards, there are C(50,5) = 2,118,760 possible 5-card boards that need to be considered. So in the table below, 2,118,760 is the denominator for the Pct Board Making Nuts calculations.

Since looping over 2,118,760 possible boards and finding the absolute nuts for each board takes a fair amount of time, I have only so far done the calculations for 11 starting hands. These results are given in the table below.

Rank	Hole Cards	Boards Making Nuts	Pct Boards Making Nuts
1	ATs	82,442	3.89%
2	AJs	80,995	3.82%
3	AQs	79,517	3.75%
4	AKs	79,023	3.73%
5	JTs	66,353	3.13%
6	A9s	63,639	3.00%
7	A6s	63,621	3.00%
8	A8s	63,479	3.00%
9	A7s	63,473	3.00%
10	JTo	62,276	2.94%
11	A5s	62,153	2.93%

Time permitting I will try to do another set of starting hands tomorrow and update the table when I have new results.

Update with more results.

Rank	Hole Cards	Boards Making Nuts	Pct Boards Making Nuts
1	ATs	82,442	3.89%
2	AJs	80,995	3.82%
3	AQs	79,517	3.75%
4	AKs	79,023	3.73%
5	JTs	66,353	3.13%
6	A9s	63,639	3.00%
7	A6s	63,621	3.00%
8	A8s	63,479	3.00%
9	A7s	63,473	3.00%
10	JTo	62,276	2.94%
.
11	A5s	62,153	2.93%
12	A4s	60,694	2.86%
13	A3s	59,204	2.79%
14	A2s	57,653	2.72%
T15	QJs	56,566	2.67%
T15	QTs	56,566	2.67%
17	KTs	55,994	2.64%
18	KJs	55,907	2.64%
19	KQs	55,848	2.64%
T20	QJo	49,676	2.34%
T20	QTo	49,676	2.34%
.
22	T9s	41,848	1.98%
23	T9o	38,789	1.83%
T24	KQo	37,122	1.75%
T24	KJo	37,122	1.75%
T24	KTo	37,122	1.75%
T27	98s	33,057	1.56%
T27	87s	33,057	1.56%
T27	76s	33,057	1.56%
T27	65s	33,057	1.56%
.
31	KK	32,750	1.55%
32	AA	31,118	1.47%
T33	98o	30,044	1.42%
T33	87o	30,044	1.42%
T33	76o	30,044	1.42%
T33	65o	30,044	1.42%
37	54s	29,451	1.39%
T38	J9s	28,213	1.33%
T38	T8s	28,213	1.33%
40	AKo	27,044	1.28%
.
41	54o	26,444	1.25%
T42	J9o	26,189	1.24%
T42	T8o	26,189	1.24%
44	AQo	25,981	1.23%
45	AJo	25,966	1.23%
46	ATo	25,951	1.22%
T47	97s	23,067	1.09%
T47	86s	23,067	1.09%
T47	75s	23,067	1.09%
50	K5s	22,613	1.07%

Your results are somewhat different to bachfan's. How does KK make the nuts more often than AA?

KK will make with the nuts with top set on a no-straight board more often than AA will (with an A on board, more straights are possible).

Yes, some of those figures do not agree with bachfan's earlier results. My guess is that I screwed up my logic somewhere.

If I have time I'll try to look into it tonight when, in all likelihood, I'll have to re-run that set of starting hands.

Since your percentages are lower than Bachfan's, my 1st thought was you were missing things like both AA and AT can be the nuts on the same board, since AT has a quad blocker, but things appear to be more serious with your app e.g. 76o > 65o, since 76o can make more nut full houses.

Another possible misstep can happen by missing flushes with a str8 flush blocker.

All these possibilities might be causing the lower percentages you are generating. It's tricky.

Thanks for your comment.

I am definitely capturing (or I am intending to capture) ALL the hands that are the nuts (or effective nuts) on each board. So that is not obviously the reason my code is finding fewer nut boards than Bachfan.

And, yes, I tried to capture Axs being the nuts (the effective nuts) when the x blocks any possible straight flushes. But maybe I screwed that up.

I haven't turned my attention to identifying where my code is screwing up but I do have a question.

It may be obvious, but why "should" 76o make the nuts on more possible boards than 65o can? It is not immediately obvious to me.

There are more boards with 776 and two cards under 6 than there are boards with 665 and two cards under 5.

FiveThirtyEight gave their answer today. Of course it was JT suited. But they also gave credit - and links - to Two Plus Two Forums for tackling this previously.

This must be with the caveat that most full houses aren’t the “solo nuts” ?!

Edit: nvm article answer was ATs.

Yes, typo on my part. Wish I could blame autocorrect but I'm just an idiot.

I have done only two small things on this topic in the last few days.

1. I have reviewed my code and am appalled and aghast. I over-looked many cases (undercounting) and screwed up the logic in some cases (error in counting). Jeezo Beezo. Not good.

2. I think I now better understand what Bachfan did those many years ago. I was puzzled why he did a simulation over more trials than there were possible boards. If he had the logic to determine the nuts for any board, then he would not need to do this.

Aha. That is the clue. My belief is that Bachfan did not have such logic but he had access to a really fast and powerful NLHE game player that could determine the winner of any NLHE deal (ties included).

So I think that Bachfan generated millions and millions of random boards and by pitting each possible starting hand vs. all other starting hands (he undoubtedly was clever in how he did this) for that board, he could tally how many boards for which each starting hand was the nuts. Maybe this was obvious to everybody except me. To be honest, I have not gone back and reviewed the earlier thread since I have been too busy on other projects. Perhaps this was discussed in the earlier thread.

Anyway, as has been pointed out in this thread above, writing code to determine the nuts for any possible 5-card board in NLHE (using logic, not brute force) is not an easy task. If I have time, I will look into "fixing" my woefully inadequate program over the weekend.

Don't be too hard on yourself. I wouldn't know where to start with this sort of thing!

Bachfan is the writer of propokertools, btw

I finally found the time to do a proper job of tallying on how many possible boards each starting hand in NLHE can make the nuts. My earlier efforts were inadequate and filled with mistakes.

Below is a table of every starting hand and how many NUTS boards it can make with the breakdown of the numbers in each category of nuts (royal flush, straight flush, etc.).

Bachfan previously reported on frequencies based upon simulations of millions of boards for each starting hand. Using the "opposite" approach, I wrote a complex program based upon dozens of logic gates to determine the set of starting hands which would be the nuts for any given 5-card board.

Another program then loops over all possible starting hands, constructs all the possible boards given the starting hand, and then calls the "inner" function that returns all the Nut hands given that board. The "outer" function then tallies for how many boards the specific starting hand is the nuts.

While I have tried to be careful in developing the underlying logic and the copious programs, mistakes are known to creep in (as my earlier attempt surely demonstrated). I hope that everything below is correct but there is always the possibility that something is amiss. Feel free to point out anything that looks suspect.

For comparability (and sanity), the table shows the percentage that I have found via logic-based brute force methods versus the percentage that Bachfan found via his simulations. Statistically, based upon the number of trials he used in his simulations, his results were likely to be within .01 percent of the actual figure.

Rank	Starting Hand	Royal Flush Nut_Boards	Straight Flush Nut_Boards	Four of a Kind Nut_Boards	Full House Nut_Boards	Flush Nut_Boards	Straight Nut_Boards	Three of a Kind Nut_Boards	TOTAL NUT_BOARDS	Nut_Board Pct	Bachfan_Sim Pct
1	ATs	1,084	135	2,560	4,158	59,850	24,324	0	92,111	4.35%	4.34%
2	AJs	1,084	90	2,575	5,160	56,714	24,324	0	89,947	4.25%	4.24%
3	AQs	1,084	45	2,590	6,246	53,635	24,324	0	87,924	4.15%	4.14%
4	AKs	1,084	0	2,605	7,416	51,013	24,324	0	86,442	4.08%	4.08%
5	A9s	49	180	2,554	3,114	63,607	5,100	0	74,604	3.52%	3.53%
6	A6s	49	133	2,536	882	63,607	5,100	0	72,307	3.41%	3.41%
7	JTs	1,084	3,105	2,071	3,744	0	62,124	0	72,128	3.40%	3.40%
8	A8s	49	133	2,548	2,220	60,897	5,100	0	70,947	3.35%	3.34%
9	A7s	49	133	2,542	1,476	60,897	5,100	0	70,197	3.31%	3.31%
10	A5s	49	1,123	2,530	333	59,850	5,100	0	68,985	3.26%	3.26%
11	JTo	94	315	2,116	3,900	230	62,142	0	68,797	3.25%	3.25%
12	A4s	49	1,075	2,533	135	56,714	5,100	0	65,606	3.10%	3.10%
13	KQs	1,084	1,035	2,173	6,120	16,719	36,924	0	64,055	3.02%	3.02%
14	QJs	1,084	2,070	2,101	4,860	3,848	49,524	0	63,487	3.00%	3.00%
15	KJs	1,084	1,080	2,158	5,070	16,808	36,924	0	63,124	2.98%	2.98%
16	QTs	1,084	2,115	2,086	3,930	3,848	49,524	0	62,587	2.95%	2.95%
17	KTs	1,084	1,125	2,143	4,104	16,984	36,924	0	62,364	2.94%	2.95%
18	A3s	49	946	2,536	24	53,635	5,100	0	62,290	2.94%	2.94%
19	A2s	49	0	2,539	0	51,013	5,100	0	58,701	2.77%	2.78%
20	QJo	94	225	2,140	5,040	1,283	49,542	0	58,324	2.75%	2.76%
21	QTo	94	270	2,113	4,036	1,045	49,542	0	57,100	2.69%	2.70%
22	KQo	94	135	2,206	6,324	3,408	36,942	0	49,109	2.32%	2.32%
23	T9s	49	4,140	2,050	2,772	0	38,700	0	47,711	2.25%	2.25%
24	KJo	94	180	2,179	5,192	2,568	36,942	0	47,155	2.23%	2.23%
25	KTo	94	225	2,158	4,158	2,327	36,942	0	45,904	2.17%	2.16%
26	T9o	49	313	2,092	2,904	0	38,700	0	44,058	2.08%	2.08%
27	AKo	94	45	2,632	7,644	7,488	24,342	0	42,245	1.99%	1.99%
28	AA	94	0	16,624	0	10,028	5,100	9,300	41,146	1.94%	1.94%
29	AQo	94	90	2,605	6,384	6,164	24,342	0	39,679	1.87%	1.88%
30	AJo	94	135	2,584	5,222	5,317	24,342	0	37,694	1.78%	1.78%
31	KK	94	90	16,162	0	4,600	5,100	11,400	37,446	1.77%	1.77%
32	98s	4	3,013	2,038	1,944	0	30,000	0	36,999	1.75%	1.74%
33	ATo	94	180	2,569	4,158	5,070	24,342	0	36,413	1.72%	1.71%
34	87s	4	3,013	2,026	1,260	0	30,000	0	36,303	1.71%	1.71%
35	76s	4	3,013	2,014	720	0	30,000	0	35,751	1.69%	1.69%
36	65s	4	3,013	2,002	333	0	30,000	0	35,352	1.67%	1.67%
37	J9s	49	3,150	2,065	2,934	609	26,100	0	34,907	1.65%	1.65%
38	98o	4	266	2,080	2,052	0	30,000	0	34,402	1.62%	1.63%
39	87o	4	266	2,068	1,344	0	30,000	0	33,682	1.59%	1.59%
40	T8s	49	3,195	2,044	2,082	0	26,100	0	33,470	1.58%	1.58%
41	76o	4	266	2,056	780	0	30,000	0	33,106	1.56%	1.56%
42	65o	4	266	2,044	375	0	30,000	0	32,689	1.54%	1.54%
43	J9o	49	268	2,089	3,024	231	26,100	0	31,761	1.50%	1.50%
44	54s	4	3,007	1,999	135	0	26,400	0	31,545	1.49%	1.48%
45	T8o	49	313	2,068	2,156	0	26,100	0	30,686	1.45%	1.45%
46	K9s	49	1,170	2,137	3,222	17,771	5,100	0	29,449	1.39%	1.39%
47	K5s	49	178	2,113	513	20,946	5,100	0	28,899	1.36%	1.36%
48	54o	4	221	2,032	153	0	26,400	0	28,810	1.36%	1.36%
49	K4s	49	133	2,116	207	20,765	5,100	0	28,370	1.34%	1.34%
50	K3s	49	88	2,119	36	20,586	5,100	0	27,978	1.32%	1.32%
51	K2s	49	45	2,122	0	20,435	5,100	0	27,751	1.31%	1.31%
52	K8s	49	225	2,131	2,322	17,921	5,100	0	27,748	1.31%	1.31%
53	97s	4	3,058	2,032	1,374	0	21,000	0	27,468	1.30%	1.30%
54	86s	4	3,058	2,020	810	0	21,000	0	26,892	1.27%	1.26%
55	K7s	49	178	2,125	1,566	17,724	5,100	0	26,742	1.26%	1.26%
56	75s	4	3,058	2,008	405	0	21,000	0	26,475	1.25%	1.25%
57	64s	4	3,013	2,005	135	0	21,000	0	26,157	1.23%	1.23%
58	QQ	94	180	15,646	0	2,074	5,100	3,000	26,094	1.23%	1.23%
59	K6s	49	178	2,119	954	17,539	5,100	0	25,939	1.22%	1.23%
60	Q9s	49	2,160	2,080	3,084	4,463	14,100	0	25,936	1.22%	1.23%
61	97o	4	266	2,056	1,432	0	21,000	0	24,758	1.17%	1.17%
62	86o	4	266	2,044	852	0	21,000	0	24,166	1.14%	1.15%
63	75o	4	266	2,032	435	0	21,000	0	23,737	1.12%	1.12%
64	64o	4	221	2,029	153	0	21,000	0	23,407	1.10%	1.10%
65	53s	4	3,007	2,002	24	0	17,400	0	22,437	1.06%	1.06%
66	J8s	49	2,205	2,059	2,208	609	14,100	0	21,230	1.00%	1.00%
67	JJ	94	270	15,180	0	460	5,100	0	21,104	1.00%	0.99%
68	Q9o	49	223	2,092	3,130	1,046	14,100	0	20,640	0.97%	0.97%
69	TT	94	360	14,720	0	0	5,100	0	20,274	0.96%	0.96%
70	T7s	49	2,250	2,038	1,476	0	14,100	0	19,913	0.94%	0.94%
71	53o	4	176	2,023	28	0	17,400	0	19,631	0.93%	0.93%
72	J8o	49	268	2,071	2,246	231	14,100	0	18,965	0.90%	0.90%
73	T7o	49	313	2,050	1,506	0	14,100	0	18,018	0.85%	0.85%
74	43s	4	1,972	2,005	24	0	13,800	0	17,805	0.84%	0.84%
75	A9o	49	133	2,554	3,114	5,061	5,100	0	16,011	0.76%	0.75%
76	43o	4	131	2,026	28	0	13,800	0	15,989	0.75%	0.75%
77	99	4	266	14,456	0	0	600	0	15,326	0.72%	0.72%
78	Q8s	49	1,214	2,074	2,322	4,468	5,100	0	15,227	0.72%	0.72%
79	A8o	49	133	2,548	2,218	5,066	5,100	0	15,114	0.71%	0.71%
80	88	4	266	14,194	0	0	600	0	15,064	0.71%	0.71%
81	77	4	266	13,926	0	0	600	0	14,796	0.70%	0.70%
82	96s	4	2,200	2,026	888	0	9,600	0	14,718	0.69%	0.69%
83	66	4	266	13,652	0	0	600	0	14,522	0.69%	0.69%
84	A7o	49	133	2,542	1,470	5,066	5,100	0	14,360	0.68%	0.68%
85	85s	4	2,200	2,014	465	0	9,600	0	14,283	0.67%	0.68%
86	55	4	266	13,398	0	0	600	0	14,268	0.67%	0.67%
87	44	4	176	13,358	0	0	600	0	14,138	0.67%	0.67%
88	33	4	86	13,312	0	0	600	0	14,002	0.66%	0.66%
89	74s	4	2,155	2,011	177	0	9,600	0	13,947	0.66%	0.66%
90	22	4	0	13,262	0	0	600	0	13,866	0.65%	0.66%
91	A6o	49	133	2,536	870	5,061	5,100	0	13,749	0.65%	0.65%
92	63s	4	2,110	2,008	24	0	9,600	0	13,746	0.65%	0.65%
93	Q7s	49	270	2,068	1,566	4,471	5,100	0	13,524	0.64%	0.64%
94	A5o	49	133	2,557	375	5,070	5,100	0	13,284	0.63%	0.63%
95	A4o	49	88	2,548	153	5,079	5,100	0	13,017	0.61%	0.62%
96	K9o	49	178	2,143	3,222	2,324	5,100	0	13,016	0.61%	0.61%
97	A3o	49	43	2,545	28	5,089	5,100	0	12,854	0.61%	0.60%
98	Q6s	49	223	2,062	954	4,464	5,100	0	12,852	0.61%	0.60%
99	96o	4	266	2,038	910	0	9,600	0	12,818	0.60%	0.60%
100	A2o	49	0	2,548	0	5,101	5,100	0	12,798	0.60%	0.61%
101	Q5s	49	223	2,056	513	4,653	5,100	0	12,594	0.59%	0.59%
102	85o	4	266	2,026	481	0	9,600	0	12,377	0.58%	0.58%
103	Q4s	49	178	2,059	207	4,647	5,100	0	12,240	0.58%	0.58%
104	K8o	49	178	2,131	2,322	2,322	5,100	0	12,102	0.57%	0.57%
105	74o	4	221	2,023	187	0	9,600	0	12,035	0.57%	0.57%
106	Q3s	49	133	2,062	36	4,641	5,100	0	12,021	0.57%	0.57%
107	Q2s	49	90	2,065	0	4,636	5,100	0	11,940	0.56%	0.56%
108	63o	4	176	2,020	28	0	9,600	0	11,828	0.56%	0.56%
109	K7o	49	178	2,125	1,566	2,324	5,100	0	11,342	0.54%	0.53%
110	Q8o	49	223	2,080	2,322	1,046	5,100	0	10,820	0.51%	0.52%
111	K6o	49	178	2,119	954	2,327	5,100	0	10,727	0.51%	0.51%
112	J7s	49	1,259	2,053	1,566	609	5,100	0	10,636	0.50%	0.50%
113	K5o	49	178	2,113	513	2,315	5,100	0	10,268	0.48%	0.49%
114	52s	4	2,104	2,005	0	0	6,000	0	10,113	0.48%	0.47%
115	Q7o	49	223	2,068	1,566	1,046	5,100	0	10,052	0.47%	0.48%
116	K4o	49	133	2,116	207	2,318	5,100	0	9,923	0.47%	0.47%
117	K3o	49	88	2,119	36	2,320	5,100	0	9,712	0.46%	0.46%
118	K2o	49	45	2,122	0	2,321	5,100	0	9,637	0.45%	0.45%
119	T6s	49	1,304	2,032	954	0	5,100	0	9,439	0.45%	0.45%
120	Q6o	49	223	2,062	954	1,046	5,100	0	9,434	0.45%	0.44%
121	42s	4	1,972	2,008	0	0	5,400	0	9,384	0.44%	0.44%
122	J7o	49	268	2,059	1,566	231	5,100	0	9,273	0.44%	0.44%
123	J6s	49	315	2,047	954	609	5,100	0	9,074	0.43%	0.43%
124	Q5o	49	223	2,056	513	1,043	5,100	0	8,984	0.42%	0.42%
125	J6o	49	268	2,047	954	231	5,100	0	8,649	0.41%	0.41%
126	Q4o	49	178	2,059	207	1,043	5,100	0	8,636	0.41%	0.41%
127	J5s	49	268	2,041	513	615	5,100	0	8,586	0.41%	0.40%
128	T6o	49	313	2,038	954	0	5,100	0	8,454	0.40%	0.40%
129	Q3o	49	133	2,062	36	1,043	5,100	0	8,423	0.40%	0.40%
130	Q2o	49	90	2,065	0	1,043	5,100	0	8,347	0.39%	0.39%
131	J4s	49	223	2,044	207	615	5,100	0	8,238	0.39%	0.39%
132	J5o	49	268	2,041	513	231	5,100	0	8,202	0.39%	0.39%
133	52o	4	133	2,020	0	0	6,000	0	8,157	0.38%	0.38%
134	T5s	49	360	2,026	513	0	5,100	0	8,048	0.38%	0.38%
135	J3s	49	178	2,047	36	615	5,100	0	8,025	0.38%	0.38%
136	T5o	49	313	2,026	513	0	5,100	0	8,001	0.38%	0.38%
137	J2s	49	135	2,050	0	615	5,100	0	7,949	0.38%	0.37%
138	J4o	49	223	2,044	207	231	5,100	0	7,854	0.37%	0.37%
139	T4s	49	268	2,029	207	0	5,100	0	7,653	0.36%	0.36%
140	T4o	49	268	2,029	207	0	5,100	0	7,653	0.36%	0.36%
141	J3o	49	178	2,047	36	231	5,100	0	7,641	0.36%	0.36%
142	J2o	49	135	2,050	0	231	5,100	0	7,565	0.36%	0.35%
143	42o	4	88	2,023	0	0	5,400	0	7,515	0.35%	0.35%
144	T3s	49	223	2,032	36	0	5,100	0	7,440	0.35%	0.35%
145	T3o	49	223	2,032	36	0	5,100	0	7,440	0.35%	0.35%
146	T2s	49	180	2,035	0	0	5,100	0	7,364	0.35%	0.35%
147	T2o	49	180	2,035	0	0	5,100	0	7,364	0.35%	0.35%
148	32s	4	940	2,011	0	0	3,000	0	5,955	0.28%	0.28%
149	32o	4	43	2,026	0	0	3,000	0	5,073	0.24%	0.24%
150	95s	4	1,258	2,020	513	0	600	0	4,395	0.21%	0.21%
151	84s	4	1,213	2,017	207	0	600	0	4,041	0.19%	0.19%
152	73s	4	1,168	2,014	36	0	600	0	3,822	0.18%	0.18%
153	62s	4	1,123	2,011	0	0	600	0	3,738	0.18%	0.18%
154	95o	4	266	2,026	513	0	600	0	3,409	0.16%	0.16%
155	94s	4	268	2,023	207	0	600	0	3,102	0.15%	0.15%
156	94o	4	221	2,023	207	0	600	0	3,055	0.14%	0.14%
157	84o	4	221	2,023	207	0	600	0	3,055	0.14%	0.14%
158	83s	4	223	2,020	36	0	600	0	2,883	0.14%	0.14%
159	93s	4	176	2,026	36	0	600	0	2,842	0.13%	0.13%
160	93o	4	176	2,026	36	0	600	0	2,842	0.13%	0.13%
161	83o	4	176	2,020	36	0	600	0	2,836	0.13%	0.13%
162	73o	4	176	2,020	36	0	600	0	2,836	0.13%	0.14%
163	72s	4	178	2,017	0	0	600	0	2,799	0.13%	0.14%
164	92s	4	133	2,029	0	0	600	0	2,766	0.13%	0.13%
165	92o	4	133	2,029	0	0	600	0	2,766	0.13%	0.13%
166	82s	4	133	2,023	0	0	600	0	2,760	0.13%	0.13%
167	82o	4	133	2,023	0	0	600	0	2,760	0.13%	0.13%
168	72o	4	133	2,017	0	0	600	0	2,754	0.13%	0.13%
169	62o	4	133	2,017	0	0	600	0	2,754	0.13%	0.13%

Let me know if you have any questions or comments.

Wow. This is simply amazing. Quite a triumph! I know any complement I have to offer is shallow, since I can not appreciate all the effort you put into this. I can say for certain, that I greatly admire a logical solution versus a brute force simulation, when such a logical solution is possible.