My home game has recently introduced Short Deck Omaha into its rotation, a game not for the faint of heart. Very lively, very swingy, very fun.

As everybody knows by now, Short Deck is played with the 2's thru 5's removed, so the deck consists of 36 cards (Ace-Six). Short Deck is sometimes referred to as Six Plus but that name isn't used much anymore.

The question arose how many different starting hands are there in Short Deck Omaha. And we mean taking into account "suit isomorphisms". In regular 52-card HoldEm there are 169 (= 13*13) different starting hands where the C(52,2) = 1,326 boils down to 169 when you care about suited or unsuited hands, but don't care about the specific suits. Similarly, in short deck HoldEm there are 81 (= 9*9) different starting hands where you only care about suited or unsuited cards.

AhJh is suit-isomorphic to AdJd (they are both in the "AJ suited" bucket). And Ks8d is suit-isomorphic to Kc8h (they are both in the "K8 offsuit" bucket).

In both Long Deck Omaha (full 52-cards) and Short Deck Omaha (36-card deck), of course, players are dealt four hole cards so the "interactions" between suits and ranks are a bit more complex than in HoldEm. For example, AdKcTs7h is suit-isomorphic to AsKhTd7c, but AdKcTd7h is not suit-isomorphic to AsKhTd7s (the first hand has a suited AT whereas the second hand has a suited A7 and we care about that difference).

In Long Deck Omaha there are C(52,4) = 270,725 different starting hands (where each card is treated as unique) and in Short Deck Omaha there are C(36,4) = 58,905 different starting hands (again, where each card is treated as unique).

But what is the equivalent number for Omaha to the 169 suit-isomorphic number in Long Deck Holdem or the 81 in Short Deck HoldEm (for either Long Deck Omaha or Short Deck Omaha)?

I will first derive the Long Deck Omaha number (I am sure this derivation has been done many times by many different people). Then I will extend the derivation to Short Deck Omaha.


Number of Different Long Deck Omaha Suit-Isomorphic Starting Hands

There are five cases to consider and we will walk through each case.

Case 1: Quads
There are clearly C(13,1) = 13 different possible quad starting hands, one for each rank from Ace to Deuce. And each quads hand has exactly one way for the suits to appear. The suit-isomorphism concept doesn't come into play here.

Case 2: Trips (trips with a singleton)

There are C(13,2)*C(2,1) = 156 different rank combos for a 4-card hand with 3 cards of the same rank and the fourth card of a different rank.

For concreteness, suppose the ranks are KKKQ. How many different suit-isomorphism sub-cases are there and how many different unique hands exist in each subcase?

Subcase 2.1: Rainbow = (K)(K)(K)(Q) where the parentheses surround cards of the same suit. Here there are C(4,3) = 4 possible unique hands.

Subcase 2.2: One suit in common = (KQ)(K)(K). Here there are C(4,3)*3 = 12 possible unique hands.

Subtotal = 4+12 = 16

Case 3: Two Pair

There are C(13,2) = 78 different rank combos for a 4-card hand with 2 cards of one rank and the other two cards of a different rank.

For concreteness, suppose the ranks are KKQQ. How many different suit-isomorphism sub-cases are there and how many different unique hands exist in each subcase?

Subcase 3.1: Rainbow = (K)(K)(Q)(Q). Here there are C(4,2) = 6 possible unique hands.

Subcase 3.2: One suit in common = (KQ)(K)(Q). Here there are C(4,2)*2*2 = 24 possible unique hands.

Subcase 3.3: Two suits in common = (KQ)(KQ). Here there are C(4,2) = 6 possible unique hands.

Subtotal = 6+24+6 = 36

Case 4: One Pair (one pair and two singletons):

There are C(13,3)*C(3,1) = 858 different rank combos for a 4-card hand with 2 cards of one rank and the other two cards having two different ranks.

For concreteness, suppose the ranks are KKQJ. How many different suit-isomorphism sub-cases are there and how many different unique hands exist in each subcase?

Subcase 4.1: (K)(K)(Q)(J). Here there are C(4,2)*2 = 12 possible unique hands.

Subcase 4.2: (QJ)(K)(K). Here there are C(4,2)*2 = 12 possible unique hands.

Subcase 4.3: (KQ)(K)(J). Here there are C(4,2)*2*2 = 24 possible unique hands.

Subcase 4.4: (KJ)(K)(Q). Here there are C(4,2)*2*2 = 24 possible unique hands.

Subcase 4.5: (KQ)(KJ). Here there are C(4,2)*2 = 12 possible unique hands.

Subcase 4.6: (KQJ)(K). Here there are C(4,2)*2 = 12 possible unique hands.

Subtotal = 12+12+24+24+12+12 = 96

Case 5: No Pair (four singletons):

There are C(13,4) = 715 different rank combos for a 4-card hand with four different ranks.

For concreteness, suppose the ranks are KQJT. How many different suit-isomorphism sub-cases are there and how many different unique hands exist in each subcase?

Subcase 5.1: (K)(Q)(J)(T). Here there are 4! = 24 possible unique hands.

Subcase 5.2: (KQ)(J)(T). Here there are 4*3*2 = 24 possible unique hands.

Subcase 5.3: (KJ)(Q)(T). Here there are 4*3*2 = 24 possible unique hands.

Subcase 5.4: (KT)(Q)(J). Here there are 4*3*2 = 24 possible unique hands.

Subcase 5.5: (QJ)(K)(T). Here there are 4*3*2 = 24 possible unique hands.

Subcase 5.6: (QT)(K)(J). Here there are 4*3*2 = 24 possible unique hands.

Subcase 5.7: (JT)(K)(Q). Here there are 4*3*2 = 24 possible unique hands.

Subcase 5.8: (KQ)(JT). Here there are 4*3 = 12 possible unique hands.

Subcase 5.9: (KJ)(QT). Here there are 4*3 = 12 possible unique hands.

Subcase 5.10: (KT)(QJ). Here there are 4*3 = 12 possible unique hands.

Subcase 5.11: (KQJ)(T). Here there are 4*3 = 12 possible unique hands.

Subcase 5.12: (KQT)(J). Here there are 4*3 = 12 possible unique hands.

Subcase 5.13: (KJT)(Q). Here there are 4*3 = 12 possible unique hands.

Subcase 5.14: (QJT)(K). Here there are 4*3 = 12 possible unique hands.

Subcase 5.15: (KQJT). Here there are 4 possible unique hands.

Subtotal = 24+24+24+24+24+24+24+12+12+12+12+12+12+12+4 = 256

Let's put this information into a table.

Table of Info on Number of Long Deck Omaha Starting Hands

Hand Category	# of Rank Combos	# of Suit-Rank Subcases	__Product__	_____	# of Rank Combos	# of Unique Hands per Rank Combo	__Product__
Quads	13	1	13	.	13	1	13
Trips	156	2	312	.	156	16	2,496
Two Pair	78	3	234	.	78	36	2,808
One Pair	858	6	5,148	.	858	96	82,368
No Pair	715	15	10,725	.	715	256	183,040
.
TOTAL			16,432				270,725

The first "Total" in the table above tells us that there are 16,432 different "suit-isomorphic" starting hands in Long Deck Omaha. The second "Total" in the table above tells us that there are 270,725 different starting hands in Long Deck Omaha when each card is treated as unique (meaning that we do not do any suit-isomorphic collapsing of hands). Of course, we encountered the 270,725 figure at the top of this post since this is exactly C(52,4). This shows that our breakdown into different rank categories and suit-rank subcases when combined into a whole gives the correct number, so we can have confidence that we did the breakdowns correctly.


Number of Different Short Deck Omaha Suit-Isomorphic Starting Hands

Now let's turn our attention to Short Deck Omaha. After a bit of thought, it is clear that the number of suit-rank subcases and number of unique hands per rank-combo are the same for both Long Deck and Short Deck. What is different, of course, is the number of rank-combos in each Hand Category since the number of ranks in Short Deck is collapsed to 9 (from 13 in Long Deck).

I will simply report the number of Short Deck rank-combos for the five categories below.

Quads: C(9,1) = 9

Trips: C(9,2)*C(2,1) = 72

Two Pair: C(9,2) = 36

One Pair: C(9,3)*C(3,1) = 252

No Pair: C(9,4) = 126

Let's make a new table reflecting these new Short Deck rank-combos.

Table of Info on Number of Short Deck Omaha Starting Hands

Hand Category	# of Rank Combos	# of Suit-Rank Subcases	__Product__	_____	# of Rank Combos	# of Unique Hands per Rank Combo	__Product__
Quads	9	1	9	.	9	1	9
Trips	72	2	144	.	72	16	1,152
Two Pair	36	3	108	.	36	36	1,296
One Pair	252	6	1,512	.	252	96	24,192
No Pair	126	15	1,890	.	126	256	32,256
.
TOTAL			3,663				58,905

So we see from the table that there are a total of 3,663 different suit-isomorphic starting hands in Short Deck Omaha. The table also show that there are a grand total of 58,905 different unique starting hands in Short Deck Omaha which we can confirm by double-checking that C(36,4) = 58,905.

My ultimate goal is to eventually run a simulation to determine the best starting hands in Short Deck Omaha, but I fear that the computing time would not be feasible given how many deals (say of 6-max) would be required given that there are 3,663 different suit-isomorphic starting hands in Short Deck Omaha.

Can you elaborate on which metric you wish to use to rank the hands? Maybe I can throw out some ideas to help.

My thought was to replicate what I did (and I am sure many others have done so before me) in the case of NLHE. I simulated millions and millions of 6-max NLHE deals, both long deck and short deck, and tabulated how often each hand won a deal.

Omaha simulations are more time-consuming than NLHE simulations, of course, since there are many more 5-card hands to evaluate for each player in Omaha (60 vs 21).

Another confounding effect is that there are only 169 suit-isomorphic hands in NLHE and their relative weights (4, 6, 12) are not too disparate so that it takes a reasonable number of deals (say 10 million) to get a clear picture of which suit-isomorphic hands are the best starting hands in NLHE. In Short Deck Omaha the 169 figure becomes 3,663 and the (4, 6, 12) relative "weights" become (1, 4, 6, 12, 24). This seems to elevate the required number of deals to get a clear picture of the best Short Deck Omaha starting hands to become very large.

I plan on setting up the machinery later this week and I hope to post preliminary results (if any) at that time. I suppose it is more likely that I'll simply post the runtime projections and give up at that time.

I think I have coded things properly and kicked off a simulation of 1 million Short Deck Omaha 6-Max deals last night (all hands go to showdown on all deals). I will present the preliminary results in two separate tables.

The first table will list the top five starting hands (suit-isomorphic buckets) based upon raw tallies of number of deals won (each hand that ties gets 1/N).

Table 1: Best Short Deck Omaha 6-Max Starting Hands based upon Raw Tallies (very preliminary)

__Rank per Raw Tallies__	Starting Hand (suit-iso bucket)	# of Unique Hands in suit-iso bucket
1	Single-suited Aces & Kings	24
2	Single-suited Aces & Queens	24
3	Single-suited Aces & Tens	24
4	Single-suited Aces & Jacks	24
5	Single-suited Kings & Tens	24

The second table will list the top five starting hands (suit-isomorphic buckets) based upon adjusted tallies of number of deals won where the raw tallies are divided by the number of unique hands in each suit-isomorphic bucket.

Table 2: Best Short Deck Omaha 6-Max Starting Hands based upon Adjusted Tallies (very preliminary)

Rank per Adjusted Tallies	Starting Hand (suit-iso bucket)	# of Unique Hands in suit-iso bucket
1	Double-suited Aces & Jacks	6
2	Double-suited Aces & Kings	6
3	Double-suited Aces & Queens	6
4	Double-suited Aces & Tens	6
5	Double-suited (AT) (QJ)	12

On first viewing these results look credible. Meaning that my simulation is very likely doing what I intended. Not too much should be put into the rankings based upon this preliminary simulation of only 1,000,000 deals.

I suppose it is clear to everybody the difference between the two tables. The first table based upon raw tallies gives a significant "advantage" to suit-isomorphic buckets containing many unique starting hands. This is the analog to NLHE where AK-offsuit wins more hands than AK-suited, but this is merely due to there being more AK-offsuit starting hands in NLHE than AK-suited starting hands (12 vs 4). Of course, the raw tallies need to be adjusted. When the number of times AK-offsuit wins is divided by the number of unique AK-offsuit starting hands (12) and the number of times AK-suited wins is divided by the number of unique AK-suited starting hands (4), AK-suited is seen to be a stronger starting hand than AK-offsuit.

The same "adjustment" is required in Omaha as well. Some Omaha starting hand suit-iso buckets contain many more unique starting hands than other starting hand suit-iso buckets. It is not surprising that the top five starting hand buckets in Table 1 all contain 24 unique starting hands (24 being the max possible). When the relevant adjustment is made, these single-suited hand buckets correctly slide down the ranking and double-suited hands correctly move up.

I don't want to spend any time discussing the specific results (other than to say they look reasonable). As discussed in a previous post in this thread, I presume that many many millions of deals will be required to get a solid ranking (and even then the specific ranks among the top 10 hands, say, will probably not be conclusive). That is, I imagine that the top 10 (top 20, top 50, etc.) hands will be identified as a group but where each hand falls within the group will be less well-defined.

I will now kick off additional simulations.

Here are the results of a simulation of 10 million Short Deck Omaha 6-Max deals (all hands go to showdown on all deals). To remove any uncertainty on this issue, these simulations use the "new" short deck rules in which a flush beats a full house but a straight beats three-of-a-kind. I will present the results in two separate tables.

The first table will list the top 25 starting hands (suit-isomorphic buckets) based upon raw tallies of number of deals won (each hand that ties gets 1/N).

Table 1: Best Short Deck Omaha 6-Max Starting Hands based upon Raw Tallies (10 million deals)

__Rank per Raw Tallies__	Starting Hand (suit-iso bucket)	Raw Tally	# of Unique Hands in suit-iso bucket	Adjusted Tally
1	(AK) (A) (K)	7196	24	300
2	(AQ) (A) (Q)	7026	24	293
3	(KQ) (K) (Q)	6559	24	273
4	(AT) (A) (T)	6518	24	272
5	(AQ) (J) (T)	6447	24	269
6	(AJ) (A) (J)	6444	24	268
7	(KJ) (Q) (T)	6442	24	268
8	(KQ) (J) (T)	6430	24	268
9	(AQ) (A) (J)	6396	24	267
10	(KT) (Q) (J)	6381	24	266
11	(AJ) (Q) (T)	6343	24	264
12	(AQ) (A) (T)	6339	24	264
13	(AT) (Q) (J)	6335	24	264
14	(AT) (A) (J)	6332	24	264
15	(QJ) (K) (T)	6331	24	264
16	(AT) (A) (Q)	6328	24	264
17	(JT) (K) (Q)	6326	24	264
18	(QT) (K) (J)	6323	24	263
19	(AJ) (A) (Q)	6314	24	263
20	(KJ) (K) (J)	6308	24	263
21	(AJ) (K) (Q)	6293	24	262
22	(AT) (K) (J)	6290	24	262
23	(AJ) (A) (T)	6240	24	260
24	(AJ) (K) (T)	6214	24	259
25	(KT) (K) (J)	6206	24	259

Recall that the parentheses surround cards in the same suit. So the first entry in the table (AK) (A) (K) is single-suited aces-and-kings. The second entry in the table (AQ) (A) (Q) is single-suited aces-and-queens. These two appear clearly ahead of all other starting hand suit-iso buckets.

The second table will list the top 25 starting hands (suit-isomorphic buckets) based upon adjusted tallies of number of deals won where the raw tallies are divided by the number of unique hands in each suit-isomorphic bucket.

Table 2: Best Short Deck Omaha 6-Max Starting Hands based upon Adjusted Tallies (10 million deals)

Rank per Adjusted Tallies	Starting Hand (suit-iso bucket)	Raw Tally	# of Unique Hands in suit-iso bucket	Adjusted Tally
1	(AK) (AK)	1993	6	332
2	(AQ) (AQ)	1975	6	329
3	(AJ) (AJ)	1920	6	320
4	(AJ) (AT)	3808	12	317
5	(AT) (AT)	1891	6	315
6	(KQ) (KQ)	1873	6	312
7	(KJ) (QT)	3675	12	306
8	(AQ) (AT)	3631	12	303
9	(KT) (QJ)	3609	12	301
10	(AK) (A) (K)	7196	24	300
11	(KJ) (KJ)	1792	6	299
12	(AQ) (AJ)	3573	12	298
13	(KQ) (JT)	3561	12	297
14	(AT) (QJ)	3547	12	296
15	(AT) (KJ)	3530	12	294
16	(AK) (AT)	3528	12	294
17	(KJ) (KT)	3524	12	294
18	(AQ) (A) (Q)	7026	24	293
19	(AJ) (QT)	3511	12	293
20	(AQ) (JT)	3509	12	292
21	(AK) (AQ)	3501	12	292
22	(AJ) (KT)	3497	12	291
23	(KQ) (KJ)	3486	12	291
24	(QJ) (QT)	3484	12	290
25	(AK) (KQ)	3474	12	289

To repeat what I said above, I am sure it is clear to everybody the difference between the two tables. The first table based upon raw tallies gives a significant "advantage" to suit-isomorphic buckets containing many unique starting hands. This is the analog to NLHE where AK-offsuit wins more hands than AK-suited, but this is merely due to there being more AK-offsuit starting hands in NLHE than AK-suited starting hands (12 vs 4). Of course, the raw tallies need to be adjusted. When the number of times AK-offsuit wins is divided by the number of unique AK-offsuit starting hands (12) and the number of times AK-suited wins is divided by the number of unique AK-suited starting hands (4), AK-suited is seen to be a stronger starting hand than AK-offsuit.

The same "adjustment" is required in Omaha as well. Some Omaha starting hand suit-iso buckets contain many more unique starting hands than other starting hand suit-iso buckets. It is not surprising that the top 25 starting hand buckets in Table 1 all contain 24 unique starting hands (24 being the max possible). When the relevant adjustment is made, these single-suited hand buckets correctly slide down the ranking and double-suited hands correctly move up.

The first entry in Table 2 shows that the best starting hand in Short Deck Omaha (based upon this simulation) is Double-Suited Aces & Kings. The Raw Tally shows that Double-Suited Aces & Kings won 1,993 of the 10 million deals (if N hands tie they each get 1/N). And when adjusted for how many unique hands are double-suited aces-and-kings (6), the Adjusted Tally is 332. The interpretation is that a specific double-suited aces-and-kings hand, say [As Ah Ks Kh] won 332 of the 10 million 6-max short deck omaha deals.

There is a lot of information to be gleaned from the above tables. Since I am a novice in short deck omaha, I will not presume to interpret the reasons and nuances behind the above rankings.

As discussed in a previous post in this thread, I presume that many many millions of deals will be required to get a solid ranking (and even then the specific ranks among the top 10 hands, say, will probably not be conclusive). That is, I imagine that the top 10 (top 20, top 50, etc.) hands will be identified as a group but precisely where each hand falls within the group will be less well-defined.

I plan to launch another simulation of another 10 million deals later today. When that simulation is completed I am hoping that the starting hand rankings will be reasonably stable so that we can place confidence in these results. (My guess is that it may take 50 million or even 100 million deals to reach the level of confidence in the results that we are used to in NLHE.)

Great work! Maybe the next table can also have confidence intervals?

The second simulation of 10 million Short Deck Omaha 6-Max deals (all hands go to showdown on all deals) has completed. To remove any uncertainty on this issue, these simulations use the "new" short deck rules in which a flush beats a full house but a straight beats three-of-a-kind. I will present the top 25 starting hands (suit-isomorphic buckets) based upon the Adjusted Tallies from the combined simulations of 20 million deals.

Table 2: Best Short Deck Omaha 6-Max Starting Hands based upon Adjusted Tallies (20 million deals)

Rank per Adjusted Tallies	Starting Hand (suit-iso bucket)	Raw Tally	Unique Hands in Bucket	Adjusted Tally
1	(AK) (AK)	3,967	6	661
2	(AQ) (AQ)	3,963	6	660
3	(AJ) (AJ)	3,858	6	643
4	(AJ) (AT)	7,639	12	637
5	(AT) (AT)	3,799	6	633
6	(KQ) (KQ)	3,757	6	626
7	(KJ) (QT)	7,332	12	611
8	(AQ) (AT)	7,224	12	602
9	(KT) (QJ)	7,160	12	597
10	(AQ) (AJ)	7,154	12	596
11	(AK) (A) (K)	14,288	24	595
12	(KJ) (KJ)	3,533	6	589
13	(KQ) (JT)	7,047	12	587
14	(AJ) (KT)	7,036	12	586
15	(AT) (KJ)	7,028	12	586
16	(AJ) (QT)	7,015	12	585
17	(AK) (AT)	7,013	12	584
18	(AK) (AJ)	7,009	12	584
19	(AQ) (JT)	7,002	12	584
20	(AQ) (KT)	6,994	12	583
21	(AT) (QJ)	6,987	12	582
22	(KJ) (KT)	6,987	12	582
23	(AK) (AQ)	6,986	12	582
24	(QJ) (QJ)	3,474	6	579
25	(AQ) (KJ)	6,938	12	578

At this juncture I will note that the two 10-million simulations gave similar results, but far from identical results. Consequently there is a fair amount of movement in the tables between the 10-million deal results and the 20-million deal results, especially as you move to the middle and lower portions of the tables. I don't want to make a big deal of it, but I will note that the top nine starting hands are the exact same (in the same order 1-9) in the two tables.

So I will refrain from making any substantive comments on the hand rankings appearing above. The results are not yet sufficiently stable and we are still seeing the effects of "variance" (how often a specific hand is dealt, what other hands it is up against in that deal, how that deal's board runs out determining which hand wins, etc.). Of course, the law-of-large-numbers applies so that the hand rankings will eventually stabilize reflecting starting hands' true underlying worthiness.

I will kick off another 10 million deals later tonight and will report back the combined 30-million results when it finishes in a few days.

The third simulation of 10 million Short Deck Omaha 6-Max deals (all hands go to showdown on all deals) has completed. These simulations use the "new" short deck rules in which a flush beats a full house but a straight beats three-of-a-kind. I will present the top 25 starting hands (suit-isomorphic buckets) based upon the Adjusted Tallies from the combined simulations of 30 million deals.

Table: Best Short Deck Omaha 6-Max Starting Hands based upon Adjusted Tallies (30 million deals)

Rank per Adjusted Tallies	Starting Hand (suit-iso bucket)	Raw Tally	Unique Hands in Bucket	Adjusted Tally
1	(AK) (AK)	5,988	6	998
2	(AQ) (AQ)	5,847	6	974
3	(AJ) (AJ)	5,687	6	948
4	(AJ) (AT)	11,317	12	943
5	(AT) (AT)	5,650	6	942
6	(KQ) (KQ)	5,604	6	934
7	(KJ) (QT)	10,858	12	905
8	(AQ) (AT)	10,844	12	904
9	(AQ) (AJ)	10,792	12	899
10	(AK) (A) (K)	21,447	24	894
11	(KT) (QJ)	10,713	12	893
12	(KJ) (KJ)	5,316	6	886
13	(AK) (AQ)	10,578	12	881
14	(AT) (KJ)	10,576	12	881
15	(AK) (AJ)	10,575	12	881
16	(AJ) (KT)	10,562	12	880
17	(AJ) (QT)	10,558	12	880
18	(AT) (QJ)	10,549	12	879
19	(AK) (AT)	10,542	12	879
20	(AQ) (JT)	10,507	12	876
21	(KQ) (JT)	10,497	12	875
22	(AT) (KQ)	10,466	12	872
23	(QJ) (QJ)	5,204	6	867
24	(KJ) (KT)	10,389	12	866
25	(AQ) (KJ)	10,372	12	864

There is still a fair amount of movement in the tables between the 20-million deal results and the 30-million deal results, especially as you move to the middle and lower portions of the tables. I don't want to make a big deal of it, but I will note that the top nine starting hands are the exact same (in the same order 1-9) in the two tables. I will kick off another 10 million deals later tonight and will report back the combined 40-million results when the simulation finishes.

The original idea was to run this "meta simulation" (over all possible hands/deals) partly to serve as a preliminary foray to identify the candidates for the best 25 starting hands. This simulation is of the Broad/Shallow variety.

The idea was then to run a whole bunch of Narrow/Deep simulations on specific hands. That is, we would next determine the expected winning percentage of double-suited Aces & Kings vs five random hands. Do the same for double-suited Aces & Queens, etc.

My plans are up in the air, but I am thinking that I'll plow through the "meta simulations" until we have results for 50 million deals. Maybe I'll post the top 100, or top 1000, or even the complete ranking of 3663 starting hands knowing that there is still likely to be significant variance around where a hand is ranked based upon "only" 50 million deals. One of the advantages of the broad/shallow meta-simulation is that we have win results for each of the 3663 starting hands and it would be a shame to throw that information away.

After that has been put aside, then I may undertake specific hand simulations (vs five other random hands). Unfortunately, I think these simulations will also suffer from runtime issues. I would love to do 25 or 50 or 100 of these (ideally all 3663), but that seems infeasible. We'll have to see what is possible when we get to that point.

The results of a simulation of 40 million Short Deck Omaha 6-Max deals.

Table: Best Short Deck Omaha 6-Max Starting Hands based upon Adjusted Tallies (40 million deals)

Rank per Adjusted Tallies	Starting Hand (suit-iso bucket)	Raw Tally	Unique Hands in Bucket	Adjusted Tally
1	(AK) (AK)	8,004	6	1,334
2	(AQ) (AQ)	7,770	6	1,295
3	(AJ) (AJ)	7,568	6	1,261
4	(AJ) (AT)	14,941	12	1,245
5	(AT) (AT)	7,442	6	1,240
6	(KQ) (KQ)	7,398	6	1,233
7	(AQ) (AT)	14,458	12	1,205
8	(AQ) (AJ)	14,443	12	1,204
9	(KJ) (QT)	14,301	12	1,192
10	(AK) (A) (K)	28,581	24	1,191
11	(KT) (QJ)	14,271	12	1,189
12	(AT) (QJ)	14,118	12	1,177
13	(AK) (AQ)	14,112	12	1,176
14	(AK) (AJ)	14,107	12	1,176
15	(KQ) (JT)	14,093	12	1,174
16	(AT) (KJ)	14,082	12	1,174
17	(AJ) (QT)	14,075	12	1,173
18	(KJ) (KJ)	7,026	6	1,171
19	(AJ) (KT)	14,050	12	1,171
20	(AT) (KQ)	14,037	12	1,170
21	(AK) (AT)	14,030	12	1,169
22	(AQ) (JT)	14,005	12	1,167
23	(AJ) (KQ)	13,901	12	1,158
24	(KJ) (KT)	13,838	12	1,153
25	(QJ) (QJ)	6,899	6	1,150

I will kick off another 10 million deals later tonight and will report back the combined 50-million results when the simulation finishes.

See the earlier posts for my thoughts on next steps.

Here are the complete results of a simulation of 50 million Short Deck Omaha 6-Max deals. To remove any uncertainty on this issue, these simulations use the "new" short deck rules in which a flush beats a full house but a straight beats three-of-a-kind.

The following tables list the adjusted tally and estimated equity for all possible 3,663 starting hands in short deck omaha. Some of the starting hands may look a little odd due to the way I display hands with pairs or trips. Since there is a limit on the size of posts, the tables will appear in separate posts.


Table 1: Best Short Deck Omaha 6-Max Starting Hands (based upon 50 million deals): Ranks 1-400

___ Rank ___	Starting Hand (suit-iso bucket)	Adjusted Tally	Estimated Equity (%)
1	(AK) (AK)	1664	32.7
2	(AQ) (AQ)	1627	31.9
3	(AJ) (AJ)	1579	31.0
4	(AT) (AT)	1555	30.5
5	(AJ) (AT)	1544	30.3
6	(KQ) (KQ)	1537	30.2
7	(AQ) (AT)	1514	29.7
8	(AQ) (AJ)	1500	29.4
9	(KJ) (QT)	1498	29.4
10	(KT) (QJ)	1491	29.3
11	(AK) (A) (K)	1491	29.3
12	(AJ) (KT)	1473	28.9
13	(AK) (AQ)	1472	28.9
14	(AK) (AJ)	1471	28.9
15	(AK) (AT)	1468	28.8
16	(AT) (QJ)	1467	28.8
17	(AJ) (QT)	1466	28.8
18	(KJ) (KJ)	1462	28.7
19	(AT) (KJ)	1462	28.7
20	(KQ) (JT)	1460	28.7
21	(AT) (KQ)	1458	28.6
22	(AQ) (JT)	1456	28.6
23	(KT) (KT)	1447	28.4
24	(QJ) (QJ)	1446	28.4
25	(KJ) (KT)	1444	28.4
26	(AQ) (A) (Q)	1439	28.3
27	(AJ) (KQ)	1438	28.2
28	(AQ) (KJ)	1433	28.1
29	(A9) (A9)	1432	28.1
30	(AK) (JT)	1424	28.0
31	(AQ) (KT)	1421	27.9
32	(KA) (KT)	1419	27.9
33	(AT) (A9)	1415	27.8
34	(KA) (KJ)	1409	27.7
35	(AK) (QT)	1407	27.6
36	(KA) (KQ)	1407	27.6
37	(AJ) (A9)	1407	27.6
38	(KQ) (KJ)	1406	27.6
39	(QA) (QK)	1405	27.6
40	(AK) (QJ)	1404	27.6
41	(QT) (QT)	1402	27.5
42	(QJ) (QT)	1401	27.5
43	(KQ) (KT)	1397	27.4
44	(QA) (QJ)	1387	27.2
45	(AJ) (A8)	1380	27.1
46	(AT) (A8)	1379	27.1
47	(QK) (QJ)	1379	27.1
48	(KQ) (K) (Q)	1375	27.0
49	(JQ) (JT)	1371	26.9
50	(JK) (JQ)	1370	26.9
51	(AJ) (A) (J)	1369	26.9
52	(JA) (JQ)	1365	26.8
53	(QK) (QT)	1364	26.8
54	(AQ) (A9)	1363	26.8
55	(QA) (QT)	1361	26.7
56	(JT) (JT)	1357	26.7
57	(TQ) (TJ)	1357	26.7
58	(JA) (JT)	1357	26.6
59	(KT) (Q) (J)	1356	26.6
60	(A8) (A8)	1355	26.6
61	(JA) (JK)	1355	26.6
62	(JK) (JT)	1353	26.6
63	(KT) (K9)	1350	26.5
64	(AT) (Q) (J)	1349	26.5
65	(AT) (A) (T)	1347	26.4
66	(QT) (J9)	1345	26.4
67	(KQ) (J) (T)	1343	26.4
68	(KJ) (Q) (T)	1341	26.3
69	(AQ) (J) (T)	1338	26.3
70	(TA) (TJ)	1333	26.2
71	(AQ) (A8)	1332	26.1
72	(TK) (TJ)	1330	26.1
73	(AK) (A9)	1330	26.1
74	(Q9) (JT)	1330	26.1
75	(K) (QT) (J)	1330	26.1
76	(AT) (A) (J)	1329	26.1
77	(AJ) (Q) (T)	1329	26.1
78	(AT) (A7)	1328	26.1
79	(AT) (J9)	1325	26.0
80	(K) (QJ) (T)	1324	26.0
81	(KJ) (K) (J)	1323	26.0
82	(QJ) (T9)	1319	25.9
83	(AK) (A7)	1316	25.8
84	(AT) (Q9)	1316	25.8
85	(AT) (A6)	1314	25.8
86	(TA) (TQ)	1314	25.8
87	(TK) (TQ)	1313	25.8
88	(KJ) (K9)	1313	25.8
89	(AK) (A8)	1312	25.8
90	(AJ) (A7)	1310	25.7
91	(AK) (A6)	1310	25.7
92	(K9) (JT)	1310	25.7
93	(A9) (JT)	1309	25.7
94	(K) (Q) (JT)	1308	25.7
95	(AJ) (A) (T)	1308	25.7
96	(AJ) (A) (Q)	1307	25.7
97	(AJ) (T9)	1307	25.7
98	(AT) (K9)	1306	25.6
99	(AQ) (A) (J)	1304	25.6
100	(AT) (K) (J)	1304	25.6
101	(AT) (A) (Q)	1303	25.6
102	(KT) (J9)	1302	25.6
103	(K9) (QT)	1301	25.6
104	(AJ) (A6)	1301	25.5
105	(A) (A) (K) (K)	1299	25.5
106	(AQ) (A) (T)	1297	25.5
107	(KT) (Q9)	1296	25.4
108	(KA) (K8)	1294	25.4
109	(AQ) (A7)	1294	25.4
110	(AJ) (K) (T)	1292	25.4
111	(KJ) (T9)	1292	25.4
112	(KT) (K8)	1291	25.4
113	(QJ) (Q) (J)	1290	25.3
114	(AQ) (K) (J)	1290	25.3
115	(TA) (TK)	1290	25.3
116	(AQ) (A6)	1289	25.3
117	(KT) (K) (J)	1285	25.2
118	(KT) (K) (T)	1282	25.2
119	(A) (QJ) (T)	1281	25.2
120	(A7) (A7)	1281	25.2
121	(K9) (K9)	1281	25.2
122	(AK) (J) (T)	1281	25.1
123	(AJ) (K) (Q)	1281	25.1
124	(A) (QT) (J)	1280	25.1
125	(AJ) (A) (K)	1279	25.1
126	(A9) (KT)	1279	25.1
127	(A9) (KJ)	1278	25.1
128	(AT) (K) (Q)	1278	25.1
129	(K9) (QJ)	1277	25.1
130	(AJ) (Q9)	1277	25.1
131	(AK) (Q) (J)	1274	25.0
132	(A9) (QJ)	1274	25.0
133	(AT) (A) (K)	1273	25.0
134	(AK) (A) (J)	1273	25.0
135	(QT) (Q9)	1273	25.0
136	(AQ) (A) (K)	1273	25.0
137	(A9) (QT)	1273	25.0
138	(QT) (J8)	1272	25.0
139	(KA) (K9)	1270	24.9
140	(AJ) (K9)	1268	24.9
141	(AQ) (K) (T)	1267	24.9
142	(AQ) (K9)	1267	24.9
143	(KJ) (K) (T)	1266	24.9
144	(AK) (A) (Q)	1264	24.8
145	(KQ) (K9)	1264	24.8
146	(A9) (A8)	1264	24.8
147	(K8) (JT)	1262	24.8
148	(QT) (Q) (J)	1262	24.8
149	(Q8) (JT)	1261	24.8
150	(KQ) (T9)	1261	24.8
151	(A) (Q) (JT)	1261	24.8
152	(A) (KT) (J)	1261	24.8
153	(AK) (A) (T)	1260	24.7
154	(A9) (KQ)	1258	24.7
155	(AK) (Q) (T)	1258	24.7
156	(QA) (Q) (K)	1258	24.7
157	(AT) (K8)	1257	24.7
158	(KJ) (K8)	1257	24.7
159	(KA) (K7)	1256	24.7
160	(AQ) (T9)	1256	24.7
161	(K) (K) (AJ)	1254	24.6
162	(KA) (K) (Q)	1254	24.6
163	(A) (KQ) (J)	1254	24.6
164	(KJ) (T8)	1253	24.6
165	(AQ) (J9)	1252	24.6
166	(QA) (Q9)	1251	24.6
167	(KT) (K) (Q)	1250	24.6
168	(A8) (QT)	1250	24.5
169	(A) (KJ) (T)	1247	24.5
170	(A) (KT) (Q)	1246	24.5
171	(KJT) (Q)	1246	24.5
172	(AT) (J8)	1246	24.5
173	(A) (KJ) (Q)	1246	24.5
174	(KT) (J8)	1245	24.5
175	(AJ) (Q8)	1245	24.4
176	(QT) (Q) (T)	1245	24.4
177	(KQ) (J9)	1245	24.4
178	(KJ) (Q9)	1244	24.4
179	(KA) (K) (J)	1244	24.4
180	(A6) (A6)	1244	24.4
181	(AK) (T9)	1244	24.4
182	(KQ) (K) (J)	1242	24.4
183	(QK) (Q) (T)	1240	24.3
184	(KJ) (K) (Q)	1240	24.3
185	(A) (A) (QJ)	1240	24.3
186	(A8) (JT)	1239	24.3
187	(JT) (J) (T)	1238	24.3
188	(KT) (K7)	1238	24.3
189	(KQ) (K) (T)	1237	24.3
190	(KT) (Q8)	1237	24.3
191	(A8) (KT)	1237	24.3
192	(A) (A) (JT)	1236	24.3
193	(JT) (J9)	1236	24.3
194	(A) (KQ) (T)	1236	24.3
195	(K) (K) (AT)	1236	24.3
196	(KA) (K) (T)	1235	24.3
197	(QJ) (Q9)	1235	24.2
198	(JQ) (J) (T)	1234	24.2
199	(K) (K) (JT)	1234	24.2
200	(A) (A) (KQ)	1234	24.2
201	(A8) (KQ)	1232	24.2
202	(QK) (Q) (J)	1231	24.2
203	(A) (K) (QJ)	1230	24.2
204	(Q) (Q) (JT)	1230	24.1
205	(QJ) (Q) (T)	1230	24.1
206	(A9) (A7)	1229	24.1
207	(J) (J) (QT)	1228	24.1
208	(AQ) (K8)	1228	24.1
209	(JK) (J) (Q)	1228	24.1
210	(A) (A) (KJ)	1227	24.1
211	(KQJ) (T)	1227	24.1
212	(KA) (K6)	1227	24.1
213	(JT) (J) (Q)	1227	24.1
214	(K) (K) (AQ)	1226	24.1
215	(QJ) (T8)	1226	24.1
216	(QA) (Q) (J)	1226	24.1
217	(Q) (Q) (AK)	1225	24.1
218	(KQ) (K) (A)	1225	24.1
219	(KT) (K6)	1225	24.0
220	(QK) (Q9)	1224	24.0
221	(TQ) (T) (J)	1224	24.0
222	(Q) (Q) (KT)	1224	24.0
223	(T) (T) (QJ)	1223	24.0
224	(Q) (Q) (KJ)	1223	24.0
225	(AT) (Q8)	1222	24.0
226	(A) (K) (QT)	1222	24.0
227	(QJ) (Q) (K)	1221	24.0
228	(A8) (KJ)	1221	24.0
229	(Q) (Q) (AJ)	1220	24.0
230	(KQT) (J)	1220	24.0
231	(AK) (J9)	1220	24.0
232	(QA) (Q) (T)	1218	23.9
233	(AK) (Q9)	1218	23.9
234	(K) (K) (QJ)	1218	23.9
235	(TJ) (T) (Q)	1218	23.9
236	(A) (A) (QT)	1218	23.9
237	(TJ) (T9)	1216	23.9
238	(KJ) (K) (A)	1215	23.9
239	(KT) (K) (A)	1215	23.8
240	(A9) (A) (T)	1214	23.8
241	(AQ) (J8)	1214	23.8
242	(A) (K) (JT)	1213	23.8
243	(QT) (J7)	1213	23.8
244	(AQT) (J)	1213	23.8
245	(AJT) (A)	1213	23.8
246	(AJ) (K8)	1213	23.8
247	(A8) (QJ)	1213	23.8
248	(QK) (Q) (A)	1213	23.8
249	(JK) (J) (T)	1212	23.8
250	(A) (A) (KT)	1212	23.8
251	(KJ) (K6)	1211	23.8
252	(AT) (Q7)	1211	23.8
253	(K) (K) (Q) (Q)	1210	23.8
254	(JA) (J) (Q)	1210	23.8
255	(AT) (A) (9)	1210	23.8
256	(J) (J) (AQ)	1210	23.8
257	(Q) (Q) (AT)	1210	23.8
258	(AJT) (Q)	1210	23.8
259	(AJ) (T8)	1210	23.7
260	(K) (QJT)	1209	23.7
261	(K) (K) (QT)	1209	23.7
262	(QT) (Q8)	1207	23.7
263	(Q9) (J) (T)	1207	23.7
264	(JQ) (J9)	1207	23.7
265	(KQ) (K8)	1206	23.7
266	(AT) (J7)	1206	23.7
267	(QT) (J) (9)	1206	23.7
268	(QT) (Q) (K)	1205	23.7
269	(K8) (QJ)	1205	23.7
270	(AT) (K7)	1205	23.7
271	(KQ) (K7)	1205	23.7
272	(K7) (JT)	1204	23.6
273	(TK) (T) (J)	1203	23.6
274	(J) (J) (KQ)	1203	23.6
275	(A9) (A) (J)	1203	23.6
276	(KJ) (Q8)	1203	23.6
277	(J) (J) (KT)	1202	23.6
278	(K8) (QT)	1202	23.6
279	(JQ) (J) (K)	1202	23.6
280	(A9) (A6)	1201	23.6
281	(AQ) (T8)	1200	23.6
282	(AQJ) (T)	1200	23.6
283	(Q7) (JT)	1200	23.6
284	(AJ) (K7)	1199	23.5
285	(JA) (J) (T)	1198	23.5
286	(K) (Q) (J) (T)	1197	23.5
287	(KJ) (K7)	1197	23.5
288	(KT) (J7)	1197	23.5
289	(A) (A) (Q) (Q)	1196	23.5
290	(JT) (J8)	1196	23.5
291	(KT) (Q7)	1194	23.5
292	(AK) (Q8)	1193	23.4
293	(AQ) (K7)	1192	23.4
294	(J) (J) (AT)	1192	23.4
295	(KQ) (T8)	1191	23.4
296	(TA) (T) (J)	1191	23.4
297	(QA) (Q8)	1191	23.4
298	(A7) (KJ)	1191	23.4
299	(A7) (KT)	1190	23.4
300	(T) (T) (KJ)	1190	23.4
301	(K9) (J) (T)	1190	23.4
302	(AJ) (A) (9)	1189	23.3
303	(JA) (J9)	1189	23.3
304	(K7) (QT)	1188	23.3
305	(Q) (J9) (T)	1187	23.3
306	(KQ) (K6)	1185	23.3
307	(A7) (QT)	1185	23.3
308	(AJ) (Q7)	1185	23.3
309	(AK) (J8)	1184	23.3
310	(A7) (KQ)	1184	23.3
311	(QJ) (T) (9)	1184	23.3
312	(KT) (Q6)	1183	23.2
313	(A8) (A7)	1183	23.2
314	(JT) (J) (K)	1183	23.2
315	(AT) (J) (9)	1183	23.2
316	(AJ) (T7)	1182	23.2
317	(QJ) (T7)	1181	23.2
318	(AQJ) (A)	1180	23.2
319	(QK) (Q8)	1178	23.1
320	(A9) (A) (9)	1178	23.1
321	(KQ) (J8)	1177	23.1
322	(A9) (J) (T)	1177	23.1
323	(KT) (J) (9)	1177	23.1
324	(K) (K) (J) (J)	1176	23.1
325	(QT) (J6)	1176	23.1
326	(TK) (T9)	1175	23.1
327	(QT) (Q7)	1175	23.1
328	(KJ) (T) (9)	1175	23.1
329	(A) (A) (J) (J)	1175	23.1
330	(A6) (KQ)	1175	23.1
331	(JK) (J9)	1174	23.1
332	(Q) (J) (T9)	1173	23.0
333	(A7) (JT)	1172	23.0
334	(KJ) (Q7)	1171	23.0
335	(QJ) (Q8)	1171	23.0
336	(A7) (QJ)	1171	23.0
337	(AT) (A) (8)	1170	23.0
338	(AQT) (A)	1169	23.0
339	(AT) (Q6)	1169	22.9
340	(AK) (T8)	1168	22.9
341	(JA) (J8)	1168	22.9
342	(QT) (Q) (A)	1168	22.9
343	(A9) (A) (Q)	1168	22.9
344	(A) (QJT)	1167	22.9
345	(T) (T) (AJ)	1167	22.9
346	(K7) (QJ)	1167	22.9
347	(Q) (JT) (9)	1166	22.9
348	(QJ) (Q) (A)	1166	22.9
349	(JA) (J) (K)	1166	22.9
350	(JQ) (J) (A)	1165	22.9
351	(Q6) (JT)	1165	22.9
352	(K6) (QT)	1165	22.9
353	(AK) (Q7)	1164	22.9
354	(A8) (A6)	1164	22.9
355	(Q9) (Q9)	1164	22.9
356	(AJT) (K)	1163	22.8
357	(KJT) (K)	1163	22.8
358	(TA) (T9)	1162	22.8
359	(AKJ) (T)	1162	22.8
360	(KJ) (T7)	1162	22.8
361	(TJ) (T) (K)	1162	22.8
362	(TQ) (T9)	1162	22.8
363	(AKT) (J)	1161	22.8
364	(TK) (T) (Q)	1160	22.8
365	(AJ) (T) (9)	1160	22.8
366	(K6) (QJ)	1160	22.8
367	(QK) (Q7)	1157	22.7
368	(A6) (KT)	1157	22.7
369	(QK) (Q6)	1156	22.7
370	(A9) (A) (K)	1155	22.7
371	(QA) (Q7)	1155	22.7
372	(A) (KJT)	1154	22.7
373	(TA) (T) (Q)	1154	22.7
374	(QJT) (Q)	1154	22.7
375	(A8) (A) (T)	1154	22.7
376	(AQ) (K6)	1153	22.6
377	(AJ) (Q6)	1153	22.6
378	(AT) (K6)	1152	22.6
379	(KQ) (T7)	1152	22.6
380	(A7) (A6)	1152	22.6
381	(AQ) (T7)	1152	22.6
382	(A6) (QT)	1152	22.6
383	(AJ) (K6)	1151	22.6
384	(K9) (K) (T)	1150	22.6
385	(A8) (A) (J)	1150	22.6
386	(AT) (J6)	1150	22.6
387	(QA) (Q6)	1150	22.6
388	(K8) (K8)	1149	22.6
389	(KJ) (T6)	1149	22.6
390	(KT) (K) (9)	1148	22.5
391	(K6) (JT)	1148	22.5
392	(AKJ) (A)	1148	22.5
393	(K) (J9) (T)	1148	22.5
394	(KT) (J6)	1147	22.5
395	(AQ) (J7)	1147	22.5
396	(AJ) (T6)	1147	22.5
397	(AQ) (A) (9)	1147	22.5
398	(K9) (K8)	1147	22.5
399	(JK) (J8)	1146	22.5
400	(QT) (Q6)	1146	22.5

Table 2: Best Short Deck Omaha 6-Max Starting Hands (based upon 50 million deals): Ranks 401-800

___ Rank ___	Starting Hand (suit-iso bucket)	Adjusted Tally	Estimated Equity (%)
401	(QJ) (T6)	1145	22.5
402	(AKQ) (A)	1143	22.4
403	(KQT) (K)	1143	22.4
404	(T) (T) (KQ)	1142	22.4
405	(A6) (KJ)	1142	22.4
406	(TJ) (T8)	1141	22.4
407	(AKT) (Q)	1141	22.4
408	(A6) (QJ)	1140	22.4
409	(KJ) (Q6)	1139	22.4
410	(J9) (J9)	1139	22.4
411	(TA) (T8)	1138	22.3
412	(AKQ) (J)	1138	22.3
413	(AK) (J7)	1138	22.3
414	(JK) (J) (A)	1138	22.3
415	(AT) (A) (7)	1138	22.3
416	(K7) (K7)	1138	22.3
417	(TQ) (T) (K)	1137	22.3
418	(JT) (J) (A)	1137	22.3
419	(AJ) (A) (8)	1137	22.3
420	(T) (T) (AQ)	1136	22.3
421	(AQ) (J6)	1136	22.3
422	(AKJ) (Q)	1136	22.3
423	(AQJ) (K)	1135	22.3
424	(A) (KQJ)	1134	22.3
425	(K) (JT) (9)	1133	22.3
426	(J) (J) (AK)	1133	22.2
427	(AQ) (T) (9)	1133	22.2
428	(KQJ) (K)	1132	22.2
429	(JQ) (J8)	1132	22.2
430	(9Q) (9T)	1132	22.2
431	(AQ) (A) (8)	1132	22.2
432	(JQT) (J)	1132	22.2
433	(A) (A) (J) (T)	1132	22.2
434	(KQ) (J7)	1132	22.2
435	(A9) (Q) (T)	1131	22.2
436	(AT) (Q) (9)	1131	22.2
437	(A8) (A) (Q)	1131	22.2
438	(KT) (Q) (9)	1131	22.2
439	(AK) (A) (9)	1131	22.2
440	(A9) (Q) (J)	1131	22.2
441	(KQ) (J6)	1130	22.2
442	(QJ) (Q6)	1130	22.2
443	(AQT) (K)	1129	22.2
444	(TQ) (T8)	1129	22.2
445	(AQ) (T6)	1129	22.2
446	(K9) (Q) (J)	1129	22.2
447	(AK) (Q6)	1129	22.2
448	(9A) (9J)	1129	22.2
449	(K9) (Q) (T)	1128	22.2
450	(AK) (T7)	1128	22.2
451	(QJ) (Q7)	1128	22.2
452	(KAJ) (K)	1128	22.1
453	(AKT) (A)	1128	22.1
454	(9J) (9T)	1128	22.1
455	(Q) (Q) (J) (J)	1128	22.1
456	(K) (K) (T) (T)	1128	22.1
457	(AKQ) (T)	1127	22.1
458	(Q8) (J) (T)	1126	22.1
459	(A) (Q) (J) (T)	1126	22.1
460	(KQ) (T6)	1126	22.1
461	(KQ) (T) (9)	1126	22.1
462	(QKT) (Q)	1125	22.1
463	(A6) (JT)	1125	22.1
464	(A7) (A) (T)	1125	22.1
465	(A) (KQT)	1124	22.1
466	(QKJ) (Q)	1123	22.1
467	(JKQ) (J)	1123	22.0
468	(T9) (T9)	1121	22.0
469	(TA) (T) (K)	1120	22.0
470	(K) (J) (T9)	1119	22.0
471	(QT) (J) (8)	1119	22.0
472	(TQJ) (T)	1118	21.9
473	(K) (Q9) (T)	1118	21.9
474	(KAQ) (K)	1117	21.9
475	(K9) (K) (J)	1117	21.9
476	(A8) (A) (K)	1117	21.9
477	(Q) (J8) (T)	1117	21.9
478	(JT) (J7)	1116	21.9
479	(QJ) (T) (8)	1115	21.9
480	(K) (K) (J) (T)	1115	21.9
481	(9A) (9K)	1115	21.9
482	(9A) (9Q)	1113	21.9
483	(9K) (9T)	1113	21.9
484	(QJT) (9)	1113	21.8
485	(Q9) (Q) (T)	1112	21.8
486	(QT) (Q) (9)	1112	21.8
487	(J8) (T9)	1111	21.8
488	(AQ) (J) (9)	1111	21.8
489	(J9) (T8)	1111	21.8
490	(A9) (K8)	1111	21.8
491	(KAT) (K)	1111	21.8
492	(AJ) (A) (7)	1110	21.8
493	(JKT) (J)	1110	21.8
494	(JQ) (J7)	1110	21.8
495	(K) (QT) (9)	1110	21.8
496	(TJ) (T) (A)	1110	21.8
497	(JK) (J7)	1110	21.8
498	(A) (A) (Q9)	1109	21.8
499	(AJ) (Q) (9)	1109	21.8
500	(AT) (J) (8)	1109	21.8
501	(KJ) (Q) (9)	1109	21.8
502	(JT) (98)	1109	21.8
503	(A) (A) (Q) (T)	1108	21.8
504	(JA) (J7)	1107	21.7
505	(A9) (K) (T)	1107	21.7
506	(AT) (A) (6)	1107	21.7
507	(KQ) (J) (9)	1107	21.7
508	(A8) (A) (8)	1106	21.7
509	(QAK) (Q)	1106	21.7
510	(A7) (A) (J)	1105	21.7
511	(AK) (T6)	1105	21.7
512	(K) (K) (A9)	1105	21.7
513	(QT9) (J)	1104	21.7
514	(Q) (JT) (8)	1104	21.7
515	(K8) (J) (T)	1103	21.7
516	(KT) (J) (8)	1103	21.7
517	(KA) (K) (9)	1103	21.7
518	(9A) (9T)	1102	21.6
519	(AK) (A) (8)	1102	21.6
520	(A) (A) (K9)	1102	21.6
521	(J9) (J) (T)	1101	21.6
522	(A8) (J) (T)	1101	21.6
523	(Q) (Q) (T) (T)	1101	21.6
524	(KT) (K) (8)	1101	21.6
525	(AJ) (T) (8)	1101	21.6
526	(JQ) (J6)	1100	21.6
527	(AJ) (A) (6)	1100	21.6
528	(A9) (J8)	1100	21.6
529	(KJ) (K) (9)	1099	21.6
530	(K9) (K7)	1098	21.6
531	(Q8) (Q8)	1098	21.6
532	(K) (Q9) (J)	1098	21.6
533	(A) (A) (T) (T)	1098	21.6
534	(A) (A) (T9)	1098	21.6
535	(KJ) (T) (8)	1098	21.6
536	(QJ9) (T)	1098	21.6
537	(JT) (J) (9)	1098	21.6
538	(AT) (98)	1098	21.6
539	(TJ) (T7)	1097	21.5
540	(AT) (K) (9)	1096	21.5
541	(AQ) (A) (7)	1096	21.5
542	(A) (K) (J) (T)	1096	21.5
543	(A) (JT) (9)	1095	21.5
544	(A8) (K9)	1095	21.5
545	(K6) (K6)	1095	21.5
546	(A7) (A) (K)	1095	21.5
547	(A) (A) (Q) (J)	1094	21.5
548	(TKJ) (T)	1094	21.5
549	(A6) (A) (T)	1094	21.5
550	(Q) (Q) (J) (T)	1094	21.5
551	(A9) (K) (J)	1094	21.5
552	(Q) (JT9)	1094	21.5
553	(A7) (A) (Q)	1094	21.5
554	(A9) (T8)	1093	21.5
555	(KQJT)	1093	21.5
556	(AK) (A) (7)	1092	21.4
557	(TK) (T8)	1092	21.4
558	(JT) (J6)	1091	21.4
559	(Q9) (T8)	1091	21.4
560	(A) (J9) (T)	1090	21.4
561	(K8) (K) (T)	1090	21.4
562	(AJ) (K) (9)	1090	21.4
563	(QAJ) (Q)	1089	21.4
564	(J) (J) (Q) (T)	1089	21.4
565	(K) (K) (Q) (J)	1089	21.4
566	(AQ) (A) (6)	1089	21.4
567	(AK) (J6)	1089	21.4
568	(A8) (T9)	1089	21.4
569	(K) (Q) (T9)	1088	21.4
570	(QAT) (Q)	1087	21.3
571	(A) (J) (T9)	1087	21.3
572	(A9) (Q8)	1086	21.3
573	(J) (J) (T) (T)	1086	21.3
574	(A) (K) (Q) (J)	1085	21.3
575	(A9) (K) (Q)	1085	21.3
576	(K9) (K) (A)	1085	21.3
577	(TQ) (T) (A)	1084	21.3
578	(A) (K) (Q) (T)	1084	21.3
579	(A) (A) (J9)	1084	21.3
580	(A6) (A) (K)	1083	21.3
581	(K9) (K) (Q)	1083	21.3
582	(AT) (Q) (8)	1083	21.3
583	(A6) (A) (Q)	1083	21.3
584	(AK) (T) (9)	1083	21.3
585	(AK) (A) (6)	1083	21.3
586	(K9) (K) (9)	1082	21.2
587	(KT9) (J)	1082	21.2
588	(K9) (K6)	1081	21.2
589	(K) (K) (Q) (T)	1081	21.2
590	(K) (K) (T9)	1080	21.2
591	(9Q) (9J)	1080	21.2
592	(K) (K) (A) (Q)	1080	21.2
593	(TK) (T7)	1080	21.2
594	(A7) (A) (7)	1080	21.2
595	(JK) (J6)	1079	21.2
596	(AK) (J) (9)	1079	21.2
597	(TJ) (T) (9)	1079	21.2
598	(Q) (J) (T8)	1079	21.2
599	(JAQ) (J)	1078	21.2
600	(A) (QT) (9)	1078	21.2
601	(A8) (Q) (T)	1078	21.2
602	(TK) (T) (A)	1077	21.2
603	(A) (Q9) (J)	1077	21.1
604	(Q9) (Q) (J)	1077	21.1
605	(Q8) (J9)	1076	21.1
606	(K) (Q) (J9)	1076	21.1
607	(JA) (J6)	1076	21.1
608	(A6) (A) (J)	1076	21.1
609	(9K) (9J)	1076	21.1
610	(KQ) (K) (9)	1076	21.1
611	(Q9) (J8)	1075	21.1
612	(TQ) (T7)	1075	21.1
613	(T) (T) (J9)	1075	21.1
614	(TA) (T7)	1075	21.1
615	(KA) (K) (8)	1075	21.1
616	(Q) (Q) (K) (T)	1075	21.1
617	(AQJT)	1075	21.1
618	(KA) (K) (7)	1074	21.1
619	(AJ) (Q) (8)	1074	21.1
620	(A8) (J9)	1073	21.1
621	(QT) (98)	1073	21.1
622	(K) (K) (A7)	1073	21.1
623	(A) (Q9) (T)	1073	21.1
624	(K8) (J9)	1073	21.1
625	(K) (K) (A8)	1072	21.0
626	(A) (A) (K8)	1072	21.0
627	(K) (QJ) (9)	1071	21.0
628	(JAT) (J)	1071	21.0
629	(T) (T) (Q) (J)	1071	21.0
630	(T) (T) (AK)	1070	21.0
631	(Q) (Q) (K) (J)	1070	21.0
632	(K9) (T8)	1070	21.0
633	(KJ9) (T)	1070	21.0
634	(AK) (Q) (9)	1069	21.0
635	(K8) (T9)	1069	21.0
636	(A) (K9) (T)	1069	21.0
637	(T9) (T) (J)	1069	21.0
638	(Q8) (T9)	1069	21.0
639	(KT) (J) (7)	1069	21.0
640	(A8) (Q) (J)	1068	21.0
641	(Q7) (J) (T)	1067	21.0
642	(A8) (Q9)	1066	20.9
643	(Q) (Q) (T9)	1066	20.9
644	(K8) (K) (A)	1066	20.9
645	(A) (A) (T8)	1066	20.9
646	(AJ) (98)	1065	20.9
647	(K) (J) (T8)	1065	20.9
648	(AT9) (A)	1064	20.9
649	(K8) (K) (J)	1064	20.9
650	(K) (JT) (8)	1064	20.9
651	(QJ) (Q) (9)	1064	20.9
652	(K8) (Q) (T)	1063	20.9
653	(A) (A) (K) (Q)	1063	20.9
654	(K7) (K) (T)	1063	20.9
655	(A) (A) (K) (J)	1063	20.9
656	(K9) (J8)	1062	20.9
657	(TQ) (T6)	1062	20.8
658	(KT) (Q) (8)	1062	20.8
659	(K) (K) (A) (J)	1062	20.8
660	(A) (K9) (J)	1061	20.8
661	(J) (J) (T9)	1060	20.8
662	(J9) (T7)	1060	20.8
663	(J) (J) (K) (Q)	1060	20.8
664	(KJT) (9)	1060	20.8
665	(KT) (98)	1060	20.8
666	(KJ) (Q) (8)	1060	20.8
667	(A) (A) (K) (T)	1060	20.8
668	(KJ) (T) (7)	1060	20.8
669	(K9) (T7)	1059	20.8
670	(A) (KT) (9)	1059	20.8
671	(A) (A) (J8)	1059	20.8
672	(KA) (K) (6)	1058	20.8
673	(QA) (Q) (9)	1057	20.8
674	(K) (K) (Q9)	1057	20.8
675	(A9) (T7)	1057	20.8
676	(KQ) (T) (8)	1057	20.8
677	(J7) (T9)	1057	20.7
678	(Q) (Q) (K9)	1056	20.7
679	(K8) (Q9)	1056	20.7
680	(A) (QJ) (9)	1056	20.7
681	(KT) (K) (7)	1055	20.7
682	(AJ9) (A)	1055	20.7
683	(K8) (K7)	1055	20.7
684	(8A) (8K)	1054	20.7
685	(AQ) (T) (8)	1054	20.7
686	(QT) (J) (7)	1054	20.7
687	(A) (K9) (Q)	1053	20.7
688	(K7) (K) (A)	1053	20.7
689	(A) (KJ) (9)	1053	20.7
690	(AQ) (K) (9)	1053	20.7
691	(AT) (J) (7)	1053	20.7
692	(K) (J8) (T)	1053	20.7
693	(QJ) (T) (7)	1052	20.7
694	(AJ9) (T)	1052	20.7
695	(K8) (Q) (J)	1052	20.7
696	(A) (A) (Q8)	1052	20.7
697	(AT) (K) (8)	1052	20.7
698	(TJ) (T6)	1052	20.7
699	(Q9) (Q8)	1052	20.7
700	(A8) (K) (T)	1051	20.6
701	(AJT) (9)	1051	20.6
702	(Q8) (Q) (T)	1051	20.6
703	(9K) (9Q)	1051	20.6
704	(QT) (Q) (8)	1051	20.6
705	(JQ) (J) (9)	1051	20.6
706	(AQ) (J) (8)	1050	20.6
707	(Q) (J) (T) (9)	1050	20.6
708	(AJ) (T) (7)	1050	20.6
709	(A) (A) (K7)	1050	20.6
710	(TAJ) (T)	1049	20.6
711	(K) (K) (J9)	1049	20.6
712	(A) (Q) (J9)	1049	20.6
713	(K) (Q8) (T)	1048	20.6
714	(TQ) (T) (9)	1048	20.6
715	(K7) (J) (T)	1047	20.6
716	(Q) (Q) (A) (K)	1047	20.6
717	(A9) (K7)	1047	20.6
718	(Q9) (Q) (K)	1047	20.6
719	(AQ9) (A)	1047	20.6
720	(K7) (K) (J)	1046	20.5
721	(KJ) (K) (8)	1045	20.5
722	(TK) (T6)	1045	20.5
723	(K) (QT) (8)	1045	20.5
724	(K8) (K6)	1044	20.5
725	(A7) (J) (T)	1044	20.5
726	(QA) (Q) (8)	1044	20.5
727	(Q) (Q) (J9)	1043	20.5
728	(J9) (J) (Q)	1043	20.5
729	(K) (K) (A) (T)	1043	20.5
730	(TKQ) (T)	1042	20.5
731	(A) (Q) (T9)	1042	20.5
732	(A7) (K9)	1041	20.4
733	(K) (JT9)	1041	20.4
734	(KQ) (T) (7)	1041	20.4
735	(QK) (Q) (9)	1041	20.4
736	(J) (J) (Q9)	1041	20.4
737	(A7) (T9)	1041	20.4
738	(A) (A) (K6)	1040	20.4
739	(JT) (97)	1040	20.4
740	(J) (J) (K) (T)	1040	20.4
741	(K6) (K) (A)	1040	20.4
742	(Q) (JT) (7)	1040	20.4
743	(A) (J8) (T)	1039	20.4
744	(AK) (T) (8)	1039	20.4
745	(K8) (K) (Q)	1039	20.4
746	(JA) (J) (9)	1039	20.4
747	(TA) (T6)	1038	20.4
748	(K) (K) (T8)	1038	20.4
749	(KQ) (J) (8)	1038	20.4
750	(A) (A) (Q7)	1038	20.4
751	(K7) (T9)	1038	20.4
752	(A9) (Q7)	1038	20.4
753	(KT) (Q) (7)	1038	20.4
754	(QT8) (J)	1037	20.4
755	(A) (JT) (8)	1037	20.4
756	(K9) (Q8)	1037	20.4
757	(AK) (J) (8)	1037	20.4
758	(QJ) (98)	1037	20.4
759	(Q) (JT8)	1036	20.3
760	(Q9) (T7)	1036	20.3
761	(KJ) (K) (7)	1036	20.3
762	(AT) (Q) (7)	1035	20.3
763	(K7) (Q) (T)	1035	20.3
764	(KT9) (K)	1035	20.3
765	(Q) (J) (T7)	1035	20.3
766	(KT) (K) (6)	1034	20.3
767	(A) (K8) (T)	1034	20.3
768	(KQ) (K) (8)	1034	20.3
769	(A) (J) (T8)	1034	20.3
770	(AQ) (K) (8)	1033	20.3
771	(AT9) (J)	1033	20.3
772	(AJ8) (A)	1033	20.3
773	(A8) (K) (J)	1033	20.3
774	(KJ) (98)	1033	20.3
775	(A8) (K) (Q)	1032	20.3
776	(AQ) (98)	1032	20.3
777	(J) (J) (K9)	1032	20.3
778	(AT) (97)	1032	20.3
779	(A) (K) (T9)	1031	20.2
780	(A9) (A) (8)	1031	20.2
781	(K7) (K) (Q)	1031	20.2
782	(JAK) (J)	1030	20.2
783	(QT) (97)	1030	20.2
784	(A9) (J7)	1030	20.2
785	(Q9) (J7)	1030	20.2
786	(A9) (K6)	1030	20.2
787	(TK) (T) (9)	1030	20.2
788	(A) (KQ) (9)	1030	20.2
789	(Q) (Q) (A9)	1030	20.2
790	(A7) (Q9)	1029	20.2
791	(Q9) (Q7)	1029	20.2
792	(Q) (Q) (A) (J)	1029	20.2
793	(K) (K) (A6)	1029	20.2
794	(A7) (J9)	1028	20.2
795	(QJT) (8)	1027	20.2
796	(AKJT)	1027	20.2
797	(AK) (Q) (8)	1026	20.1
798	(QJ) (T) (6)	1026	20.1
799	(A) (A) (J7)	1026	20.1
800	(J) (J) (A9)	1026	20.1