The Nuts Game (Quiz)

Imagine the following 991 handed hold'em game: You get 2 cards from a complete deck, 5 cards (flop, turn and river) are dealt face down on the table. The dealer now gives each of the other 990 players two cards such that all possible combinations of the remaining 45 cards are covered.


All call the single $1 blind, you are on the button. State laws limit the size of the pot to $991, so there will be no further betting.


1. What is the best hand you could wish for in this situation?


2. Which hands would you fold?


3. How would you rank the following hands for EV?


AA, JJ, AKs, ATs, A9s, KJs, JTs, AK, JT, 72


4. How would you rank the above hands for nuts potential?


results follow


cu


Ignatius

I'll give #1 a shot !


" 1. What is the best hand you could wish for in this situation? "


Any suited Ace !?


You can also play AA - I guess - and maybe KK and QQ.


Anything else is pretty much garbage.

1. What is the best hand you could wish for in this situation?


The bare minimum winning hand will be top set (when the board isn't paired and no flush or straight is possible) and 3 people will be holding it every single time. For example if the As is on the board, three people will be holding AcAd, AdAh, and AcAh. These three players will split the pot. Therefore, AA, as usual, is a top candidate.


When the board is paired (let's say with AhAd), then the one person who holds AcAh will win the pot by himself. This adds to the value of holding AA or any pair.


If the board has three-of-a-kind on it (let's say 4h4c4d), then four people will split the pot. The four people will be the ones hold Ac4s,Ad4s,Ah4s,As4s. This gives some value to holding any Ax.


If the board has four-of-a-kind on it, then everybody who holds an Ace will split the pot. That will be a lot of people. Again, this gives value to any Ax.


When a flush will win the pot, you must almost always be holding AKs. Even AQs is a nearly worthless flush hand since somebody else will be holding AKs in the same suit. Of course, if the K of that suit is on the board, then AQs will win.


When a straight will win the pot, then connectors (suited irrelevent!), one-gappers, and two-gappers will be the winners. Three-gappers will never win.


My answer: AA, of course. It will win more top sets and more top quads than any other pair. It will also win many pots when there is a four-flush on the board and occasionally win when the board is TJQKx. The only hand you could make a reasonable arguement for would be AKs but I don't think it's even close.


2. Which hands would you fold?


Let's see. You're getting 990 to 1 pot odds on your call so you don't have to play just the best starting hands. You play hands that will win once out of every 990 times that won't split the pot with other players. Since top set will always split the pot three ways, you can play pairs from AA down to a certain point where the odds are 330 to 1 that no overcards will come when your card is on the board. Also, no straight or flush can be possible. It is impossible for a 9 to be the high card on the board without a straight being possible. Therefore 99 can only win with Quads. So the playable pairs are AA,KK,QQ,JJ,TT as far as the top set qualifier is concerned. What are the odds of a T being high card on the board and no straight being possible. I suspect it's high enough to make TT playable since in will too often be beat by a higher set or a straight. The same may be true of JJ and QQ.


My answer: Well, I can't tell you all the hands I'd fold so I'll tell you the hands I'd definitely play. They are AA and KK. And probably QQ and AKs (for the flush). Everything else gets folded.


3. How would you rank the following hands for EV?

AA, JJ, AKs, ATs, A9s, KJs, JTs, AK, JT, 72


Only AA and AKs will be positive. Everything else will lose money.


4. How would you rank the above hands for nuts potential?


In order: 1.AA, 2.AKs, 3.JJ, 4.AKo, 5.ATs, 5.A9s, 5.KJs, 5. JTs, 5.JTo, 5.72o


It's awfully late. I hope all this thinking about a topic which will do my game no good will help me fall asleep.

It's impossible for a 9 to be the high card on the board without a straight being possible. The same can also be said of Ts and Js. The ideal non-straight board would be:


2,3,7,8,Q


You must have 3 gaps between the 3 and 7 and the 8 and Q to ensure there are no straights. That means JJ is unplayable since you'll have to make Quads to win.


This changes my answer to Question 2 since it's extremely unlikely for a Q-high board to not have a possible straight on it. This may be true of KK too.


So the playable hands are AA, maybe KK, and maybe AKs. Hmmm. just realized something. Since you're the only one holding that suited Ace, you could play Axs. It isn't the least bit important that you have the K. The second card isn't important. Oops. Of course, your flushes will lose everytime there is a pair on the board.


So the playable hands are: AA, maybe KK, maybe Axs.

I've put way too much thought into this quiz. I wish I could just fall asleep. Oh, well. I just keep realizing new stuff.


If you flop top set with no straights or flushes possible, you will win the whole pot since there is only 1 Ace distributed amon the other 990 hands. This adds value to AA and KK but I think QQ is still unplayable due to the threats of straights. Not a big deal but I wasted some space in my original post about splitting pots 3-ways. Just part of a too-long thought process.


So my final (please!) answers:


1. What is the best hand you could wish for in this situation?


a. AA for it's set value

b. KK for it's set value

c. Axs for it's flush value


2. Which hands would you fold?


Fold all hands except AA,KK, and Axs


3. How would you rank the following hands for EV?


1.AA, for it's set value

2.A9s, for it's flush value (interferes with more straight flushes than ATs and AKs)

3.ATs, for it's flush value (see above)

4.AKs, for it's flush value

5.JTs, for straight value (and straight flush value)

6.JT, for straight value

7.KJs, for straight value

8.AK, for straight value

9.JJ , for quad value

10.72, for nothing


4. How would you rank the above hands for nuts potential?


My rankings would be the same as in Question 3.


I'm going to bed. Hope I don't come up with something else.

> 1. What is the best hand you could wish for in this situation?


ATs looks pretty good.


> 2. Which hands would you fold?


I would play any suited A for sure. K suited with a straight card is probably profitable, due to all of the dead hands out there among your 990 opponents. In fact, any suited K MIGHT be profitable, since it makes the nut flush when the A of that suit appears. Any connectors and maybe even 1-gappers will also likely be profitable, again only because of the huge overlay of almost dead money with hands like J2o and such being in there for $1 each.


The hard part of this question is figuring out just how weak your hand can be, given the 990:1 payoff for winning. I simply do not know how often hands like J9o can make the nuts.


> 3. How would you rank the following hands for EV?


ATs, AKs, A9s, JTs, JT, KJs, AK, AA, JJ, 72


> 4. How would you rank the above hands for nuts potential?


Shouldn't it be the same rank? I mean, in this game, nut potential is all that matters, if I understand things correctly.


Later, Greg Raymer (FossilMan)

I make the following observations:


[] Somebody will have the nuts so it must be you.


[] Nobody can have your cards duplicated, so when you make quads or the Ace flush it will be the only one.


[] I notice that their are relatively few non-paired boards that do not also make a straight; some of which are: KQ873 through Q8732. Thus, only KK and QQ can make top set when its the nuts and not very often. Otherwise, you need to make quads with a pocket pair to have the nuts.


[] When you make the nut 2-card straight you will split it 10 ways.


I didn't do any counting, but suspect ... Axs is clearly the best hand you can get. A-Face suited are a little better. Kxs is also profitable since you will snag your Ace often enough. I doubt you can snag the A and K to make Qxs profitable. KQo through 43o are profitable, KJo-64o, and KTo and AT are also profitable. KK isn't profitable (I suspect), other pairs are obviously not.


- Louie

The probability of quads is roughly .008, and even quad 2's have a nearly 80% chance of standing up. So all pocket pairs are great and 2's are almost as good as Aces.


People are underestimating the value of flush draws - Qxs is fine. Of course Jxs is poor. Louie makes a great point that straights are only getting 99:1, so AKo is poor.

I'm not sure we're understanding the game the same way.


The way I understand the game, Qxs is an extremely thin hand.


You have to flop three queens, or three x's, (either of those with no straight flush possibility), or the board has to contain the ace of your suit, the king of your suit, at least one more of your suit, no pair, and no three-to-a-straight-flush.


If it did not have, say, the ace of your suit, then many of the other 990 players have Axs in your suit, and you lose.


The reason pocket aces are better than pocket deuces is that it is possible to win with three aces when the board contains one ace, no pair, and no straight or flush possibility. That is a possibility not available to pocket deuces. Pocket deuces can only win by making quads.


Part of the reason (that my intuition says that) AK offsuit sucks is that, when you make the most likely straight (board reads QJTxy, x less than T, y less than x, no three-flush), you have to split the pot with nine other people.


The whole computation is going to be a lot of work for little reward, unfortunately.


--JMike

You're forgetting that straights split the pot with nine other people. If you have Js Th and make the nut straight when the board reads


AKQxy

KQ9xy

Q98xy

987xy


then you split the pot with Jh Ts, Jh Td, Jh Tc, Jd Ts, Jd Td, Jd Tc, Jc Ts, Jc Td, and Jc Tc.


God forbid you make the nut straight when the board reads AKQTx because then you split the pot with everyone who holds one or two Jacks. I think three of your opponents have two Jacks and 126 others have one Jack?


That's why the EV differs from the probability of making the nuts.


--JMike

what about JTs? makes more straights and straight flushes... and any straight that uses both J and T is the nut straight. just a thought..

i agree with JMike. the other 990 players have every other possible hand. in other words, all starting hands are out. every single possible one, with distinction by suit, etc. none are duplicated exactly, but Js9h will split its pots with Jd9s, Jd9h, Js9d, etc. in other words, there are 12 different J9o possibilities. same goes for all other non-pair/non-suited hands. and why wouldn't, say, AsKs beat AsQs when the board comes Ks2s9s3dJh? the way i understand the game, every possible starting hand is out, so some hole cards will be duplicated (in fact many hole cards will be duplicated). this brings situations like having AsKsKs4s8s being a possible hand with 2 spadeKings (one in the hole, the other on board.) so the AsKs will still win, leaving Axs very little possibility of winning. someone else said something about KK and QQ being the only hands that can be the nuts if they make a set. this person forgot AA. how you forget AA and still call yourself a poker player, i don't know. (i dont mean disrespect, im just amazed somebody could forget AA...) also, no hands are worth playing if both cards don't play to make the nuts (if only 1 hole card is used to make the nuts, then that hand will split pots with every other possible hand that contains that single card. specifically, 50 other hands). this is 19.8-to-1 pot odds. there are 311,875,200 possible boards, meaning you have to have 15751272.8-to-1 odds to make that profitable. hands that will not split will be the best hands to play..

It is perverse that 45 other players will have your queen with a different hole card. This takes away some pretty central intuitions from regular hold'em. It is much tougher for Qxs to make the nut flush except for QJs. But at 990:1 odds, there seems enough chance for AKxs to flop.

You and I are understanding the game differently as well.


The way I understand it, not only can't your cards be duplicated in the other hands, but the cards on the board can't be duplicated in the other hands.


There are 1,326 starting hands, but 990 starting hands if you're selecting from only 45 cards.


--JMike

I'm liking ATs. It wins when:


(AKQJTs: 1 winner) Ks Qs Js x y

(AAAA: 1 winner) A A A x y, no 3-straight-flush

(AAAA: 1 winner) A A A x x

(TTTT: 1 winner) T T T x y, no 3-straight-flush

(TTTT: 1 winner) T T T x x, x less than ten


(AAATT: 10 winners) A A T x y, x and y less than ten, no 3-straight-flush


(flush: 1 winner) three- or more flush, no 3-straight-flush, no pair


(AKQJT: 10 winners) K Q J x y, x and y less than ten, no 3-flush


(AKQJT: 45 winners) A K Q J x, x less than ten, no 3-flush


(AKQJT: 45 winners) K Q J T x, x less than ten, no 3-flush


(AKQJT: 991 winners) A K Q J T


Computing the exact chances of each of these boards coming up

is an enormous pain in the ass. I think the most likely situation is the flush. Just doing something mindlessly dumb like "pick three of the remaining eleven flush cards and two of the thirty-nine non-flush cards" is probably about five percent high. You accidentally include some three-straight-flushes and some boards with pairs, but you're not counting any four-flushes. So ATs makes the nut flush about 110,000 out of 2,118,760 times or about five percent of the time. That's going to be the biggest chunk of the hand's wins.


AA wins when:


(AAAA: 1 winner) two aces on board, no 3-straight-flush


(flush: 1 winner) four- or five-flush, no 3-straight-flush, either suit


(AKQJT: 45 winners) KQJTx, x less than ten, no 3-flush


(AKQJT: 991 winners) AKQJT


(AAA: 1 winner) A-high board, no pair, 3-straight, or 3-flush


somewhat fewer than 20,000 boards you're going to make the four-flush.


Two aces on board will appear about 17,000 times.


The one-ace-on-board-no-pair-no-straight combinations, I think, are


AK984

AK983

AK982

AK974

AK973

AK972

AK964

AK963

AK962

AK873

AK872

AK863

AK862

AK762


AQ974

AQ973

AQ972

AQ964

AQ963

AQ962

AQ873

AQ872

AQ863

AQ862

AQ762


AJ964

AJ963

AJ962

AJ863

AJ862

AJ762


So there are 31 combinations, 2 remaining aces, four of each of the other cards, so 31*2*4*4*4*4 = 15872, of which maybe, what, five percent have three-flushes? So call it 15000 or so boards with which AA wins making three aces.


So it looks like AA is going to win the pot outright something like 50,000 times, or about two-and-a-half percent of the time.


Any other pocket pair wins less often than pocket aces (can't make top straight if the pair is less than tens, if the board is A A x x y then pocket x's lose).


Ax suited, where x is less than ten, wins less often than ATs -- it can't make top straight as often.


Any other suited combination wins much less often than Ax suited -- the higher flush cards must appear on the board to make top flush.


SO I think I like ATs the best.

See next message for the hands I'd fold.

i see. but how can you have that many players all with a hand, and not have the board duplicate the cards either? you wouldn't have a board...

I fold every hand that is a three-gapper or wider, where both cards are nine or lower. I play all other hands.


Let's consider the worst possible class of hands -- a hand that is not a pair, where the cards don't combine to form any nut straight chances, and both cards are less than ten. It doesn't matter much whether the cards are suited because, if neither card is jack or higher, you'll never make the nut flush anyway. The hands in this class are not equal but they're probably as close as dammit.


Let's use 72 as the template.

72 wins the whole pot when the board reads


7 7 7 x y no 3-straight-flush (1056 - SF1) times

7 7 7 z z where z less than 7 30 times

2 2 2 x y (1056 - SF2) times


I think there are 54 straight flush possibilities in 7 7 7 x y and 27 straight flush possibilities in 2 2 2 x y so

72 offsuit wins the whole pot 2,061 times.


72 will split the pot with the whole rest of the field a few other times:


1000 or so times when the board reads AKQJT with no 3-flush

4 times the board is an AKQJT straight flush

4 times the board reads AAAAK

48 times the board reads xxxxA


so let's call that equivalent to winning the whole pot one more time. 72 offsuit now wins the whole pot 2,062 times out of 2,118,760 boards, or one in 1,027. That's damn close to playable but not quite.


I think the smallest extra EV we can add to this hand is to make one of the cards ten or higher. Let's see when T2 offsuit can win outright:


T T T x y, no 3-straight-flush. (1056 - 54) times.

T T T z z, z less than ten. 48 times.

2 2 2 x y, no 3-straight-flush. (1056 - 27) times.


Total: 2079 + 1 for board being the nuts = 2080.


Additionally, T2 offsuit wins 1/45 of the pot when the board reads


A K Q J x, x less than ten, no three-flush. This happens a little less than 8000 times. So we're effectively winning another 170 pots.


Effective total wins = 2250. That's one in 942 times.

That's positive EV!


Making one card a ten is the smallest improvement I can think of. Note that making the hand a two-gapper (e.g. 74) yields over 10,000 boards where the hand wins one-tenth of the pot.


I think I'm done with this problem, as getting into any more detail involves a lot of picky counting that I'm not doing unless I get paid for it /images/smile.gif


--JMike

It is a bit of a mind-bender, the way this problem is set up.


Think of it this way -- for every combination of you, the board, and ONE of your opponents, none of the cards are duplicated.


Subject to that constraint, every possible opponent is in the pot. All 990 of them.

1. What is the best hand you could wish for in this situation?


While JMike correctly pointed out the difference between nuts potential and EV, both, he and Fossil Man considered ATs to be the best hand in the Nuts Game. While ATs does have has the greatest nuts potential (4.343% or 92015 of 2118760 possible boards), A9s, due to its potential to sabotage more hostile straight flushes, is slightly better in terms of EV ($30.52 vs. $30.18 for ATs) although it will be the nuts only 3.519% of the time. Dynasty got that one right, but considered AA to be even better (in fact, AA is the 13th best hand with an EV of $15.78

but any suited ace is better)


2. Which hands would you fold?


The key to this question is to realize that the Nuts Game is not fair: While each of the 990 opponents faces 86 hands which hold one of his cards, you enjoy the privilege of holding two cards which can not appear in somebody else's hand. The resulting bias is so strong that there are hardly any hands not worth playing. The only hands with negative EV are 92s, 92, 82s, 82, 72 and 62, but even those will not lose more than 2 cents in the long run.


(My first computer simulation of the Nuts Game gave a +EV result of 0.44 cents even for 72o and 62o, but this was due to a glitch in the program which considered straight flushes with a gunshot straight flush draw on board to be the best hand, even when higher straight flushes using 2 hole cards were out. It that case you could have called blind with any hand.)


JMike was closest to the solution by considering the greatest number of hands as playable.


3. How would you rank the following hands for EV?


pos EV

A9s 1. +30.52

ATs 2. +30.18

AKs 10. +26.52

AA 13. +15.78

KJs 16. +11.06

JJ 27. +6.46

AK 32. +5.79

JTs 40. +5.08

JT 49. +3.46

72 168. -0.02


4. How would you rank the above hands for nuts potential?


Fossil Man got the ranking right:


pos. nut-boards prob.


1. ATs 92015 4.343%


4. AKs 86346 4.075%


5. A9s 74550 3.519%


7. JTs 72122 3.404%


11. JT 68793 3.247%


15. KJs 63118 2.979%


27. AK 42181 1.991%


28. AA 40990 1.935%


67. JJ 21060 0.994%


168. 72 2754 0.130%


Actually, it has been this very question which made me spend two days in writing the computer simulation. (That, and a crazy pineapple pot limit hand I lost to a guy who flopped the nut straight with 53o and kicked me out of the tourney).


The 21 hands with the highest nuts potential are AXs, suited broadway cards, JT, QJ and QT. These are about 8% of all hold'em starting hands. They will river the nuts in one of 23 (ATs) to 37 (QT) cases. The next best hand (KQ) will only end up as the nuts for one of 46 boards.


I'm still not sure how these results can be used in actual play - one might argue that nuts potential plays a role in loose passive pot limit games or limit structures with higher river bets. Also, holding the nuts should become more important in games like pineapple, where the quality of hands is much higher than in ordinary hold'em. Any thoughts how the nuts potential of otherwise marginal hands should affect your preflop play?


cu


Ignatius

Hey!


As close as I can tell, A9s breaks up something like 750 straight flushes that are not analogously broken up by ATs.


ATs either is the straight flush, or breaks up the straight flush for 9 three-flush combinations.


A9s either is the straight flush, or breaks up the straight flush, for eleven three-flush combinations.


The two "extra" three-flush combinations that A9s breaks up represent something like 370 boards each.


For example, when the board is T87xy, to avoid double-counting, accidentally putting a pair on the board, or making a straight flush, the remaining cards cannot be tens, eights, sevens, or the KQJ654 of the suit. We also have to be sure x and y don't match. I think that yields 373 boards.


So we have something on the order of 750 extra straight flushes that A9s breaks up and wins with.


There are a couple small benefits, and a large benefit, of having AT vs. having A9. There's one small loss: AT has slightly fewer ways of making quads and getting beat by a straight flush than A9 does. This is counterbalanced by the slightly extra number of times AT can win 1/10 of the pot with AAATT vs. the number of times A9 can win 1/10 of the pot with AAA99. (In the former case, the board has to be AATxy, x and y less than ten. In the latter case, the board has to be AA9xy, x and y less than nine.)


But the huge benefit that ATs has over A9s is that ATs wins 1/10 of the pot when the board reads:


K Q J x y, x and y less than ten, no three-flush.


A9s has no analogous win (winning 1/10 of the pot when three straight cards hit the board).


K Q J x y, x and y less than ten, happens 28672 times.

I _think_ that a board made up of five "live" cards is a three-flush 47/256 of the time. So KQJxy with no three-flush happens 23408 times. So ATs wins 2340.8 pots that A9s does not.

That's a lot more than the straight flush breakup benefit of having A9s. Those extra 1000 pots are not going to be absorbed by weird things like the 45 extra times A9s wins with four straight flush cards on the board than ATs does etc.


ATs is therefore better than A9s.


If this is still controversial, I can grind out a long message listing off a simplified discussion of when each hand wins, although I haven't ground out the numbers to the conclusion. Seeing the wins side-by-side and a slightly simplified computation of the straight flush breakup factor was enough for me. 'course I could be making a mistake, but I'm pretty convinced I'm not...


--JMike

but the only variable is my opponent's hand for each constant hand and board. that means that none of my opponents' cards can duplicate the board or my hand. i guess that puts Axs back on the list. and AA KK and maybe QQ, because they only split the pot with one hand when they split. otherwise i can't see any other hand being worth it.

hi JMike!


Well, the result took me by surprise, too. So let's take a closer look:


> As close as I can tell, A9s breaks up something

> like 750 straight flushes that are not analogously

> broken up by ATs.


In both cases, there are 76197 unpaired boards which will give XYs a flush or straight flush:


66825 3-flushes (comb(11,3)*comb(10,2)*3*3)

8910 4-flushes (comb(11,4)*9*3)

462 suited boards (comb(11,5))


[ comb(n,k):=n!/k!(n-k)! ]


From all comb(11,3)=165 different 3-flushes, 3*comb(4,2)+3=21 (for A9s) and 4*comb(4,2)+3=27 (for ATs) will give an opponent straight flush. This amounts to 6*405=2430 additional pots lost to straight flush for ATs on unpaired 3-flush boards or 3.64%.


We can expect the relative difference to be considerably higher for 4-flushes and suited boards. The exact numbers are:


x suited cards: won by A9s vs. won by ATs

3 suited cards: 57915 vs. 55485

4 suited cards: 5670 vs. 4968

5 suited cards: 188 vs. 145


for a total difference of 3175 boards, resulting in an EV loss of $1.485.


> K Q J x y, x and y less than ten, happens 28672 times.


Don't forget that x and y have to be of different rank, so the number of unpaired rainbow KQJxy boards is


comb(8,2)*(1024-4-4*3*5-4*3*3*comb(5,2))=28*600=16800


Those will win 1/10 pot each and increase EV by $0.785.


> AT has slightly fewer ways of making quads and

> getting beat by a straight flush than A9 does.


No, because the straight flush would have to be of a different suit, so it doesn't matter that ATs is a three-gapper. What does help, however, is the lower probability of ATs to lose to higher quads (as e.g. A9s with TT999), but this is only worth 0.28 cents.


> This is counterbalanced by the slightly extra

> number of times AT can win 1/10 of the pot with

> AAATT vs. the number of times A9 can win 1/10

> of the pot with AAA99. (In the former case, the

> board has to be AATxy, x and y less than ten. In

> the latter case, the board has to be AA9xy, x and

> y less than nine.)


There are 3*3*16*comb(8,2)=4032 AATxy and 3*3*16*comb(7,2)=3024 AA9xy boards (some boards will give an opponent straight flush, but this effect will equal out). Each will win 1/3 or the pot, resulting in an EV difference of $0.157.


All else being equal, the above would result in A9s being a $0.54 favorite over ATs.


ATs higher chance to win with royal flush when the board pairs and AKQJx split-pots bring that down to $0.34, but A9s is still the best hand.


cu


Ignatius

For those interested in the complete solution of the Nuts Game, here is a table of all starting hands. It contains some minor corrections (a brute force simulation revealed some subtle bugs in my original program), none of which have had any influence on the solutions posted above (esp. A9s is still the best hand in terms of EV). If there are still errors, please let me know!


The table is sorted by nuts potential, since it is the only figure which might have some relevance for less-than-991-handed play. The other fields are hand, nuts-probability, EV and rank according to EV:


1. ATs 4.347% +29.94 (2)

2. AJs 4.245% +28.61 (6)

3. AQs 4.150% +27.33 (8)

4. AKs 4.080% +26.26 (10)

5. A9s 3.521% +30.28 (1)

6. A6s 3.413% +29.92 (3)

7. JTs 3.404% +5.03 (40)

8. A8s 3.349% +28.86 (4)

9. A7s 3.313% +28.74 (5)

10. A5s 3.256% +28.55 (7)

11. JT 3.247% +3.41 (49)

12. A4s 3.096% +27.03 (9)

13. KQs 3.023% +11.09 (15)

14. QJs 2.996% +5.94 (29)

15. KJs 2.979% +10.99 (16)

16. QTs 2.954% +5.81 (31)

17. KTs 2.943% +10.94 (17)

18. A3s 2.940% +25.52 (11)

19. A2s 2.771% +23.85 (12)

20. QJ 2.753% +3.46 (48)

21. QT 2.695% +3.20 (51)

22. KQ 2.318% +4.02 (43)

23. T9s 2.252% +3.89 (45)

24. KJ 2.226% +3.47 (47)

25. KT 2.167% +3.21 (50)

26. T9 2.079% +2.13 (78)

27. AK 1.994% +5.51 (34)

28. AA 1.942% +15.63 (13)

29. AQ 1.873% +4.71 (41)

30. AJ 1.779% +4.15 (42)

31. KK 1.767% +14.11 (14)

32. 98s 1.746% +2.99 (53)

33. AT 1.719% +3.88 (46)

34. 87s 1.713% +2.89 (55)

35. 76s 1.687% +2.80 (57)

36. 65s 1.669% +2.74 (59)

37. J9s 1.648% +3.15 (52)

38. 98 1.624% +1.74 (81)

39. 87 1.590% +1.63 (85)

40. T8s 1.580% +2.75 (58)

41. 76 1.563% +1.54 (86)

42. 65 1.543% +1.48 (89)

43. J9 1.499% +1.65 (83)

44. 54s 1.489% +2.54 (62)

45. T8 1.448% +1.42 (93)

46. K9s 1.390% +9.32 (22)

47. K5s 1.364% +9.93 (18)

48. 54 1.360% +1.25 (98)

49. K4s 1.339% +9.78 (19)

50. K3s 1.320% +9.66 (20)

51. K2s 1.310% +9.57 (21)

52. K8s 1.310% +8.81 (24)

53. 97s 1.296% +2.51 (64)

54. 86s 1.269% +2.42 (68)

55. K7s 1.262% +8.58 (25)

56. 75s 1.250% +2.35 (71)

57. 64s 1.235% +2.30 (74)

58. QQ 1.232% +8.82 (23)

59. K6s 1.224% +8.40 (26)

60. Q9s 1.224% +3.96 (44)

61. 97 1.169% +1.22 (99)

62. 86 1.141% +1.13 (103)

63. 75 1.120% +1.06 (108)

64. 64 1.105% +1.00 (109)

65. 53s 1.059% +2.11 (79)

66. J8s 1.002% +2.03 (80)

67. JJ 0.996% +6.49 (27)

68. Q9 0.974% +1.46 (90)

69. TT 0.957% +6.10 (28)

70. T7s 0.940% +1.65 (82)

71. 53 0.927% +0.79 (113)

72. J8 0.895% +0.96 (110)

73. T7 0.850% +0.75 (114)

74. 43s 0.840% +1.46 (91)

75. A9 0.756% +2.88 (56)

76. 43 0.755% +0.61 (118)

77. 99 0.723% +5.86 (30)

78. Q8s 0.719% +2.98 (54)

79. A8 0.713% +2.74 (60)

80. 88 0.711% +5.74 (32)

81. 77 0.698% +5.62 (33)

82. 96s 0.695% +1.50 (87)

83. 66 0.685% +5.49 (35)

84. A7 0.678% +2.63 (61)

85. 85s 0.674% +1.43 (92)

86. 55 0.673% +5.37 (36)

87. 44 0.667% +5.31 (37)

88. 33 0.661% +5.24 (38)

89. 74s 0.658% +1.37 (95)

90. 22 0.654% +5.18 (39)

91. A6 0.649% +2.53 (63)

92. 63s 0.649% +1.33 (96)

93. Q7s 0.638% +2.42 (67)

94. A5 0.627% +2.47 (65)

95. A4 0.614% +2.43 (66)

96. K9 0.614% +1.64 (84)

97. A3 0.607% +2.39 (69)

98. Q6s 0.607% +2.30 (73)

99. 96 0.605% +0.60 (119)

100. A2 0.604% +2.38 (70)

101. Q5s 0.594% +2.32 (72)

102. 85 0.584% +0.53 (123)

103. Q4s 0.578% +2.26 (75)

104. K8 0.571% +1.50 (88)

105. 74 0.568% +0.47 (128)

106. Q3s 0.567% +2.22 (76)

107. Q2s 0.564% +2.19 (77)

108. 63 0.558% +0.43 (133)

109. K7 0.535% +1.38 (94)

110. Q8 0.511% +0.91 (111)

111. K6 0.506% +1.29 (97)

112. J7s 0.502% +1.07 (106)

113. K5 0.485% +1.22 (100)

114. 52s 0.477% +1.15 (102)

115. Q7 0.474% +0.80 (112)

116. K4 0.468% +1.16 (101)

117. K3 0.458% +1.12 (104)

118. K2 0.455% +1.10 (105)

119. T6s 0.446% +0.71 (115)

120. Q6 0.445% +0.70 (116)

121. 42s 0.443% +1.07 (107)

122. J7 0.438% +0.43 (132)

123. J6s 0.428% +0.54 (122)

124. Q5 0.424% +0.64 (117)

125. J6 0.408% +0.34 (136)

126. Q4 0.408% +0.57 (121)

127. J5s 0.405% +0.45 (130)

128. T6 0.399% +0.25 (139)

129. Q3 0.398% +0.53 (124)

130. Q2 0.394% +0.51 (125)

131. J4s 0.389% +0.39 (134)

132. J5 0.387% +0.27 (138)

133. 52 0.385% +0.24 (140)

134. T5s 0.380% +0.20 (142)

135. J3s 0.379% +0.35 (135)

136. T5 0.378% +0.18 (144)

137. J2s 0.375% +0.33 (137)

138. J4 0.371% +0.21 (141)

139. T4 0.361% +0.11 (149)

140. T4s 0.361% +0.11 (148)

141. J3 0.361% +0.17 (145)

142. J2 0.357% +0.15 (146)

143. 42 0.355% +0.19 (143)

144. T3 0.351% +0.07 (152)

145. T3s 0.351% +0.07 (151)

146. T2 0.348% +0.05 (155)

147. T2s 0.348% +0.05 (154)

148. 32s 0.281% +0.47 (127)

149. 32 0.239% +0.06 (153)

150. 95s 0.207% +0.58 (120)

151. 84s 0.191% +0.51 (126)

152. 73s 0.180% +0.47 (129)

153. 62s 0.176% +0.45 (131)

154. 95 0.161% +0.12 (147)

155. 94s 0.146% +0.07 (150)

156. 84 0.144% +0.05 (157)

157. 94 0.144% +0.05 (156)

158. 83s 0.136% +0.03 (158)

159. 93 0.134% +0.01 (160)

160. 93s 0.134% +0.01 (159)

161. 73 0.134% +0.01 (162)

162. 83 0.134% +0.01 (161)

163. 72s 0.132% +0.01 (163)

164. 92 0.131% -0.01 (165)

165. 92s 0.131% -0.01 (164)

166. 82 0.130% -0.01 (167)

167. 82s 0.130% -0.01 (166)

168. 62 0.130% -0.01 (169)

169. 72 0.130% -0.01 (168)