This is going to be a long and winding post regarding Six Plus Hold-Em (also called Short Deck).

My daughter and I were watching some of the recent Poker After Dark live stream in which several of the players were discussing Six Plus Hold-Em, a poker variant in which all of the 2's, 3's, 4's, and 5's are removed from the deck. Sixes are the lowest card (hence the name Six Plus Hold-Em). Of course, there are now 36 cards in the deck rather than the traditional 52.

The players mentioned that the hand rankings of Six Plus Hold-Em are slightly different from the traditional poker hand rankings. My daughter asked me why they would make any changes.

Of course, the traditional poker hand rankings are: royal flush, straight flush, four of a kind, full house, flush, straight, three of a kind, two pair, one pair, high card. These hand rankings reflect how rare a certain type of hand is.

For example, when dealing five cards from a traditional 52-card deck there are fewer flushes than straights, and, thus, a flush is ranked higher than a straight in traditional poker hand rankings. Note that the frequencies change slightly in 7-card hands vs. 5-card hands (these respective frequencies can be easily found on the internet), a point we will come back to shortly.

So the first thing my daughter and I did was to tabulate all 5-card Six Plus Hold-Em hands, using the hand ranking that the players mentioned on the live stream. Of course, a hand is categorized in the highest category it achieves, something that will become relevant when we turn our attention to 7-card hands below.

5-Card Six Plus Hand Frequencies

Category 1: Royal Flush
Clearly there are 4 possible royal flushes (one in each suit).

Category 2: Straight Flush (excluding royal flush)
High card can be K,Q,J,T,9 (note that 9876A is considered a straight in Six Plus poker), so that there are 4*5 = 20 straight flushes.

Category 3: Four of a Kind
9 ranks, other card can be any of (36-4) cards, so 9*32 = 288

Category 4: Flush (not royal or straight flush)
4*[C(9,5)-6] = 4*(126-6) = 4*120 = 480

Category 5: Full House
C(9,2)*2*C(4,3)*C(4,2) = 1,728

Category 6: Three of a Kind (excluding full house)
C(9,1)*C(4,3)*C(8,2)*4*4 = 16,128

Category 7: Straight (not royal or straight flush)
6*(4^5) - (4+20) = 6,120

Category 8: Two Pair (not full house)
C(9,2)*C(4,2)*C(4,2)*(36-8) = 36,288

Category 9: One Pair (not two pair, etc.)
C(9,1)*C(4,2)*C(8,3)*(4^3) = 193,536

Category 10: High Card (no straight, flush, straight-flush, or royal-flush)
C(9,5)*(4^5) - (4+20+480+6120) = 122,400

TOTAL NUMBER OF POSSIBLE HANDS = C(36,5) = 376,992

So, based on the above, it does look like it makes sense to move Flush ahead of Full House in the Six Plus poker hand rankings, but moving Three of a Kind ahead of Straight looks a little funny.

7-Card Six Plus Hand Frequencies

Since the game the players were discussing was Six Plus Hold-Em, we need to look at the frequencies of 7-card hands rather than, or in addition to, 5-card frequencies.

Here are the detailed derivations that my daughter and I slogged through (undoubtedly there are easier or more condensed ways to do this).

Category 1: Royal Flush
4*C((36-5),2) = 1,860

Category 2: Straight Flush (excluding royal flush)
4*5*C((36-6),2) = 8,700

Category 3: Four of a Kind
9*C((36-4),3) = 44,640

Category 4: Flush (not royal or straight flush)
Seven cards are in same suit = 4*C(9,7) = 144
Six cards are in same suit = 4*C(9,6)*C(27,1) = 9,072
Five cards are in same suit = 4*C(9,5)*C(27,2) = 176,904
So we have (144+9072+176904) - (1860+8700) = 175,560

Category 5: Full House
Ranks are [3,3,1] = C(9,2)*C(4,3)*C(4,3)*C(7,1)*C(4,1) = 16,128
Ranks are [3,2,2] = C(9,3)*3*C(4,3)*C(4,2)*C(4,2) = 36,288
Ranks are [3,2,1,1] = C(9,2)*2*C(4,3)*C(4,2)*C(7,2)*C(4,1)*C(4,1) = 580,608
Total = 633,024

Category 6: Three of a Kind (no full house, flush, straight-flush, or royal flush)
Ranks need to be [3,1,1,1,1] = C(9,1)*C(4,3)*C(8,4)*(4^4) - 4*C(9,5)*5*C(3,2) = 637,560

Category 7: Straight (no three of a kind, flush, straight-flush, or royal flush)
Ranks are [2,2,1,1,1] = 6*C(5,2)*C(4,2)*C(4,2)*C(4,1)*C(4,1)*C(4,1) = 138,240

Ranks are [2,1,1,1,1,1] & pair is part of straight:
Ace high straight = 5*C(4,2)*(4^4)*C(4,1)*4 = 122,880
King-Nine high straight = 5*5*C(4,2)*(4^4)*C(3,1)*4 = 460,800

Ranks are [2,1,1,1,1,1] & pair is not part of straight:
Ace high straight = (4^5)*C(4,1)*C(4,2) = 24,576
King-Nine high straight = 5*(4^5)*C(3,1)*C(4,2) = 92,160

(Subtotal of ranks are [2,1,1,1,1,1] = 700,416 (we will use this below))

Ranks are [1,1,1,1,1,1,1]:
Ace high straight = (4^5)*C(4,2)*(4^2) = 98,304
King-Nine high straight = 5*(4^5)*C(3,2)*(4^2) = 245,760

(Subtotal of ranks are [1,1,1,1,1,1,1] = 344,064)

Subtotal of all straights = 1,182,720

Now we need to subtract off straights that are also flushes:

Ranks are [2,2,1,1,1] = 4*6*C(5,2)*C(3,1)*C(3,1) = 2,160

(Subtotal of straights & flushes with ranks [2,2,1,1,1] = 2,160)

Ranks are [2,1,1,1,1,1] with pair part of straight:
Ace high straight with 6 suited cards, 5 in straight = 4*5*3*4 = 240
Ace high straight with 5 suited cards, 5 in straight = 4*5*3*4*3 = 720
Ace high straight with 5 suited cards, 4 in straight, pair in suit = 4*5*3*4*3*4 = 2,880
Ace high straight with 5 suited cards, 4 in straight, pair not in suit = 4*5*C(3,2)*4 = 240

King-Nine high straight with 6 suited cards, 5 in straight = 4*5*5*3*3 = 900
King-Nine high straight with 5 suited cards, 5 in straight = 4*5*5*3*3*3 = 2,700
King-Nine high straight with 5 suited cards, 4 in straight, pair in suit = 4*5*5*3*4*3*3 = 10,800
King-Nine high straight with 5 suited cards, 4 in straight, pair not in suit = 4*5*5*C(3,2)*3 = 900

Ranks are [2,1,1,1,1,1] with pair not part of straight:
Ace high straight with 6 suited cards, 5 in straight = 4*4*1*3 = 48
Ace high straight with 5 suited cards, 5 in straight = 4*4*C(3,2) = 48
Ace high straight with 5 suited cards, 4 in straight, pair in suit = 4*5*3*4*1*3 = 720
Ace high straight with 5 suited cards, 4 in straight, pair not in suit = 0 (impossible)

King-Nine high straight with 6 suited cards, 5 in straight = 4*5*3*1*3 = 180
King-Nine high straight with 5 suited cards, 5 in straight = 4*5*3*C(3,2) = 180
King-Nine high straight with 5 suited cards, 4 in straight, pair in suit = 4*5*5*3*3*1*3 = 2,700
King-Nine high straight with 5 suited cards, 4 in straight, pair not in suit = 0 (impossible)

(Subtotal of straights & flushes with ranks [2,1,1,1,1,1] = 23,256)

Ranks are [1,1,1,1,1,1,1]:
Ace high straight with 7 suited cards, 5 in straight = 4*C(4,2) = 24
Ace high straight with 6 suited cards, 5 in straight = 4*C(4,2)*2*1*3 = 144
Ace high straight with 6 suited cards, 4 in straight = 4*5*3*C(4,2) = 360
Ace high straight with 5 suited cards, 5 in straight = 4*C(4,2)*3*3 = 216
Ace high straight with 5 suited cards, 4 in straight = 4*5*3*C(4,2)*2*1*3 = 2,160
Ace high straight with 5 suited cards, 3 in straight = 4*C(5,2)*3*3*C(4,2) = 2,160

King-Nine high straight with 7 suited cards, 5 in straight = 4*5*C(3,2) = 60
King-Nine high straight with 6 suited cards, 5 in straight = 4*5*C(3,2)*2*1*3 = 360
King-Nine high straight with 6 suited cards, 4 in straight = 4*5*5*3*C(3,2) = 900
King-Nine high straight with 5 suited cards, 5 in straight = 4*5*C(3,2)*3*3 = 540
King-Nine high straight with 5 suited cards, 4 in straight = 4*5*5*3*C(3,2)*2*1*3 = 5,400
King-Nine high straight with 5 suited cards, 3 in straight = 4*5*C(5,2)*3*3*C(3,2) = 5,400

(Subtotal of straights & flushes with ranks [1,1,1,1,1,1,1] = 17,724)

Putting it all together, we have:
= 1,182,720 - (2,160+23,256+17,724)
= 1,139,580

Category 8: Two Pair (no full house, flush or straight)

Ranks are [2,2,2,1] = C(9,3)*C(4,2)*C(4,2)*C(4,2)*C(6,1)*C(4,1) = 435,456

Ranks are [2,2,1,1,1] = C(9,2)*C(4,2)*C(4,2)*C(7,3)*C(4,1)*C(4,1)*C(4,1) = 2,903,040

Now subtract off flushes:
Ranks are [2,2,1,1,1] = 4*C(9,5)*C(5,2)*C(3,1)*C(3,1) = 45,360

Now subtract off straights:
Ranks are [2,2,1,1,1] = 138,240 (from above)

Now add back in all hands with ranks [2,2,1,1,1] that are both straights and flushes:
We know this is 2,160 (from above)

Putting it all together, we have:
= (435,456 + 2,903,040) - (45,360 + 138,240) + 2,160
= 3,157,056

Category 9: One Pair (no straight or flush)
Ranks need to be [2,1,1,1,1,1] = C(9,1)*C(4,2)*C(8,5)*(4^5) = 3,096,576

Now subtract off flushes with ranks [2,1,1,1,1,1]:
Pair is not part of flush = 4*C(9,5)*C(4,1)*C(3,2) = 6,048
Pair is part of flush, other card is suited = 4*C(9,6)*6*C(3,1) = 6,048
Pair is part of flush, other card is not suited = 4*C(9,5)*5*C(3,1)*C(4,1)*C(3,1) = 90,720

(Subtotal of flushes with ranks [2,1,1,1,1,1] is 102,816)

Now subtract off straights with ranks [2,1,1,1,1,1]:
We know this is 700,416 (from above).

Now add back in hands with ranks [2,1,1,1,1,1,] that are both straights and flushes:
We know this is 23,256 (from above).

Putting it all together, we have:
= 3,096,576 - (102,816 + 700,416) + 23,256
= 2,316,600

Category 10: High Card (no straight or flush)
Ranks need to be [1,1,1,1,1,1,1] = C(9,7)*(4^7) = 589,824

Now subtract off flushes with ranks [1,1,1,1,1,1,1]:
Seven cards in suit = 4*C(9,7) = 144
Six cards in suit = 4*C(9,6)*3*3 = 3,024
Five cards in suit = 4*C(9,5)*C(4,2)*3*3 = 27,216

(Subtotal of flushes with ranks [1,1,1,1,1,1,1] is 30,384)

Now subtract off straights with ranks [1,1,1,1,1,1,1]:
Ace high straight = (4^5)*C(4,2)*(4^2) = 98,304
King-Nine high straight = 5*(4^5)*C(3,2)*(4^2) = 245,760

(Subtotal of straights with ranks [1,1,1,1,1,1,1] is 344,064)

Now add back hands with ranks [1,1,1,1,1,1,1] that are both straights and flushes:
We know this is 17,724 (from above)

Putting it all together, we have:
= 589,824 - (30,384 + 344,064) + 17,724
= 233,100

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Okay, let's create a table to display all of these 5-card and 7-card frequencies (I'll also add in columns for percentages to make comparisons easier):

Six Plus Category	Five Card Freq	Five Card Pct	Seven Card Freq	Seven Card Pct
Royal Flush	4	0.00	1,860	0.02
Straight Flush	20	0.01	8,700	0.10
Four of a Kind	288	0.08	44,640	0.53
Flush	480	0.13	175,560	2.10
Full House	1,728	0.46	633,024	7.58
Three of a Kind	16,128	4.28	637,560	7.64
Straight	6,120	1.62	1,139,580	13.65
Two Pair	36,288	9.63	3,157,056	37.82
One Pair	193,536	51.34	2,316,600	27.75
High Card	122,400	32.47	233,100	2.79
TOTAL	376,992	100.00	8,347,680	100.00

Whether looking at the 5-card hand frequencies or the 7-card hand frequencies, we see that elevating Flush ahead of Full House makes perfectly good sense in Six Plus poker. Of course, by compressing the deck, flushes are harder to come by whereas full houses are easier to come by.

You will also see that when 7-card hands are considered, it makes perfectly good sense to elevate Three of a Kind ahead of Straights (three of a kind being more rare) in Six Plus Hold-Em. This was not apparent when we looked only at 5-card hands.

I hope that these derivations may provide some value for others.

Any comments are more than welcome.

I found this very valuable, thank you.

However, I don't believe the results for 7 card hands have meaning with respect to rarity. When calculating your straights, you are already assuming the hand strengths and throwing away three-of-a-kind combos, making three-of-a-kind seem rarer.

Wow. Thanks for this whosnext.

Yeah, thank you for the effort!

So should a hand with no pair count as more than a pair? It would make for a really interesting game.

-d

Very good message.

A question, since when flush should be higher than full house? In 4 low decks (44 cards) or 5 low decks (40 cards)?

"So should a hand with no pair count as more than a pair? It would make for a really interesting game."

I think that is an excellent question. I saw an interview with Dwan, where he stated that someplace had to make the changes in the hand ranking due to a "law" or something that required more rare hands to beat less rare hands. It seems like this would qualify and would totally jack up every player even worse than just the flush and 3 of a kind rank changes. Looking at the 7 card percentages a high card would beat a full house!! That would be crazy.

So, going by the rule that rare hands are worth more, the true 7-card rating would be this:

Hand____________Seven Card Pct
Royal Flush____________0.02
Straight Flush__________0.1
Four of a Kind__________0.53
Flush_________________2.1
High Card_____________2.79
Full House____________ 7.58
Three of a Kind_________7.64
Straight______________13.65
One Pair______________27.75
Two Pair______________37.82


My 10 high beats your Aces over Queens full house, sorry bud!

This would be a super mind bending game to play.

ACtually it would be the following:

Six Plus Category Seven Card Pct
Straight Flush 0.12
Four of a Kind 0.53
Flush 2.10
High Card 2.79
Full House 7.58
Three of a Kind 7.64
Straight 13.65
One Pair 27.75
Two Pair 37.82
TOTAL 100.00

That sounds like an insane (and insanely fun) game.
Suddenly starting hands that can't easily make straights or flushes (Q7o or somesuch) jump up in value....and you start calculating "anti-outs" to protect your no-pair hand

Interesting! Is it possible to hold an ace high flush draw and miss the flush but finish with only high card?

I don’t think this is possible as a full house requires a paired board thus you do technically have a pair

But it would beat other strong hands which is weird. I assume Aria and everywhere else where this is spread went through the necessary steps. I thought it was possible that some “law” may specify that more rare hands must be ranked higher unless they got some type of exemption

Nice post Whosnext, very valuable post.

I saw some Six Plus Hold'em game with high stakes players and Chinese gamblers (Triton game). It seems that the hands ranking is not the same that the one you describe, straight beats 3 of a Kind in their game.

It should change the derivation for Category 6 (3 of a kind) and 7 (straight).
What should be the calculation with this new rules?

Yes it’s played that way to facilitate action otherwise arguably it’s not really a game people would want to play. It’s my opinion that any game spread where trips beat straights has any future long term.

Nothing changes at with the probabilities of making certain hands though

So certainly holdings that can make straights should be a big part of your playing strategy since they are great value.

I think the change in hand rankings would change the hand frequencies since they form a hierarchy and now the hierarchy would be different.

For example, if a player can make a straight and also has trips, in one way the calculation gives him a straight and in the other calculation he would be given trips.

I will leave it to others to re-do those calculations under the different hand rankings rule.

Oh yes that’s right. Probably wouldn’t move it much though

Not much of a difference but I'm getting 1,169,940 straights and 607,200 three of a kind for the 7-card combinations. The 5-card totals are ok.
For instance, (C(9,5)-6)*5*4*253 = 607,200.

I agree and thanks for the post.

I re-calculated the frequencies using the "new" Short Deck hand rankings (where a straight beats trips) around a year ago but I guess I never got around to posting them to this thread. Today, as confirmation of the combinatoric results, I wrote a simple program to tally the respective frequencies in the two cases.

For completeness, I first repeat the frequencies using the "original" Short Deck hand rankings (trips beat a straight) from above:

Six Plus Category	Five Card Freq	Five Card Pct	Seven Card Freq	Seven Card Pct
Royal Flush	4	0.00	1,860	0.02
Straight Flush	20	0.01	8,700	0.10
Four of a Kind	288	0.08	44,640	0.53
Flush	480	0.13	175,560	2.10
Full House	1,728	0.46	633,024	7.58
Three of a Kind	16,128	4.28	637,560	7.64
Straight	6,120	1.62	1,139,580	13.65
Two Pair	36,288	9.63	3,157,056	37.82
One Pair	193,536	51.34	2,316,600	27.75
High Card	122,400	32.47	233,100	2.79
TOTAL	376,992	100.00	8,347,680	100.00

Here are the category tallies using the "new" Short Deck hand rankings (a straight beats trips):

Six Plus Category	Five Card Freq	Five Card Pct	Seven Card Freq	Seven Card Pct
Royal Flush	4	0.00	1,860	0.02
Straight Flush	20	0.01	8,700	0.10
Four of a Kind	288	0.08	44,640	0.53
Flush	480	0.13	175,560	2.10
Full House	1,728	0.46	633,024	7.58
Straight	6,120	1.62	1,169,940	14.02
Three of a Kind	16,128	4.28	607,200	7.27
Two Pair	36,288	9.63	3,157,056	37.82
One Pair	193,536	51.34	2,316,600	27.75
High Card	122,400	32.47	233,100	2.79
TOTAL	376,992	100.00	8,347,680	100.00

How come a bunch of websites claim that trips beat a straight? Like pokerlistings here:



Where did this come from? Afaik on stars, ggnetwork, and the one time I watched it on PAD, they always had it as a straight beating trips.

I do not know the history of this game or its rules. I do know that the very first time I saw it played on TV several years ago trips did beat a straight (above I call this the "original" Short Deck hand ranking) due to the frequencies shown above -- in 7-card Short Deck there are more straights than trips.

I believe that the players of Short Deck quickly realized that the game "plays" better if you go back to having a straight beat trips and they changed the hand rankings. Above in this thread I refer to these as the "new" Short Deck hand rankings.

Yes, from what I have been told most (all?) online sites that offer Short Deck utilize the hand rankings were a straight beats trips.

I was just looking it up since I only ever played it a handful of times.

It seems that the WPN is the only one that still has three of a kind beating a straight.

Stars, GG, Party, ipoker, etc. (pretty much all the other networks) all have the straight ahead.

As a side note, do you know whether they used the new rankings from the start or the old ones, and then switched, when it first appeared online (I presume on stars)?