3.186% is 11*4*4*4/22100 instead of 12*4*4*4/22100 for 3.475%, this is maybe why, so I guess we will get a new set of flops soon.

there are 8 (strategically indifferent) duplicates: flops 2 & 41 (J98r), 3 & 22 (AKQr), 4 & 13 (875r), and 39 & 53 (KK9tt). This means that the total set of 103 flops renders down to 99 flops.

yes, ofc.

have you also checked for this frequency in the actual subset (3.19%)?

I posting in this thread originally, I've though about using a slightly different approach from creating a static set of flop groupings. Instead a dynamic method for grouping flops based on individual range vs range match ups could prove more useful.

I think it would be very difficult to simulate a complete game very accurately from a single grouping set given the wide variety of ranges vs range match ups for different positions and effective stacks, multiple players to the flop and especially as stacks get deeper.

Consider creating a method that could generate tailored flop categories for specific pre-flop ranges.

Start with continuation flops that completely miss both ranges including smaller paired or tripped boards as well as any smaller SD boards that don't connect with either range which will may remove most flops depending on range sizes.

Then set aside all hands that are common to both ranges, leaving only two much smaller partial ranges to deal with.

Without being clear yet on the best method to do this you would need to find numbers for all flop combinations that improve either or both partial ranges and categorize them as value, draws, or combos. This is further reduced since many of these combos will include a wide range of flops where single card or two card blanks can also be grouped together.

Example how many flop combinations make TPGK for player A and player B ranges, how many make TP+FD with one over card, and so on.

Once you get these totals it's easy to break them down as percentage of range and percentage of flops.

I can ballpark these numbers using combinatorics with hand vs range while playing in marginal post flop situations which is easier than it sounds so it seems like a simple algorithm for generating flop combo breakdowns for a range vs range matchup shouldn't be too difficult even if it uses a brute force method for finding all flops that improve at least one of the partial ranges.

Once you get all the numbers for flop combo groupings, it's easy to translate them into percentage of flops and percentage of a player range.

This would be all the information you need for a large variety of post flop calculations.

Just a couple basic examples would be calculating how often a given made hand is ahead, or the ratio of value/draws/combos/bluffs in an opponents range.

In general, you can use these numbers to tweak your balancing strategies on each street, modify pre-flop ranges for helping to balance post flop action or find improved range vs range post flop match ups by adjusting variance.

More advanced uses may be to, improve clarity of implied odds for a percentage of hands that are likely to pay you off. Or finding fold equity when board textures favor one range over the other.

While it would be immediately useful for post hand analysis I believe it would provide all the essential components for creating a poker calculator that could provide street by street equities as well as street by street range breakdowns when providing starting ranges.

Yes I find 15 3-straight flops, summing up for 3.18552195060%

Hey guys,

So I've reread this whole thread twice and I'm still not sure if I understand it properly. Perhaps this is due to a language barrier(my natrive language is french). Anyways, I would really appreciate it if someone could explain this to me like if I was in elementary school!

Thanks!

to wrap this up,

while this should work in theory, i made a mistake by confusing rows and columns: it should be the other way around so (1/1755)/(true frequency in the full set of flops).

it seems that A32 is missing as a 3-straight flop condition in this subset.

it's an interesting idea but imo having an unbiased static set of flops should prove more useful than sets that are dependent on specific ranges, especially for comparability reasons.

For modeling poker or studying both players range interaction which were the examples given for choosing to do this, a biased dynamic set holds far more advantages to a static unbiased one because realistic ranges are both dynamic and biased.

For instance, the following flops would all be represented in the same group of one gapped rainbow.
AQJr
976r
643r

However when modeling a game we need to be able to see that they are quite different in any flop scenario.

They also uniquely interact with opponents ranges. For example consider these two ranges in a hand:

EP open with a tight value range
LP overall call with a caped wide merged range.

We should see that:

AQJr favors the aggressor.
976r favors the caller.
643r is a continuation flop that missed both.

A more subtle example:

Ad, 9h, 4h
Ad, 9d, 4h

Are both two tone flops that appear to be very similar.

However the two flops should be treated very differently with strong action since TP+NFD can only exist in the first flop.

It seems to me that there are too many Qhigh flops (19.44%) compared with Khigh flops(14.53%)Am I missing something?

Reading the topic , make me thing how swallow my poker knowledge is. Thank you and good luck to all.

I've finally given this thread a read. Tell me if I'm caught up to speed:

- You guys have determined that there are 103 properties (listed in yahq's pastebin link), thus we have matrix A of dimension 103x22100.

- You guys have calculated the true frequencies of each property (also in link), thus we have matrix B of dimension 103x1. That is, we know what each A_px should equal. (But are the frequencies definitely right, or might they benefit from fact-checking?)

- We have yet to find a nontrivial solution for x, some of whose entries we want to be 0 so as to simplify the game.

Have I correctly summed it up, and is there anything I left out?

It seems like the paired boards are far too wet. With pair+FD+OESD/GSD being much more likely than their natural counterparts.

heehaww, the 103 is the approximate number of flops needed to satisfy the possible combinations of properties at their approximate frequencies. B is the vector of flops which satisfy the list of properties (corrected frequency of monotone/two suited/rainbow flops, correct frequency for each particular rank and suit, for a pair of a given rank, for number of possible straights, etc.) given.

The current solution is probably skews the flops too wet since it ends up loading multiple properties onto flops (see my comment above on overly wet paired boards).

I'll probably do an analysis of wetness of the proposed subset vs true wetness when I get some free time.

Oops I didn't notice anyone post a solution. I'll have to read again.

How is this number of 1755 derived, please?

The easiest way is to write a small program that canonicalizes flops/boards and then simply counts the resulting unique flops.

I never bothered to do it manually yet, but lets give it a try:

• There are the 13 different trip-flops
• Paired flops have 13*12 unique rank structures, with 2 different suit structures each (two-suited or rainbow)
• Unpaired flops have 13C3 unique rank structures, with 5 different suit structures (rainbow, monotone, 3 two-suited)

> 13 + 13*12*2 + choose(13,3)*5
[1] 1755

Thank you, looks much easier then my own answer I would still like to contribute:

Why are there 1,755 distinct flops, if considering color isomorphism?

Total numer of flops possible: Draw any 3 of 52 → 52! / 3! * (52-3)! = 52*51*50/6 = 22.100

Probability of a monotone flop: Draw any first card. There are 12 other cards of the same suit. Draw a second card of the same suit from the 51 remaining deck → (12/51). Multiply with the probability to draw a third card of the 11 identical suited cards from the remaining 50 card deck (11/50). 0,05176. Multiply with total number of flops → there are 1.144 monotone flops.

Probability of a rainbow flop: Draw any first card. Draw a second card of the 3*13 cards of a different suit from the 51 cards left in the deck → (39/51). Multiply with the probability to draw a third card from the 2*13 cards of a different suit from the remaining 50 cards → (26/50). 0,39765. Multiply with total number of flops → there are 8.788 rainbow flops.

Probability of a two suits flop: Draw a second card of the same suit → (12/51). Multiply with probability to draw a third card of different suit (3*13/50). Each of these flops can occur in three suit variations, so multiply accordingly → 3*(12/51)*(39/50) = 0,5506. Multiply probability with total number of flops → there are 12.168 two suited flops.

Test: all probabilities add up to 1, and all flops counts add up to 22.100.

Now color isomorphism:

Each of the monotone flops can occur in four different suits. Devide by 4 = 286 different flops.

Two suited flop: every combo of two cards of the same suit occurs once for every color, i.e. four times. And each occurence is accompied by a third card of a different suit of which there are three. So devide 12.168 by (4*3) to get the count of unique two suited flops = 1.014.

To receive the count of unique rainbow boards, count the number of unique flops possible, if there were no color distinctions at all. It's like variations of drawing 3 cards from only 13 differently ranked cards, putting them back to the deck. (n-1+k)! / (k!(n-1)!) = 15! / 6*12! = 455

Add up 455 + 1.014 + 286 = 1.755 different flops.

Your answer is much more to the point. You were thinking from the other side, i.e. rank strunc → suit struc, while I am coming from the suit struct → ranks. I alreay realized that 455 is 5*91, but I couldn't figure why 5 until your answer.

http://math.stackexchange.com/questi...th-isomorphism fyi

Could someone please elaborate about the PioSolver subset of 184 (kuba) flops (https://piosolver.com/blogs/news?page=3) They seem to be the state of art, as many solvers (SimplePostFlop, EquiLab, PioSolver etc.) are utilizing them. How exactly are the given weights derived? And is there a list somewhere to look up which flops of the other 1.571 exactly are represented by each of these 184 flops? The weights sum up to about 43. SimplePostFlop multiplied each weight by the number of combinations possible, i.e. 4X, 12x and 24x. Their weights sum up to ca. 530. Can't make sense of. I was expecting 1.755 or 22.100.

i.e. here is 103 flop subset

https://cdn.shopify.com/s/files/1/07...46264467327999

Wondering if there is program or excel file where you can input a random flop and it will let you know what subset it falls under. Has anyone done this?

Just giving this thread some love. This is a really important notion for those trying to work on the game beyond preflop. Thanks for the heavy lifting.

Anyone knows smth about this?
This tool would be so useful

What to those numbers after the board texture represent?

What's up with the 2.15?

the weighting i believe.

Bumping this thread in 2020.

How useful is the 103 flops list in the op for working with flop texture, as in elaborating a strategy "for all flops".

What I mean is, if I setup a preflop range vs range scenario, and analize their performance against every one of the 103 flops available in the op, could I say for practical purposes that I have assessed my strategy against all posible flops?

There's a margin of error. For the purpose of practicing against the solution or just studying postflop in general 100+ flops is absolutely sufficient. If you're trying to work out preflop EV it might be off in some spots.

This page goes over benchmarks of the different subsets of flops, comparing pio and GTO+ subsets. https://www.gtoplus.com/subsets/