How it’s solved · Muck GTO

What this is (and isn’t)

Heads-up by the flop4-max and 6-max are solved as a game where every pot is heads-up by the flop: no overcalls, no limping behind, no three-way all-ins, preflop or postflop. Real tables do all three. (All-in or Fold is the exception: its 4/6-max all-ins can be three- and four-way.)
A fixed bet treeOnly the sizes listed under “The game, for the record” exist. The solutions are as good as they are inside that tree and say nothing about sizes outside it.
Stopped at 0.5% of the potEvery flop solve stops once a perfect opponent could take no more than 0.5% of the pot from it. Most land around 0.4%. Close, not zero.
Preflop is measured against tablesThe preflop numbers below are measured against postflop value tables built from sample flops, not against the full game. They say how close preflop is to perfect given those values.

76.8 billionMonte Carlo hands dealt for 4- and 6-max All-in or Fold: 56 spots, four to six runs of 300 million each. Most of them were bad. That was the point.

22,100Possible flops, every one solved heads-up in all seven pot types (1,755 once suits are folded together)

275,000Flop-and-turn solves on file at 100, 40 and 20 BB, give or take a couple hundred old ones

1,894Rented cloud machines, by the spending log

0.5%Of the pot: the most a perfect opponent can take from any flop solve before it stops

0.0045BB per hand: the most a perfect player can take from heads-up preflop, against its postflop tables

The game, for the record

RulesWorst hand wins: the lowest five of seven takes the pot, ranked in reverse, like 2-7 lowball. The ace is always high: A-2-3-4-5 is ace-high, not a straight.
Stacks and blinds100 BB, blinds 0.5 / 1. No ante, no rake, no tournament math: chips are chips.
Preflop sizesOpen 2.5 BB, 3-bet 7.5, 4-bet 22.5, then all-in. Heads-up the SB may limp, and the BB can raise a limp to 3, then ×3. At 4/6-max only the SB may limp, when it’s folded to them.
Postflop sizesFlop: bet ⅓ or ½ pot, one raise to 3×. Turn: ½ pot, no raises. River: ⅓, ½, pot or 2× pot, no raises (since October 28; ½ pot only before). All-in once the stack is no bigger than the pot.

Heads-up: all of it100% on their own solve

Preflop SolvedEvery decision, every hand. Preflop and flop were solved against each other, round after round: eight rounds, with flop values from 48–184 sample flops per pot type. It isn’t fully settled: the last round still moved the strategy about 6%. Against those flop values, a perfect opponent could take at most 0.0045 BB per hand. That’s a rounding error with a superiority complex.
Every flop, flop and turn SolvedAll 1,755 flop types, which by suit symmetry is every one of the 22,100 possible flops, in every pot: single-raised, limped, limp-raised, 3-bet, 4-bet, limp-3-bet and limp-4-bet. Every flop decision and every turn decision, each solve stopped at 0.5% of the pot or better.
Every river Solved liveStoring every river of every flop would take over 100 GB, so we don’t store them. We solve them. When a hand reaches the river, your browser takes both players’ ranges as they arrive and solves that river on the spot: 400 rounds of discounted CFR, about a second on a desktop, longer on a phone. Since October 28 the river offers ⅓, ½, pot and 2× pot. The stored flop and turn solves were made expecting ½-pot rivers, so the turn play doesn’t know about the other sizes yet.

4-max and 6-max4-max 99.98% · 6-max 98% of flops on their own solve

Preflop SolvedEvery seat, every decision. Flop values come from that table size’s own solves on 10 sample flops per spot: every seat pairing’s single-raised and 3-bet pots plus blind-vs-blind limps; 4-bet and limp-3-bet pots borrow heads-up values. Against those values, the most any seat can gain by going rogue: under 0.0015 BB per hand at 4-max; at 6-max 0.002–0.004 BB, except the hijack at 0.009 BB. The lines nobody takes (cold 4-bets, say) were filled in approximately afterward, some of them still carry large errors, and that fill-in is what moved the 6-max hijack up from 0.0016.
How much has its own solve 100 BBWeighted by how often each spot is dealt, as of the November 2 check: 4-max 99.98% of flops get the solve for your own line on your own board; 6-max 98.0% (turns 98.4%, rivers 98.5%). The rest get the nearest solved flop of the same texture (6-max 1.9%, 4-max 0.02%) or a labeled estimate, and 0.002% at 6-max say “Not solved yet”.
4-max, every flop SolvedBlind vs blind (limped, raised, limp-raised and limp-3-bet pots), every single-raised pot between any two seats and 19 3-bet lines, each solved for its own ranges, pot and bet sizes on all 1,755 flop types.
6-max, every flop SolvedBlind vs blind (limped, raised and limp-raised pots) and every single-raised pot between any two seats, on all 1,755 flop types, each with its own ranges, pot and bet sizes.
6-max 3-bet pots SolvedThe 20 most common 3-bet lines, and the limp-3-bet pot, on all 1,755 flop types. The other 63 ways a 3-bet pot happens (about 2% of flops) are each solved for that preflop line on 184 flops chosen to cover every texture (paired or not, rainbow, two-tone or monotone, high, middle and low boards) plus the 10 sample flops; on any other flop, Study shows the nearest solved flop instead, on its own board, and says so (see below).
4-bet pots Fewer flopsToo rare to solve on every flop. 4-max: 3 lines on 211 texture flops, the other 38 on a separate set of 71 texture flops. 6-max: 4 on the 211, the other 221 on the 71. On any other flop they show the nearest solved flop.
Rivers Solved liveIn your browser, same as heads-up, with the same four sizes, and only as good as the flop and turn feeding them.

Short stacks: 20 and 40 BBHeads-up 100% · 4/6-max about 96% on their own solve

Preflop SolvedHeads-up, 4-max and 6-max, same sizes as 100 BB; a raise the stack can’t cover becomes all-in. Flop values come from that depth’s own flop solves, because 20 BB poker is not 100 BB poker with less money. Measured against those values, the worst seat can gain up to about 0.01 BB per hand (6-max at 40 BB).
Heads-up, every flop SolvedAll 1,755 flop types in every pot that reaches a flop, with that depth’s own ranges and stacks: all seven pot types at 40 BB; five at 20 BB, where a 4-bet (or a limp-4-bet) is already all-in and never sees a flop. Rivers solved live.
4-max and 6-max, every flop SolvedThe most common lines are each solved for their own ranges, pot and sizes on all 1,755 flop types; the rest on 211 texture flops or a separate set of 71 texture flops, with the nearest solved flop elsewhere; rare 3-bet and 4-bet lines show as labeled estimates on their own range.
How much has its own solve 20 / 40 BBWeighted by how often each spot is dealt, as of the October 8 count: heads-up 100% on every street. 4-max and 6-max: about 96% of flops get the solve for your own line and your own board; the rest get the nearest solved flop or a labeled estimate.

All-in or Fold100% solved

Every table, every stack SolvedHeads-up, 4-max and 6-max; every stack from 2 to 20 BB, then 22, 25, 30, 35, 40, 50, 60, 75 and 100. Equal stacks, blinds 0.5 / 1. One decision each: shove or fold when it’s folded to you, call or fold when it isn’t.
Heads-up SolvedExact CFR over all 169 starting hands with exact card removal, on an all-in equity table that is itself Monte Carlo: 40,000 boards per hand-class matchup. Against that table, a perfect opponent gains under 0.00001 BB per hand, at any stack.
4-max and 6-max SolvedMonte Carlo CFR with exact card removal for every player: four to six independent runs of 300 million hands each per spot, averaged. Three- and four-way all-ins get played out like any other. Every big blind call or fold against one all-in is checked against the arithmetic: 4 of 42,588 hand-class decisions are more than 2 points off, all borderline hands, worst 0.67 BB.

Reading the charts

Fill heightHow much of that hand is in the range at this point. Full = always here; a sliver = here only sometimes.
GrayNot in the range here: the hand folded or took another line earlier. That’s not advice to fold. That’s history.
“Show hands you deviated with”Gray hands you played anyway (say, calling a hand the solution always 3-bets) get their best play here. Your opponent isn’t expecting them either, so it’s guidance, not equilibrium.
ExploitabilityHow much a perfect opponent could win from you per hand. Smaller is better. Zero is perfect. You are not zero.

Under the hood

What’s CFR?Counterfactual Regret Minimization. Translation: the solver plays itself millions of times, and after every hand it asks, at each decision, “how much better would the other options have done?” That gap is regret. Next time it leans toward whatever it regrets not doing. Repeat until neither side regrets anything. That’s GTO, and also the opposite of how you play. Discounted CFR and CFR+ are faster versions; Monte Carlo CFR samples random deals instead of checking every single one.
SolverAn open-source postflop solver (discounted CFR), rebuilt for Hold’em Low hand rankings. Its evaluator was checked against the game’s own: 200,000 hands, zero disagreements.
PreflopDiscounted CFR over all 169 starting-hand classes, exact card removal between the two players in a pot.
HardwareA swarm of cloud computers running in parallel, which Larry insists on calling “the boys,” plus one Mac in his basement for the examples and All-in or Fold. Larry hasn’t slept. Neither have the boys.
UpdatedNovember 4, 2026.

Still in the oven6-max 2% · 4-max 0.02% of flops

Everything above has a solve of its own, inside the limits at the top of the page. This is what’s borrowed, simplified or not here yet. Read it before you tell us the solver is wrong. It might be. It’ll say so here first.

4/6-max: your flop isn’t in the solved set Approximate6-max 1.9% · 4-max 0.02% of flopsHeads-up has every flop. At 4/6-max, a line solved on a set of texture flops (184 for the less common 6-max 3-bet lines, 211 or 71 for 4-bet lines and at 20 and 40 BB) shows you the nearest solved flop of the same texture: same pairing, same suits, closest ranks (low cards count more, because in Hold’em Low they make the hands). The chart is drawn on that flop’s board, marked “Nearest solved flop”, with a button back to yours. It’s the right strategy for a slightly different flop, not a guess dressed up as yours. At 100 BB that’s about 2% of flops at 6-max (the less common 3-bet and 4-bet lines) and 0.02% at 4-max. Practice only deals flops that are solved, and “Analyze in Muck GTO” grades on the nearest flop and marks the grade approximate.
4/6-max: everything else Approximate6-max 0.06% · 4-max 0.005% of flopsLines with no solve of their own borrow the heads-up solution for the same pot type, applied to the table’s ranges, and say “Estimate”; if the heads-up seats don’t match, the page says “Not solved yet” instead. At 100 BB that’s 0.06% of flops at 6-max and 0.005% at 4-max saying “Estimate”, and about 0.002% at 6-max saying “Not solved yet” (none at 4-max). Lines the solution never or almost never takes say “Rare line”: they’re solved for the hands most likely to end up there, 0.05% of hands at 6-max and 0.02% at 4-max.
4/6-max flops at 20 and 40 BB ApproximateAbout 4% of flopsAbout 4% of flops (October 8 count), on lines solved on 211 texture flops or a separate set of 71 texture flops rather than every flop: they show the nearest solved flop of the same texture, or a labeled estimate for the rarest 3-bet and 4-bet lines.
4/6-max preflop BetaUp to 0.009 BB per hand from perfectIts flop values come from 10 representative flops per seat pairing instead of heads-up’s 48–184, so it’s noisier: 4-max under 0.0015 BB per hand, 6-max up to 0.009 BB (the hijack). Only preflop is affected, and more flops can be added later without starting over.
Preflop lines nobody takes ApproximateRare by definitionAt 4/6-max, the lines the solution almost never takes (cold 4-bets, say) were filled in afterward with a re-solve that forces a little play down them. It’s approximate, and some of those lines still carry large errors.
Short-stack preflop BetaUp to 0.01 BB per handFewer preflop-and-flop rounds than 100 BB. A little less settled, mostly on hands that are close between two plays.
All-in or Fold at 4/6-max BetaRuns agree within 0.2%Sampling still leaves a thin sliver on a few close hands. Independent runs agree to within 0.2% on average, 0.35% at worst.
Turns that expect ½-pot rivers ApproximateEvery stored turnThe stored flop and turn solves assumed the river bet is ½ pot. The river now offers ⅓, ½, pot and 2× pot, solved live, but the turn play feeding it hasn’t been re-solved with those in mind.
4-bet and limp-3-bet pots, 4/6-max Approximate6-max 0.5% · 4-max 0.06% of flopsPreflop borrows heads-up values for these rare pots.
Other stack depths ApproximateAnalyze only“Analyze in Muck GTO” and Study show the nearest solved depth (20, 40 or 100 BB) and say so.
Prize money ApproximateTournaments onlyChip values only, no ICM. Near a tournament’s pay jumps, survival is worth more than chips and the right play gets tighter.
Multiway pots Not available4/6-max onlyThe solver plays two people after the flop, and the 4/6-max preflop tree never lets a third one in: no overcalls, no limping behind, no three-way all-ins. Every pot is heads-up by the flop. The biggest simplification at 4/6-max.