Paired Boards: KK4, 884, 555
Explains equity compression on paired flops: trips are rare for everyone, most hands are unimproved, and the paired cards act as blockers that shrink strong combos for both players.
Assumptions: 100bb 6-max online cash, no rake unless stated. All examples use the site range file in src/lib/poker/ranges.json as the baseline unless a hand description says otherwise. Quantitative claims are backed by commands recorded in content/mathlogs/postflop-flop/board-textures.md.
Paired Boards: KK4, 884, 555
This lesson is about Explains equity compression on paired flops: trips are rare for everyone, most hands are unimproved, and the paired cards act as blockers that shrink strong combos for both players. Works through A5s as the BTN raiser on KK4r vs BB at 100bb (ace-high is near the top of both ranges), 99 as the CO raiser on 884 two-tone (an overpair to everything that still fears exactly one card class), and pocket pairs on 555r where any pair plays as the effective nuts. Counts trips combos for each range to show why high-paired boards favor the raiser and low-paired boards tilt toward the caller. The quiz should test counting trips combos for given ranges on KK4 versus 884 and identifying which hands are effectively the nuts on 555r. The practical target is not to memorize a solver output for one board. The target is to build a repeatable flop process: identify the texture, identify the preflop ranges, decide who has the equity and nut advantages, then choose a betting or checking line that still makes sense after villain responds. Flop mistakes compound because they create the pot size and range story for the turn. A small error on the flop often becomes an expensive river guess.
The first discipline is naming what changed when the flop landed. A dry ace-high board is not merely a board with an ace; it is a board where the caller has many weak pairs and missed hands, while the raiser keeps premium aces, overpairs under the ace, and strong broadway blockers. A middling connected two-tone board is not merely scary; it is a board where straight draws, pair-plus-draws, sets, and two pairs are distributed differently from the preflop equity chart. The player who opened may still have more raw equity, but the player who called may own the top of the distribution. Those are different advantages, and this track keeps them separate.
For Board Textures, I want four habits in order: classify before betting; separate static boards from dynamic boards; ask which range owns top pair, overpairs, sets, and nutted draws; avoid treating raw equity and nut advantage as the same thing. These habits are deliberately plain because plain decisions survive at the table. If you need a complicated exception before making the normal play, the normal play is probably not stable enough yet. Begin with the texture and range facts, then add hand-specific information such as blockers, backdoor equity, stack-to-pot ratio, and opponent type.
The Core Model
Start with range construction. The preflop raiser normally keeps all six combinations of each premium pair, four suited combinations of each suited broadway, and twelve offsuit combinations of hands like AKo. The mathlog records the repo combo command: AA has 6 combos, AKs has 4, and AKo has 12. That matters because players routinely overcount pretty suited hands and undercount the offsuit broadways that dominate dry high-card boards. A range with AA, KK, QQ, JJ, TT, AKs, AQs, and AKo is 50 live combinations, only 3.8% of all starting hands, but it covers a huge amount of top-pair and overpair strength. By contrast, the middling slice 87s, 76s, 65s, 54s, 88, 77, 66, and 55 is 40 live combinations, 3.0% of starting hands, and it controls many low and connected flops.
Then add board movement. A static flop is one where the current best hand is likely to remain the best hand. K72 rainbow, A83 rainbow, and many paired high boards behave this way. A dynamic flop is one where many turn cards rearrange equity: JT9 two-tone, 876 two-tone, 654 two-tone, and many monotone boards. Static boards reward small, frequent pressure when the bettor owns the range. Dynamic boards punish thin automatic betting because the caller can continue with hands that either already beat you or have enough equity to make your bet indifferent.
The repo odds tool gives the anchor numbers. A nine-out flush draw has 19.15% to hit on the turn and 34.97% by the river. An eight-out open-ender has 17.02% on the turn and 31.45% by the river. A four-out gutshot has 8.51% on the turn and 16.47% by the river. A fifteen-out combo draw has 31.91% on the turn and 54.12% by the river. Those numbers should change how you speak about aggression. A bare gutshot cannot be treated like a monster because it is not one. A combo draw can raise and get called without disaster because more than half the runouts over two cards can rescue or improve it.
Sizing follows from the same arithmetic. A 25% pot bet needs only 20.0% folds to show immediate profit as a pure bluff, and the defender's minimum defense frequency is 80.0%. A 33% pot bet needs 24.8% folds and asks the defender to continue about 75.2%. A 75% pot bet needs 42.9% folds and drops the defender's MDF to 57.1%. This is why small c-bets appear on boards where the bettor can attack widely: the price is cheap, the fold threshold is low, and the value range does not need to be extremely polarized. Large bets require a different promise. They say the bettor has enough strong value and enough high-equity bluffs that the caller cannot continue every pair and gutshot.
Worked Hand 1
In the first hand, hero's exact cards matter less than the interaction between the board and the ranges. If the flop is high, dry, and disconnected, the opener's preflop range retains many of the hands that are hard for the caller to match. If the flop is middling, connected, or heavily suited, the caller's flatting range suddenly contains more two-pair, set, and draw density than the opener wants to admit. The correct flop decision comes from that range story. Betting because hero raised preflop is not analysis; it is a shortcut that fails on the first hostile texture.
A useful question is: what continues against my bet, and what folds? On a small bet, many ace-highs, king-highs with backdoors, low pairs, and gutters may continue because the price is so good. On a large bet, weak pairs and bare overcards are under pressure, but the continuing range is stronger and more coherent. That is the trade. If you use a large bet with a hand that hates being raised and has no turn plan, you bought fold equity at the cost of isolating yourself against the part of villain's range that was never folding. If you use a tiny bet on a wet board with a hand that needs protection, you may charge the draw too little.
Worked Hand 2
The second hand is the version that punishes autopilot. Dynamic boards invite emotional overreactions: players either blast because the board is scary or check because the board is scary. Neither sentence is a strategy. The real question is whether hero owns enough value at the top and enough equity in the bluffs. If hero has strong overpairs, sets, nut draws, and combo draws, pressure is natural. If hero has mostly ace-highs with no backdoors and capped one-pair hands, the check becomes the professional action.
Think about the call that follows a bet. When villain calls a 75% pot flop bet, the call range has already survived a price that required hero to generate 42.9% folds with pure bluffs. That caller is not arriving with the same range that called a 25% pot bet. He has more pairs with redraws, more strong top pairs, more flush draws, more open-enders, and fewer hands that were simply curious. This is why flop sizing determines turn honesty. If the flop bet was large and got called, a weak turn barrel needs a better reason than momentum.
Applying It at the Table
For this lesson, the clean routine is five steps. First, name the board in complete words: high or low, paired or unpaired, rainbow, two-tone, or monotone, connected or disconnected, static or dynamic. Second, name the preflop roles: opener, caller, three-bettor, cold-caller, blind defender, or multiway participant. Third, list the strongest hands each side can credibly hold. Do not say "sets" unless the range actually contains that pocket pair. Do not say "straights" unless the preflop action includes those connectors. Fourth, select a size that matches the distribution. Fifth, decide what happens after a raise or call before chips go in.
That last step is where many intermediate players leak. They can explain why a flop bet is reasonable but have no plan when called. A flop bet without a turn plan is often just a delayed check-fold. A check without a plan is often just a hidden fold. If you check back medium showdown value, know which turns you call and which probes you fold. If you c-bet a backdoor hand, know which turn cards create the second barrel. If you check-raise a draw, know whether you are continuing on bricks, equity cards, and cards that improve villain's range.
Common Errors
The first error is confusing hand strength with range strength. Top pair is a good hand, but top pair on a board where villain has all the sets, two pairs, and combo draws is not permission to build the biggest pot. The second error is treating blocker language as a magic spell. A blocker matters only if it removes hands from the exact continuing range. Holding the ace of a suit on a monotone board can matter a lot; holding an irrelevant low card and calling it a blocker is decorative thinking. The third error is using the same size everywhere. Small range bets and large polar bets are different tools. A player who uses one size for every flop is forcing the game to fit the habit instead of the texture.
The fourth error is overfolding to small bets. When a one-third-pot bet needs only about 24.8% folds to profit immediately, the defender cannot throw away every no-pair hand with backdoor equity. The fifth error is overdefending against large bets with hands that realize poorly. A large bet gives a worse price and creates a stronger turn range. Calling because "I have something" is not enough. You need equity, blockers, implied odds, position, or a clear plan to contest later streets.
The Lesson Rule
The rule for paired boards is simple: make the flop decision from the board and range first, then let the exact hand choose between the available lines. That order keeps you honest. If the board belongs to your range, small pressure can be correct even with hands that missed. If the board belongs to villain's nutted region, checking a hand that looks pretty may be the highest-EV play. If the board is dynamic and both players have strong hands, the hands with equity and blockers earn aggression while the middling region protects the check line.
A final table drill for Paired Boards: KK4, 884, 555: before acting, force yourself to state one value hand, one bluff or semi-bluff, one check, and one hand that folds from each player's range. This takes less than ten seconds once practiced. It prevents the most common intermediate leak, which is building a line for hero's two cards while ignoring the range that villain sees. The answer does not need to be solver-perfect. It needs to be structurally sound: value can get called by worse, bluffs can make better hands fold or retain equity when called, checks contain enough strength not to be bullied, and folds are hands that lack price, blockers, or future. When all four buckets exist, the flop plan is real. When one bucket is empty, the plan is usually an emotion wearing poker vocabulary.