The Narrowing Algorithm
Teaches the formal five-step procedure: assign a position-based preflop range, then at every subsequent action node remove or down-weight the combos inconsistent with that action, never adding combos back. Walks the full...
Assumptions: 100bb 6-max online cash, ranges per the site's canonical charts unless stated. Quantitative claims use repo tool outputs logged in content/mathlogs/hand-reading-review for this lesson. Population and solver-facing statements are practical heuristics unless tied to logged arithmetic.
The Narrowing Algorithm
Teaches the formal five-step procedure: assign a position-based preflop range, then at every subsequent action node remove or down-weight the combos inconsistent with that action, never adding combos back. Walks the full algorithm through one complete hand: MP opens 2.5x at 100bb, BTN calls, MP c-bets 33% on Ts7s2d, BTN calls, MP checks the 4h turn, BTN bets 66%, MP calls, river Kd checks through - students rebuild BTN's surviving range at all five nodes with explicit before/after combo grids. Covers the two cardinal rules (ranges only shrink; every action is evidence) and the common procedural mistake of re-widening a range when a scary card arrives. The quiz presents a four-node action sequence and asks students to identify which combos are eliminated at each specific node and which node eliminates the most combos.
The purpose of this lesson is to make The Narrowing Algorithm usable during real review, not merely recognizable as vocabulary. Expert hand reading starts with a written distribution and keeps that distribution honest as the hand develops. The bad version of review starts with the result, names the hand Villain showed, and then retrofits every street until the story sounds inevitable. The good version starts with position, stack depth, player type, and the first action. It names all plausible hand classes before caring which card happened to roll off. That habit is slower at first, but it prevents the two most expensive review leaks: false certainty and outcome worship.
The working unit is the combo. A range is not a mood, a sentence, or a single favorite hand. It is a weighted set of combinations that entered the line and survived the filters. The canonical range file gives stable anchors: the local TAG baseline opens about 17% from UTG, 26.1% from the cutoff, 42.1% from the button, and 36.3% from the small blind. Those numbers are not a claim that every opponent copies the chart. They are the default for unknown regulars, and the reviewer's job is to widen, tighten, cap, or polarize that default only when the action and evidence justify it. A 52/10 recreational caller should not receive the same preflop range as a 21/18 regular, and a nit's 3-bet should not be treated like a balanced small blind attack.
Use the narrowing algorithm in a fixed order. First assign the starting range from position, stack depth, sizing, and player type. Second, after every action, remove combos that are genuinely inconsistent with the action. Third, down-weight combos that remain possible but less common. Fourth, classify the surviving range into value, showdown value, draws, and air or bluff candidates. Fifth, make the decision against that distribution, not against the one hand that scares or tempts you. The rule is simple: ranges shrink or get reweighted; they do not magically widen because a later card makes a dramatic story possible.
The Narrowing Algorithm lives inside the module Range Narrowing Fundamentals. That matters because the lesson is not isolated advice. In this module, the student is practicing range narrowing, so every example should end with a range statement that another reviewer could audit. A useful statement sounds like this: Villain reaches the river with sets at full weight, top pair at partial weight, missed flush draws that barreled turn, and very little no-equity air because the flop call removed it. A weak statement sounds like this: Villain probably has it. The first statement can be checked with combo counts and action logic. The second statement is just fear wearing a poker vocabulary costume.
Sizing is one of the cleanest filters because it changes incentives immediately. The logged odds tool gives the core bluff thresholds: a half-pot bluff needs 33.3% folds, a 75% pot bluff needs 42.9% folds, and a pot-sized bluff needs 50.0% folds. Those values do not tell you whether the opponent will fold. They tell you what must be true for the bet to print as a pure bluff. In review, connect the sizing to the range. Small flop bets in single-raised pots often leave ranges wide. Large turn barrels compress the continuing range and remove casual floats. River overbets tend to polarize because medium-strength hands dislike risking so much when called only by better. Against recreational players, however, sizing may track hand strength more directly, so apply the population read before importing regular logic.
Pot odds create the same discipline for calls. A $10 call into a $30 pot requires 25.0% equity. That number is easy, but the hard part is building the range accurately enough that the equity estimate means something. If you give Villain every missed draw because you want to call, the number lies. If you delete all bluffs because you hate variance, the number lies in the other direction. The reviewer must ask which draws actually arrive after the preflop line, which draws bet this size, which value hands choose the same size, and whether blockers shift the ratio. The goal is not mathematical decoration. It is a clean answer to whether the price is being offered by a range you can beat often enough.
Combo bookkeeping keeps the review from becoming emotional. Premium pressure ranges can be very compact: QQ plus and AK is 34 starting combinations before blockers, only 2.6% of all 1326 starting hands. A suited connector and suited broadway bluff package such as A5s through A2s, KTs plus, QTs plus, JTs, T9s, and 98s is 48 combinations before blockers. On a specific board with dead cards, the example draw-and-value package logged in the math record falls from 42 raw combinations to 31 live combinations. Those counts are not universal ranges. They are anchors showing why blockers, board cards, and previous actions matter. Three set combos instead of six can change a river bluff-catch from close to bad.
Worked hand 1
In the first hand, the key move is to write the range before writing the verdict. Start from the preflop actor's chart or read-based range, then mark which hands interact with Ks 8d 3c 4h Qd. Value is not only the nuts. It includes hands that can bet and be called by worse at the chosen size. Draws are not only flush draws. They include open-enders, gutters with overcards, pair-plus-draws, backdoor draws that continued earlier, and hands that became blockers by the river. Air is not every missed hand in the deck; it is the subset that plausibly took the prior streets.
A disciplined review of this hand would list at least four classes. Strong value might be sets, two pair, overpairs, or top pair with a dominant kicker depending on the line. Medium showdown value might be second pair, pocket pairs between board cards, or ace-high hands that checked for pot control. Semi-bluffs might be suited broadways, combo draws, or straight draws with live overcards. Pure bluffs should be rare unless the earlier action and player type support them. Once the classes are written, the final action becomes less personal. Hero is not asking whether Villain found the one hand that wins. Hero is asking whether the whole surviving distribution makes calling, betting, raising, or folding profitable.
The important correction is to avoid re-expanding the range on a dramatic card. If the flop action removed offsuit junk and the turn action removed weak pairs, the river queen does not put every queen-x hand back into the range. It only improves queen-x combinations that actually arrived there. This is where many reviews fail. The river card is vivid, the result is painful, and the student rewrites the earlier streets to explain it. The narrowing algorithm forces the opposite sequence: earlier evidence first, new card second, final decision third.
Worked hand 2
The second hand tests whether the read can survive pressure. Hero has blockers and a texture that can support aggression, but blockers are not permission slips. A blocker matters only when it removes meaningful value combos or increases the chance that Villain has a bluff-catching hand. Holding a spade on a spade draw board may reduce some flushes later, but it does not erase sets, two pair, or stubborn top pair. Holding an ace can reduce AK and AQ, but against a loose-passive player who under-bluffs rivers, that blocker might not create enough folds. Review the blocker inside the whole range, not as a slogan.
The same is true for timing and game flow. A snap-call can deny hands that needed a difficult raise decision, and a long tank can signal an uncomfortable polar action, but timing is secondary evidence. Action and sizing come first, range construction comes second, timing modifies the weights after that. If a passive opponent check-raises a blank river after calling twice, the strong action should dominate one weak timing read. If a regular has shown repeated fast check-backs with medium hands, then a snap-check can become useful evidence, but only because it repeats across relevant spots.
Street-by-street application
Preflop review asks who owns the uncapped hands. An opener can hold AA, KK, QQ, and AK unless later action removes them. A caller often has a capped, condensed, pair-heavy or suited-heavy range because many premiums would have 3-bet or 4-bet. In a 3-bet pot, the caller can still be strong, but the exact cap depends on whether the pool flats premiums. Tournament hands add stack depth, antes, payout pressure, and bounty incentives before any postflop hand reading is graded. Cash-game hands focus more directly on chip EV, rake, position, and opponent tendencies.
Flop review asks which player has range advantage and nut advantage. On dry ace-high boards, the preflop raiser often keeps enough strong ace-x that a small bet removes little. On low connected boards, the blind defender or caller may own more two pair and straight interaction, so a large c-bet is a stronger filter. On monotone boards, many hands have equity but fewer hands want to play for stacks. A raise on the flop is usually a sharper filter than a call, especially in low and mid-stakes pools where bluff raises are underused on many textures. Still, do not delete all bluffs. Count the plausible draws and backdoors before deciding how value-heavy the raise is.
Turn review is where ranges become much clearer. A blank turn that faces a second large barrel removes a lot of flop curiosity. A card that completes obvious draws changes which value hands can bet confidently and which one-pair hands shift into bluff-catchers. A check after betting flop can cap the aggressor or induce from a trap-heavy range depending on position, player type, and board. Your job is to write what the check means, not merely whether you liked seeing it. A turn probe into a checked-back range is profitable only when the check-back actually capped the opponent enough and when your own range has enough credible value.
River review is the discipline test. By the river, the pot is large enough that small mistakes are expensive and emotional stories are tempting. Before calling a big river bet, list the value combos first, then the bluffs, then the blockers. Before bluffing, name the hands you expect to fold and compare the required fold frequency to the sizing threshold. Before thin value betting, name the worse hands that call and the better hands that raise. If a passive player raises river, respect how few bluff combos their line contains unless you have direct evidence otherwise. If an aggressive regular overbluffs a missed-draw node, call wider but document why.
Review workflow
The same process can be run in this site's analyzer after a student supplies a hand history. Paste or import the hand, confirm the parser's positions, stacks, pot sizes, and action timeline, then compare the analyzer's baseline ranges to the written range you built by hand. If the history is unavailable, do not invent the user's cards, pool sample, or showdown. Use the constructed examples here as drills, and use the analyzer only on actual pasted or uploaded hands. The tool is strongest when it checks your range reconstruction, equity movement, and node selection; it is not a replacement for writing the reasoning before the result is known.
Common errors
The first error is collapsing the range too early. Students often decide on the flop that Villain has a set, then spend the rest of the hand confirming that fear. The correction is to keep at least three hand classes alive until the action truly removes them. The second error is monotone reading: assigning only value to a line when missed draws exist, or only bluffs to a scary overbet when the value range is obvious. The third error is hero-bias. If the only assigned range makes Hero's preferred action correct, rebuild the range as if you held a different hand. The fourth error is result-first review. A won hand can still be badly played, and a lost hand can still be clean.
The final standard for The Narrowing Algorithm is auditability. Another strong player should be able to read your notes and see why each class was included, excluded, or weighted. A good note says: cutoff baseline opened 26.1%, big blind defended wide, flop call kept pairs and backdoor clubs, turn large bet removed most ace-high floats, river jam represented full houses plus missed clubs, and Hero's blocker reduced the nut flush but not boats. A bad note says: weird line, probably bluff. The first note can improve a future decision. The second note only records confusion.
Drill
Review one hand using this exact order. Write the preflop range. Write the flop survivors. Write the turn survivors. Write the river value and bluff candidates. Then, and only then, check the result or analyzer output. If the analyzer disagrees with your assignment, decide whether the tool used a baseline you should override or whether your written range was biased. Record one correction in a fix log. The correction should be specific enough to use next session: call less versus passive river raises, bluff more when button range is capped after check-back, widen value versus loose-passive callers, or stop adding missed draws that folded earlier. That is how hand reading becomes a repeatable review engine instead of a memory of the last painful pot.
Quantitative anchors
The mathlog for this lesson records the exact local commands behind the numeric claims used here. The reusable anchors are 33.3% folds for a half-pot pure bluff, 42.9% for a 75% pot pure bluff, 50.0% for a pot-sized pure bluff, 25.0% equity for calling $10 into $30, 34 combos for QQ plus and AK, 48 combos for a suited bluff package, and the checked chart sizes from ranges.json. Treat them as guardrails. The real work is connecting those numbers to the range that actually reached the node.