A Modular book punishes a bad file in a way no sample puzzle ever shows. Your buyer judges puzzle forty-one, when a line your key swears is sound runs, in three sequential cells, a 1, a 2 and a 4 — a window a classic checker waves through without a murmur, because 1, 2 and 4 are each present in their houses exactly once, and that your book's one extra law forbids absolutely, because 1 and 4 belong to the same residue group, so the window holds two digits from {1, 4, 7}, one from {2, 5, 8} and none at all from {3, 6, 9} — or when a puzzle your generator proved unique while treating each line as decoration turns out to carry a second solution your buyer found by rotating residues along a line whose cycle your checker never fixed, a solution your rules page declares impossible and your solver never looked for — or when your lines wander across so many bends, printed so faintly, that the buyer cannot tell which three cells are sequential on the line at all, so the extra law reads, from the buyer's chair, like a rumour about a book they did not buy. Modular Sudoku, in the form this book sells, is a classic frame with a strict rotation of remainders: a 9 by 9 grid, digits 1 to 9 once in every row, column and outlined 3 by 3 box, crossed by lines, and on every line every window of three sequential cells holds exactly one digit from {1, 4, 7}, exactly one from {2, 5, 8} and exactly one from {3, 6, 9}. Underneath it is a contract in five parts, each part exact. First, the row rule. Second, the column rule. Third, the box rule. Fourth, the modular rule: split the digits into three residue groups — {1, 4, 7}, {2, 5, 8} and {3, 6, 9} — and every three sequential cells along a line, counted from either end and sliding one cell at a time, contain exactly one digit from each group. Fifth, closing the contract, given integrity with uniqueness: the givens are fixed exactly as published, and exactly one completed grid satisfies every part at once, reachable by deduction alone, with no guessing. A puzzle that breaks any part is not slightly harder. It is false as stated. A good Modular book is a verified stack of windows: every window of every line re-read, on the final file, puzzle by puzzle.
Four warnings belong at the front, because they are the mistakes this genre makes most often, and all four start with puzzles Modular is not. Modular is not Entropy: an entropic line also works in sliding windows of three, and also splits the digits into three groups of three — but its groups are the size bands 1 to 3, 4 to 6 and 7 to 9, so a window holding 1, 4 and 7 is a perfect modular window and a broken entropic one, a window holding 1, 2 and 3 is a perfect modular window and an all-low entropic failure, and a generator, a checker or a rules page that confuses the two partitions produces a book that is wrong on nearly every line while passing its own audit. Modular is not German Whisper: a whisper line constrains neighbours, demanding that adjacent digits differ by at least 5, and says nothing about windows of three — a line can whisper perfectly and still run two digits from the same residue group in one window. Modular is not Renban, though the two are cousins close enough to fool a tired checker: any three consecutive digits automatically cover all three residues, so a renban line passes every modular window test by accident — but a modular line may repeat digits and need not be consecutive anywhere, so a checker that tests for renban is testing a stricter law than the book promises, and a rules page that teaches renban teaches the buyer to reject legal modular lines. And modular is not a promise about any three cells a buyer picks out: it binds sequential cells only, in windows that slide one step at a time, and it allows digits to repeat on a line — the same digit may stand three cells later, in the next cell of its own residue, wherever the classic rules permit it. A buyer who has been taught any of these four impostors by a careless rules page will solve your book under the wrong law and write the review your book did not earn.
The book is every window on every line, not the sample puzzle
A sample puzzle proves almost nothing about a Modular book, because the law lives in the windows, and a line carries far more windows than it carries givens. A line of seven cells holds five windows of three; a line of nine holds seven; and every one of those windows is a separate promise, sliding one cell at a time, so that a single misplaced digit can break two windows at once — the window ending on it and the window beginning on it — while leaving every house in the grid perfectly legal. That is the shape of the buyer's experience: they do not solve your sample, they solve your fortieth line at midnight, pencil in hand, checking a window your file may never have checked. When the window fails, the buyer does not think the puzzle is hard. They think the book is careless, and they are right, because the window is not an edge case of the product. The window is the product, repeated hundreds of times per book.
So the unit of quality in this genre is not the puzzle and it is certainly not the listing. It is the window, counted and verified: every window, on every line, in every puzzle, against the solved key and against the printed page. Everything below — the contract, the audit, the difficulty scale, the print rules — is machinery for keeping that one promise at book scale, because a book is where modular arithmetic stops being a clever constraint and starts being a manufacturing problem: hundreds of lines, thousands of windows, one standard.
The contract, part by part
Part one: the row rule. Every row of the 9 by 9 grid holds the digits 1 to 9 exactly once. No modular line, however elegant its windows, excuses a doubled digit in a row, and no buyer forgives it: the classic frame is the floor the whole book stands on.
Part two: the column rule. Every column holds the digits 1 to 9 exactly once. Lines cross columns constantly — a line running down a column shares its cells with that column, and a window that satisfies the residue law can still be impossible because the column already holds one of its digits elsewhere. The two laws are audited separately because they fail separately.
Part three: the box rule. Every outlined 3 by 3 box holds the digits 1 to 9 exactly once. A line that threads three boxes carries its windows across all three, and the box rule is what stops a line's residue cycle from being filled in by digits the box cannot accept.
Part four: the modular rule. This is the part the book exists for, and it must be stated so exactly that a buyer can audit it with a pencil. Divide the digits into three groups by their remainder when divided by three: {1, 4, 7} is the first group, {2, 5, 8} is the second, and {3, 6, 9} is the third. The membership is the whole law, so the rules page prints it without abbreviation, and prints the two traps by name: digits three apart are the same group, so 1 and 4 can never share a window, and 3, 6 and 9 are one group together — not a high band, however natural that misreading feels to a buyer arriving from entropic books. Then: on every line, take any three sequential cells — the first, second and third cells; or the second, third and fourth; sliding one cell at a time to the end of the line — and those three cells hold exactly one digit from each group, in some order. Three consequences follow, and an honest book states all three because solvers rely on them. First, no digit can repeat within two steps along a line, and neither can any digit from its group: a digit standing in a window bans its whole group from the next two cells, because those cells belong to the other two groups. Second, the groups along a line fall into a fixed repeating cycle: once two neighbouring cells fix their groups, every other cell's group is forced, three cells in the pattern repeating the length of the line — one of the six orders of the three groups, but one order only, from one end of the line to the other. Third, repetition is real but rationed: a digit may appear again on the same line exactly three, six or nine cells later, in a cell of its own group, and nowhere else, and only where the row, column and box rules allow it. A generator that forbids all repetition on a line is writing a stricter book than the rules page promises; a generator that allows two digits of one group inside a window is writing a false one. Both errors ship silently unless the audit looks for them by name.
Part five: given integrity with uniqueness. The givens printed on the puzzle page are fixed exactly as published — no given moves, appears or vanishes between the puzzle page and the key — and exactly one completed grid satisfies all five parts at once, reachable by deduction alone. Uniqueness in this genre is a claim about the modular solver, not the classic one: a puzzle can be unique under classic rules and still be a bad Modular puzzle if its windows do no work, and far worse, a puzzle can look unique to a classic checker while a second grid — identical in every house, different along a line whose residues rotate into the other cycle — satisfies the buyer's reading of the rules. The contract closes only when a solver that enforces the windows, all of them, finds exactly one grid.
Verify like a buyer: the eight-step audit
Buyers audit your book the slow way, one window at a time, and they find what your pipeline missed in the order they meet it. Audit the way they solve. These eight steps run on the final exported file — not on the generator's internal state, which has a way of remembering a cleaner book than the one you shipped — for every puzzle, with a written record per puzzle.
Step one: the line inventory. List every line in the puzzle: its cells, in order, from one end to the other, and its length. A line of length n carries n minus 2 windows; write that number down. If the inventory's window total does not match the audit's window total at the end, a line was skipped, and a skipped line is where broken windows live.
Step two: the residue map of the givens. For every given that stands on a line, write its group beside it — A, B or C — and double-check the pairs three apart, 1 with 4, 2 with 5, 3 with 6, 4 with 7, 5 with 8 and 6 with 9, one at a time. A given filed in the wrong group poisons every deduction the buyer makes from it; a 4 filed as a second-group digit in your production notes will pass windows it should fail and fail windows it should pass, in your audit and in the buyer's pencil alike.
Step three: the window walk on the key. On the solved grid, walk every line from one end to the other and test every window: one digit from {1, 4, 7}, one from {2, 5, 8}, one from {3, 6, 9}, exactly. Count windows tested against the inventory from step one. This is the step classic checkers skip entirely, and it is the only step that can catch a key that is classically perfect and modularly broken — the 1, 2, 4 window from puzzle forty-one passes every house count ever written and fails here, in seconds, if anyone walks it.
Step four: the cycle check. For each line, read the groups in the key from one end to the other and confirm they repeat in a single cycle of period three: the group in the fourth cell matches the first, the fifth matches the second, the whole line one repeating order. The cycle check catches what the window walk can miss in a tired hour — a line whose windows each pass locally while its group order quietly changes halfway, which cannot happen in a correct key and therefore proves a transcription error between solver and page.
Step five: the house recount. Count every row, column and box in the key: digits 1 to 9, once each, no omissions, no doubles. The modular audit never replaces the classic one; it stands beside it, because a key can satisfy every window on every line and still double a digit in a box the lines never cross.
Step six: the reverse recount. Walk every line backwards, from the last cell to the first, and recount the windows in that direction. The windows are the same windows, and that is the point: a forward-only walk trains the eye to see the window it expects, and the reverse pass reads digits instead of expectations. Forward and reverse totals must agree on every house and every window before the puzzle moves on.
Step seven: the uniqueness solve, with the law switched on. Run a solver that enforces all five parts — rows, columns, boxes, and every window of every line — and require exactly one solution, reachable without guessing. Then run the control every honest Modular book needs: the same puzzle through a classic-only solver. If the classic solver also finds exactly one solution and it is the same grid, your lines may be decoration on that puzzle, which is a difficulty problem, not an honesty one. If the modular solver finds a second grid, the puzzle fails the contract outright, however clean its houses look.
Step eight: the print proof. Print the puzzle page and the key page at final size, on the paper your buyers use, and trace every line with a finger: is the sequence of cells unambiguous where the line bends? Can the buyer tell, at print size, which cell follows which, and where the line ends? A line whose order cannot be followed cannot be solved under the law, because the law is defined on sequence — and a puzzle that cannot be followed as printed is broken as printed, whatever the file believes.
Difficulty is cycle work, not given counts
Every lazy Modular listing grades its puzzles the same way: count the givens, call the sparse ones hard. In this genre that number is close to meaningless, and buyers who have solved three honest books know it. A puzzle with thirty givens can be brutal if none of them stands on a line, so the residue cycles are nowhere fixed and every window is a three-way guess. A puzzle with twenty-four givens can melt in minutes if the givens on its lines pin every cycle early, after which the group of every lined cell is known and the solving reduces to placing digits inside groups — closer to a classic puzzle wearing a modular coat. The difficulty of a Modular puzzle is the work of fixing its cycles and spending them, and an honest grade measures that work, not the ink on the page.
Grade on three levers, in this order. First, the cycle-decision share: what fraction of the solve is spent determining residue cycles — choosing among the six orders on a line, propagating a fixed cycle across a bend, discovering that a house forces a lined cell's group — against the fraction spent on ordinary classic placement once the groups are known. A book whose hard puzzles are hard because their cycles stay open longest is telling the truth; a book whose hard puzzles are merely classic-hard with lines drawn on is selling the variant and delivering the original. Second, cycle leverage: how much a single fixed cycle buys. A line whose cycle, once fixed, groups nine cells across three boxes is a lever that moves the whole grid; a short line in one box barely moves its neighbourhood. Long lines with early-fixable cycles make generous easy puzzles; long lines whose cycles stay ambiguous deep into the solve make honest hard ones. Third, and weakest, classic depth: the ordinary singles, pairs and intersections the solve needs once every group is known. Classic depth counts, but it counts last, because it is the one lever the buyer could have got from a book with no lines at all.
Then say the levers on the listing. Name the share of each grade that turns on cycle work, say whether the hard section's difficulty is open cycles or classic depth, and never let a given count stand alone as a difficulty claim. The buyer who reads that listing and then solves the book will find the same book the listing described — and in a genre where the rules page, the key and the listing are three separate chances to be caught out, that agreement is the entire reputation.
Print honesty: lines a buyer can actually follow
The modular law is a law about sequence, so print is not decoration in this genre; print is where the law either exists or does not. A window is three sequential cells. If the buyer cannot tell which cells are sequential — because the line bends without a visible joint, crosses another line without a visible crossing, or fades to a grey the home printer renders as a rumour — then the buyer is not solving your puzzle badly. They are solving a different, undefined puzzle, printed by you.
Four print rules carry the genre. Line weight first: every line prints at a weight that survives a home printer at final size, tested on paper, not admired on a retina screen — a line that vanishes between two cells breaks the sequence exactly where a window needs it. Bends second: where a line turns, the turn must be drawn as a turn, through the centres of the cells it joins, so the order of cells along the line can be traced with a finger without guessing at corners; a line drawn as a smooth smear across a corner asks the buyer to invent the sequence, and two buyers will invent it differently. Endpoints third: the first and last cells of every line must be identifiable — direction does not matter to the law, since a window reads the same backwards, but membership does, and a cell the buyer cannot assign to the line is a cell whose group the law cannot reach. Crossings fourth: where two lines cross in a cell, both lines pass through that cell and the drawing must say so; where lines merely pass near each other, the drawing must not marry them. Crossed wires in print become crossed deductions in pencil.
And the key must print the identical lines. Puzzle page and key page are generated from the same line inventory — the audit's step one — never redrawn, never re-exported from a second source that might carry an older geometry. A key whose lines sit one cell off from the puzzle's lines will show windows the buyer never faced and hide the ones they did, and no errata note beginning "the puzzles themselves are correct" survives that discovery. Identical geometry, identical givens, identical cycles: the key is the puzzle, completed, and print honesty is the discipline of never letting the two drift apart.
Licensing, provenance, and whose puzzles these are
A Modular book is a generated object, and generation is exactly why provenance needs paperwork. Every puzzle in the book should trace to a record: the generator and its settings, the seed or source grid, the date, the audit record from the eight steps, and the licence under which any bought or borrowed puzzle entered the book. Variant sudoku has a living community of setters whose lines appear in packs, apps and streams; a puzzle lifted from a community pack and re-skinned with your cover is not made yours by the lift, and the community is small enough — and keeps receipts well enough — that provenance is not a courtesy. It is a survival trait.
Keep three records per puzzle and the book can defend itself. The generation record: how this grid and these lines were produced, by what tool, under what constraints, including whether uniqueness was proven with the windows enforced. The rights record: written licence or your own authorship, in a form you could show a marketplace without embarrassment. The audit record: the eight steps, dated, against the final file. When a marketplace asks where a book came from — and marketplaces increasingly ask — the seller with records answers in minutes, and the seller without them discovers that "my generator made it" is a claim, not a record.
Corrected files, errata, and the delivery rule. Every puzzle book ships with a mistake eventually; the genre only decides whether the mistake becomes a one-star review or a five-star story about a seller who fixed it. The rule is simple and it is not negotiable: when any window in any puzzle is wrong — a window holding two digits of one residue, a cycle that changes halfway down a line, a key that breaks the group law — every buyer gets the corrected file, not just the buyer who complained, and the correction goes out through the same delivery that sold the book, so it lands in the same library, the same downloads page, the same inbox thread the original came from. This is where delivery stops being plumbing and becomes part of the product: a download tool that versions your file and re-delivers the corrected edition to past buyers turns an errata notice from a hope into a fact, and it is why I deliver my own puzzle books through FileFreight — one flat monthly price for download delivery and license keys, the corrected file replaces the old one for every buyer at once, and nobody has to write in and ask. Pair the corrected file with a plain errata note inside it: puzzle number, what was wrong, what changed, in words a buyer can check against their printout. Never silently swap a file a buyer may already have printed and half-solved; a silent fix gaslights the exact buyer who found your error, and they are the buyer your next review is made of. Keep the old version archived, keep the note dated, and let the record show the book getting truer over time.
The launch checklist
Before the listing goes live, walk this list in order and do not skip backward. One: the rules page declares the law — 9 by 9 grid, digits 1 to 9, the three residue groups printed in full, every three sequential cells on a line holding one digit from each group, sliding one cell at a time — and names the neighbouring variants, entropic lines above all, only to exclude them. Two: every puzzle has passed the eight-step audit on the final exported file, with a written per-puzzle record. Three: every window of every line in every key holds one digit from each residue group, verified window by window, line by line. Four: forward and reverse recounts agree on every house and every window, and every line's residue cycle repeats with period three. Five: difficulty is graded from real solves, and the listing names the levers used. Six: puzzle pages and keys present the identical lines, and every line's sequence survives printing at final size. Seven: all puzzles are your own generation or licensed, records kept. A book that can say all seven, truthfully, has earned its listing copy — and the copy should then say the plainest true thing about it: these windows were re-read, one by one, twice.
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