Sum Cut Strategy: Row and Column Targets, Free Lines and the Crossings (2026)

1. The puzzle book in numbers: 36 boards, 1,860 cells, 756 cuts

Sum Cut ships 36 frozen boards in three tiers of twelve. Nothing is generated while you play, so the whole library can be counted:

Across the library that is 1,860 cells total: 756 are cut in the stored solutions and 1,104 stay. The cut count is fixed inside each tier (every easy board cuts exactly 7, every medium 19, every hard 37), so difficulty does not come from “how much to remove” varying board to board — it comes from the grid getting larger and the cuts taking a bigger share of it.

The numbers written in the cells are the digits 1–9, spread fairly evenly across the whole book (215 ones down to 188 fives and 188 eights). Row and column targets range from 2 to 34 on easy, 3 to 46 on medium and 2 to 55 on hard.

2. You start with everything kept, so every line begins over (or already done)

The board opens with every number in place (keep is all 1). A move never adds a number; it only cuts one out or puts a cut one back. That means the sum of a line at the start is the sum of all its numbers, which can only be equal to the target or higher than it — never lower.

Measured on the shipped book, the 504 row/column lines (2×5×12 + 2×7×12 + 2×9×12) start like this:

21 lines need zero cuts. Those are the lines whose full sum already equals the printed target: 16 on easy, 4 on medium, 1 on hard. If you cut anything out of such a line, its sum drops below the target and the line turns “under” (too low). The practical consequence is simple: at the very beginning, scan for any line that is already green/exact — every number on that line must stay.

For every other line, the opening question is not “what should I add?” but “how much must I remove?” The amount a line must shed is full sum − target, which ranges from 0 up to 46 across the book (and up to 20 on easy, 38 on medium, 46 on hard).

3. One line, one equation, two constraints: why rows and columns must talk to each other

Each row and each column is an independent equation: pick a subset of its numbers whose sum is the target. Looking at a single line in isolation, that equation often has more than one solution. The puzzle becomes determined because every cell sits on exactly one row and exactly one column, so a choice that satisfies a row can break the column crossing it.

The build’s solutions make the crossing visible. A cell is cut in 7/25, 19/49 or 37/81 positions per board, but the cuts are not spread evenly along lines: in the stored solutions a single line loses anywhere from 0 to 3 cells on easy, 0 to 6 on medium and 0 to 8 on hard (averages 1.40, 2.71 and 4.11 cuts per line). A line that keeps all its numbers is common on easy (16 of 120 lines) and rare on hard (1 of 216).

So the working habit is: use a row to propose, use the column to confirm (or reject). A line tells you “these numbers together hit my target”; the crossing line tells you whether those same cells are allowed to stay or must go.

4. Free anchors: lines that are fully determined on their own

Before you combine anything, some lines already have only one possible subset that hits their target — even when you look at that line alone, with no other information. On the shipped book that is true for:

On every easy board there is at least one such line; on hard boards some have two and some have none. When a line is fully determined, the choice inside it is not a choice at all: those specific cells must be cut and the rest of that line must stay. Locking them in gives you fixed cells on every line they cross, which is what turns a crossing column from “many possibilities” into “one”.

This is why the first pass over a fresh board is a scan, not a guess: find the lines where the numbers leave only one way to reach the target, fill them, and let the crossings shrink the rest.

5. How far deduction goes on its own (measured)

“Find a line with only one valid subset of the cells that are still undecided” is a move you can repeat: after you fix some cells, a crossing line that used to have several valid subsets may now have exactly one. The probe applied that rule iteratively to every shipped board, starting from a completely unknown grid and using only row/column target sums (no guessing, no branching):

Two readings matter:

• Easy is almost entirely a deduction exercise. Eleven boards fall to repeated “this line now has one way” forcing (in 2–4 rounds); the twelfth stalls with 16 cells still open, which is where you read the crossings more carefully or test a possibility.
• Hard gives you almost no free forcing. Most hard boards are still entirely undecided after the first line-subset pass (64–81 of 81 cells), because each line has several combinations that hit its target until crossings eliminate them. On hard, the work is comparing a row against its columns (and back) rather than waiting for a line to announce the answer.

The shipped boards were generated so that exactly one keep-subset satisfies every line at once, and the probe re-proved that uniqueness for all 36. So a hard board is never ambiguous; it just does not hand you the answer one line at a time.

6. A practical loop: shed the excess, then read the crossings

Because the start state is all-kept, a reliable loop for any tier is:

1. Mark the keep-whole lines. Any line already at its target (16/4/1 across the tiers) keeps every number — cutting one would make it under. Treat those cells as fixed.
2. Find a forced line. Look for a row or column where the remaining numbers can hit the target in only one way. Cut and keep exactly those cells. (On easy this often finishes the board; on medium it gives you a foothold.)
3. Re-read every crossing. A cell you just fixed changes the “how much is still needed” count of the other line it belongs to. Re-check that line: it may now be forced, or it may now be impossible if a guess was wrong.
4. When nothing is forced, compare one row to its columns. Pick a line with few plausible cut combinations and see which of those combinations is compatible with the columns it crosses. On hard this is the main move, because whole-line forcing rarely triggers on its own.

The status line helps you keep score without counting by hand: it shows Cut X · Y/Z lines match their target, and each target cell shows current/target until it turns into a check. A line goes “over” when it still holds too much, “done” at the target, and “under” if you cut past the target — under means at least one cell on that line must go back.

7. Hints: what they tell you, and what they cost

The Hint button fixes one wrong cell per press. “Wrong” here means different from the board’s stored solution: if the cell should stay, the hint restores it; if it should be cut, the hint cuts it. The choice is deterministic for a given board state — the engine picks from the current wrong cells with (board id × 7) mod count — so pressing Hint twice on the same position gives the same answer.

Hints are counted. The count shows in the HUD while you play and on the win card at the end. They do not change the rules or the solution, and a hint can be used before you make any other move (it starts the timer the same way a cut does). The practical split: use deduction to learn which crossings force a line, and use a hint when you want to confirm a single contested cell rather than guess-and-undo a chain.

8. Undo, restart and the history you actually get

Undo restores the board to the state before your last action and can be repeated all the way back to the opening position (each action pushes a full snapshot). It undoes a cut, a restore and a hint alike. Restart clears the board, the undo history, the hint count and the timer, and returns every number to “kept”.

There is no move limit and no penalty for undoing. On a board where several cut patterns look plausible along one line, testing a pattern and undoing it is the same information you would get by reasoning it out — the build does not distinguish the two.

9. Controls, URL pinning and what is saved

• Cut / restore: tap or click a number. A cut number stays visible but struck through; tap it again to put it back.
• Targets: the cell at the end of each row and column shows the target and, until it matches, the current sum as current/target. A checkmark means that line is exact.
• Buttons: Undo, Hint, Restart, Easy / Medium / Hard (cycles to the first unsolved board in that tier, or the first board), previous/next puzzle, and Rules.
• Keyboard: U undo, H hint, R restart, N or Right Arrow next puzzle, Left Arrow previous, Escape closes the rules or win card.

Two URL parameters pin the board: ?puzzle= followed by a board number (1–36) loads that exact board, and ?level=easy, ?level=medium or ?level=hard opens that tier. A pinned puzzle is the reproducible case — the same 36 boards, the same targets, the same single solution.

The browser stores two things locally for this game: the best time per board and which boards you have solved. Times, hint counts and progress stay on this device; there is no account and no server record.

10. What makes the tiers different (and what does not)

The rule is identical in all three tiers: every row and every column must keep exactly its target sum. The tier changes only the geometry and the share of the grid you remove:

• Easy (5×5, cut 7/25): more than half the lines on a typical board are either already exact or fully determined once crossings are applied. Eleven of twelve boards yield to repeated line-subset deduction.
• Medium (7×7, cut 19/49): fewer free anchors (1–2 fully determined lines at the start), and most boards still have 20–42 cells open after the first round of forcing. Crossings do the real work.
• Hard (9×9, cut 37/81): almost no line is free at the start (0–2), every line keeps about 4–5 numbers and cuts about 4, and pure forcing leaves 64–81 cells undecided. The solution is still unique, but you reach it by comparing candidates across rows and columns, not by waiting for a line to solve itself.

Because the cut count is constant within a tier, you cannot read difficulty from “how many to remove” on a given board — you read it from how many lines start determined and how quickly crossings collapse the rest.

11. A compact checklist

• Scan once for lines already at the target: every number on them stays.
• On every other line, think in “how much to remove”, not “what to pick” — the board starts over-high, never under.
• Prefer a line that has only one valid cut combination (now, given what is fixed) over a line that has several; a forced line creates fixed cells for its crossings.
• When a line shows under, a cell on it was cut that must stay — undo that cut before chasing others.
• On hard boards, expect to compare a row’s possible cuts against its columns; the answer is unique but rarely handed to you by a single line.
• Use Hint to settle one contested cell, Undo to back out a tested pattern, and Restart only when you want a clean board.

FAQ

Is there more than one correct solution?

No. Each of the 36 shipped boards was built so that exactly one keep-subset satisfies every row and every column target at once, and the probe re-proved that (searching for a second solution found none on all 36). If a line is satisfied but the board is not solved, at least one other line is off.

Why does a line show “under”?

“Under” means the numbers you have kept in that line add up to less than its target. Since the board starts with everything kept, you can only go under by cutting a number that needs to stay. Put that number back (tap it again, or Undo) until the line is at or above the target.

What does a line that is already correct at the start mean?

It means the sum of all numbers in that line already equals the target, so none of them can be cut. In the shipped book this happens on 21 of 504 lines (16 easy, 4 medium, 1 hard). Treat every cell on such a line as fixed from move one.

Do I have to solve lines in order?

No. Rows and columns are independent equations linked only by their shared cells. Solving (or fully determining) any line is useful because it fixes cells for the lines crossing it, but there is no required order and no penalty for working on several lines at once.

What does the Hint button actually do?

It picks one cell whose current state differs from the stored solution and sets it to the correct state — cut if it should be removed, restored if it should stay — then adds one to the hint count. The pick is deterministic for the current board position. It does not solve the board and it does not skip the win check.

Are the numbers always 1–9, and can they repeat?

Yes. Every cell holds a digit from 1 to 9, digits repeat freely, and the same digit can appear cut in one place and kept in another. The value that matters for a line is the sum of the digits you keep, not which specific digits they are except through that sum.

Why does an easy board feel obvious but a hard one does not?

The rule is the same; the geometry is not. Easy boards are 5×5 with 7 cuts and often have lines that are forced on their own or after one crossing (11 of 12 solve by repeated line-subset deduction). Hard boards are 9×9 with 37 cuts, start with almost no forced lines, and line-subset deduction alone leaves most cells undecided — you have to reconcile rows with columns directly.

Is this the same as Sumplete?

The puzzle form — delete numbers so each row and column hits a stated sum — is the same family that the site sumplete.com popularized under the name “Sumplete”. Sum Cut is a separate, Playnesta-original implementation with its own 36-board set, generator, interface and code (MIT); the name “Sumplete” is mentioned here only to identify the puzzle type for readers who know it by that name.

Do ?puzzle= and ?level= change the puzzle?

?puzzle=13 loads board 13 exactly as shipped. ?level=easy, ?level=medium or ?level=hard opens that tier and picks the first unsolved board in it (or the first board if all are solved). They select a board; they do not generate or alter one. The 36 boards and their single solutions are fixed.

What is saved, and is there a move limit?

The game stores your best time and solved flag per board in this browser’s local storage. There is no move counter, no move limit and no account. Undo is unlimited within a session and Restart resets the board but not your saved best times.