How to Solve KenKen / MathDoku: A Step-by-Step Strategy Guide

No guessing required. Just a Latin square, some arithmetic, and a few pruning tricks you can apply in order, every time.

Updated July 16, 2026

The two rules that run the whole puzzle

A KenKen (also sold under the name MathDoku) is a grid — usually 4×4, 5×5, or 6×6 — split into a normal Latin square and then chopped into irregular, bold-outlined regions called cages. Every cage carries a small label in its top-left cell, like 6+ or 3÷, telling you the target number and the operation that produces it from the digits inside.

Two rules govern everything you write down:

• Latin square rule: each row and each column contains every digit from 1 to N exactly once (N being the grid size), and no digit repeats within a row or column.
• Cage rule: the digits inside a cage, combined with the cage's operation, must equal the target number. Order inside the cage doesn't matter for addition and multiplication; it does matter for subtraction and division, which is where a lot of the real technique lives.

That's the entire rule set. There's no third rule, no hidden mechanic. Everything below is just leverage — ways of using those two rules together so you're never stuck guessing.

Start with the freebies

Every KenKen grid has at least a few single-cell cages — one box, one number, no arithmetic. If a cage label just says 3 with no symbol, that cell is 3. Fill every single-cell cage first, before you touch anything else. It costs zero deduction and it immediately removes that digit as a candidate from the rest of its row and column.

This matters more than it sounds. On a small 4×4 grid, one freebie cell can eliminate a digit from up to six other cells (three in its row, three in its column). Chain two or three freebies together and you've often narrowed half the grid before you've done a single piece of real solving. Scan the whole puzzle for these before you commit to anything else — it's the cheapest progress you'll make all game.

List the candidate combinations for every cage

For any cage with two or more cells, the next move is to write out every combination of digits (1 through N, no repeats within the cage if the cells share a row or column) that could produce the target. This is the backbone technique — almost every other trick is really just a faster way of narrowing this list.

Take a concrete case: a 3-cell cage summing to 6 on a 4×4 grid, where digits run 1–4. List every unordered set of three distinct digits from {1,2,3,4} that adds to 6:

• 1 + 2 + 3 = 6 — works
• 1 + 2 + 4 = 7 — too high
• 1 + 3 + 4 = 8, 2 + 3 + 4 = 9 — both too high

Only one combination survives: {1, 2, 3}. That's a huge result from one line of arithmetic — you now know the exact set of digits in that cage, even though you don't yet know which cell holds which one. On a 5-cell row, that instantly tells you the fifth cell (the one outside this cage) must be whatever's left — in this case 4, if the cage sits entirely inside one row.

Do this for every multi-cell cage before you try to place individual digits. A 2-cell cage marked 7+ on a 4×4 grid has exactly two possible pairs: {3,4}. A 2-cell cage marked 2× only has {1,2}. Small grids and small targets often collapse to a single combination immediately — write those in as a candidate set and move on to the next cage.

Subtraction and division: only two orderings, ever

Addition and multiplication cages give you a set of digits — you still have to figure out which cell gets which value. Subtraction and division cages are friendlier: with exactly two cells, there are at most two ways to arrange the pair, and often only one actually fits.

Say you have a 2-cell cage labeled 3− in a 4×4 grid. The two digits must differ by exactly 3. On digits 1–4, the only pair that works is {1,4} — 4 − 1 = 3. There are only two ways to place it: 4 in the left cell and 1 in the right, or the reverse. You've cut an initially open-ended arithmetic problem down to a coin flip, and often the row/column rule (see next section) picks the winner for you without any further guessing.

Division cages behave the same way. A 2÷ cage on a 4×4 grid can be {1,2} or {2,4} — both divide evenly to give 2. If {2,4} would put a second 2 in a row that already has one from a solved freebie cage, that pair is dead; you're left with {1,2}. This is why solving order matters: freebies and easy sum/product cages first, then subtraction and division cages, which lean on what you've already ruled out.

Prune with the no-repeat rule, then scan for the only place it fits

Once candidate digits sit in several cells, cross-reference every row and column. If a digit already appears once in a row, it can't appear again anywhere else in that row — strike it from every other cell's candidate list in that row and column. This single mechanical step turns a pile of candidate sets into an actual solution, and it's worth doing every time you place a new number, not just once at the start.

The companion technique is the only place it fits scan, sometimes called a hidden single. Look at one row and ask: after eliminations, which digits from 1–N still need a home, and how many cells could each one go in? If a digit can only legally go in one remaining cell — even if that cell still shows two or three other candidates — it belongs there. Write it in immediately; the other candidates were never going to survive anyway.

Run this scan across rows, then columns, then cages. A cage where three of four candidate combinations have already been eliminated by neighboring rows often collapses to one legal arrangement even though you never solved it directly — the pruning did the work for you.

Parity and sum shortcuts worth memorizing

A few shortcuts save real time once you're used to spotting them:

• Parity on subtraction/division cages: an odd subtraction target means the two digits are one odd and one even. An even target means they share parity — both odd or both even.
• Row/column sum totals: the digits 1 through N always sum to a fixed number — 10 for a 4×4 row, 15 for a 5×5 row, 21 for a 6×6 row. If a row has one cage spanning it almost entirely, subtract the known cage sum from that total to find the missing cell instantly.
• Extreme targets are gifts: a large multiplication target (12× on a 4×4 grid) usually has very few factor pairs in 1–4 — check {3,4} first, since 3×4=12 is often the only option. A tiny sum like 3+ across two cells likewise only has {1,2}.
• Product cages and the digit 1: a multiplication target that seems low for a 3-cell cage is a strong hint that 1 is involved, since 1 doesn't change a product.

None of these replace listing candidates — they just help you spot the short list faster on bigger grids.

Worked example: a 4×4 grid, cage by cage

Here's a real 4×4 grid, digits 1–4, cells named (row, column) from the top-left. Seven cages cover it: 7+ at (1,1)-(1,2); 8+ at (1,3)-(2,3)-(2,4); a freebie 2 at (1,4); 2÷ at (2,1)-(2,2); 24× at (3,1)-(3,2)-(4,1); 1− at (3,3)-(3,4); and 12× at (4,2)-(4,3)-(4,4).

• Freebie. (1,4) = 2. Row 1 and column 4 both lose 2 as a candidate.
• Candidate list, then elimination. With 2 gone, row 1's other three cells hold {1,3,4}. The 7+ cage at (1,1)-(1,2) needs two of those summing to 7 — only 3+4 works — so the leftover, (1,3), must be 1.
• Reuse a solved cell. The 8+ cage runs through (1,3) = 1. So (2,3)+(2,4) = 8 − 1 = 7, and the only pair summing to 7 is {3,4}. Order still open.
• The two-option trick, resolved by the row. Row 2 needs all of 1–4, and (2,3)/(2,4) just claimed {3,4}, leaving {1,2} for (2,1) and (2,2). The 2÷ cage on those cells had two legal pairs in isolation — {1,2} or {2,4} — but row 2 has no 4 left, so it collapses to {1,2}.
• Two more unique combinations. 24× needs three digits multiplying to 24: only {2,3,4}. 12× needs a product of 12: only {1,3,4}. Both cages collapse to one candidate set immediately.
• A cage that eats a whole row. Row 4 is (4,1) plus the entire 12× cage, and that cage's set {1,3,4} already covers three of the row's four digits. So (4,1) = 2, the one digit missing — which also leaves (3,1) and (3,2) holding {3,4} between them, since that's what remains of the 24× cage's set.
• Chain it forward. Column 1 needs {1,3,4} across (1,1), (2,1), (3,1) once (4,1) = 2 is placed. (2,1) can only be 1 or 2 (its cage's set), and the column has no room for a 2 there — so (2,1) = 1, which pins (2,2) = 2.
• Squeeze a column, twice. Column 3 needs {2,3,4} once (1,3) = 1 is placed; (2,3) and (4,3) can each only be 3 or 4, so (3,3) takes the leftover 2 — which also pins (3,4) = 1 in the 1− cage. The same squeeze on column 2 (with (2,2) = 2 placed, (1,2) and (3,2) both confined to {3,4}) pins (4,2) = 1.

Eight of sixteen cells are locked from pure deduction — no guessing, just freebies, candidate lists, and row/column elimination. The remaining eight split into four matched pairs, each narrowed from four digits down to two. That's the forced-pair situation the next section covers.

When you're stuck: forced pairs and the last-resort branch

If the loop above stalls, look for a forced pair: two cells in the same row or column that both have identical two-digit candidate lists (say, both can only be 2 or 3). Between the two of them, those digits are locked — even though you don't know which cell gets which, you know neither digit can appear anywhere else in that row or column. Strike 2 and 3 from every other cell in the line and rescan; this often unlocks the next freebie-style deduction.

Only after freebies, candidate lists, subtraction/division ordering, row/column pruning, and forced pairs are all exhausted should you consider a guess-and-check branch: pick the cell with the fewest remaining candidates, try one, and follow the logic forward until you either complete the grid cleanly or hit a contradiction (a row, column, or cage rule broken), at which point you back up and try the other candidate. On well-formed 4×4 and 5×5 puzzles this is rarely necessary — it becomes more common on 6×6 grids with longer cages.

The fastest way to internalize this loop is to run it against a live board instead of a printed one. MathDoku (KenKen) on Scotix generates fresh 4×4 and 5×5 grids each round and includes a built-in cage helper that lists valid digit combinations for whichever cell you select — useful for checking your own candidate list against the engine's while you're still building the habit. If you'd rather warm up on plain arithmetic before layering in the Latin-square constraint, CrossMath Grid is a good lower-friction starting point.

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