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How to Solve KenKen: Cages, Arithmetic and Logic
KenKen is a Latin square wrapped in arithmetic — no boxes, only rows, columns and cages — and once you translate a cage target into a short menu of possibilities, the numbers mostly order themselves.
1. Lock in the single-cell cages first
A one-cell cage is a given digit that the board has already filled and locked for you. Treat it exactly like a crossword letter you can trust: it banishes that number from its whole row and its whole column, and it is usually the seed that decides the orientation of a neighbouring cage.
2. Turn two-cell cages into short menus
The quickest wins live in the smallest cages. The two cells of a two-cell cage always sit orthogonally adjacent, so they share a row or a column and can never hold the same digit. On a 4x4 board that makes a sum target of 3 equal to 1+2, a sum of 4 equal to 1+3, and a product target of 6 equal to 2x3 and nothing else.
- Two-cell cages are the only ones that ever ask for subtraction or division; larger cages are always sums or products.
- Write each target as a pair list before you fill anything: a target of 7 on a 4x4 permits only {3,4}, while a difference of 2 allows {1,3} or {2,4}.
3. Use the Latin square to order each pair
A two-cell cage tells you the pair, and the no-repeat rule tells you the order. If the only combination is {1,4} and one of the cells already shares its row with a 4, that cell must hold the 1 while its partner takes the 4. Run the same test against the column and the cage resolves with a single placement.
4. Solve difference and division cages by matching
Minus and quotient targets care only that the larger digit minus the smaller (or divided by it) equals the target, so either cell could be the big one. List the legal pairs first, then let the surrounding rows and columns pick the orientation. A division target of 2 on a 4x4 offers 2÷1 or 4÷2; if one cell shares a line with a 1, that cell cannot be the divisor, and the cage resolves.
5. Budget each line with its magic total
Every row and column sums to the same total, because each holds the numbers 1 up to N exactly once: 10 on a 4x4, 21 on a 6x6, and 45 on a 9x9. When whole cages cover all but one cell of a line, subtract their totals from that magic number and the leftover cell answers. A row whose cages total 7 on a 4x4 must leave exactly 3 in the remaining square.
6. Keep notes that answer to three masters
Switch Notes on and pencil the surviving candidates into each cell, then erase a number the instant a row, a column or a cage kills it. Every candidate must satisfy all three at once — its line, its column and the arithmetic of its cage — so a cell left with a single candidate is a placement, not a gamble. Red highlights flag a digit that repeats in a line or a cage that breaks its target, and a red flash is an invitation to backtrack rather than push onward.
7. Attack the cage with the fewest possibilities
When the board stalls, hunt for the cage whose target still allows the smallest number of combinations and test each option against the lines around it. Easy 4x4 boards fall to this loop in minutes; the 6x6 and 9x9 grids simply demand the same discipline repeated. If a cage still refuses to close, Hint will fill the next correct cell — spend a moment working out why it is forced, because that reason is the technique you came to learn.