KenKen Strategy: Solving the Cages

Updated June 2026

KenKen takes the clean row-and-column logic of a Latin square and bolts on a layer of arithmetic. Each outlined region — a cage — comes with a target number and an operation, and your job is to fill the grid so both the math and the no-repeats rule are satisfied. The trick to solving quickly is to convert every cage into a short list of possible number sets, then let the grid eliminate down to one.

The rules

• Latin square. On an N-by-N grid, each row and each column contains the numbers 1 to N exactly once.
• Cages. Each outlined cage shows a target and an operation. The numbers you write in the cage must combine, using that operation, to make the target.
• Repeats within a cage are allowed — as long as they do not break the row/column rule. Two cells in the same cage may hold the same number only if they are in different rows and different columns.

Free numbers: single-cell cages

Always start with the freebies. A cage made of a single cell has no operation — the target is the answer. Write every single-cell cage in immediately. On many boards these few given digits are enough to start chaining row and column eliminations before you touch any arithmetic.

Turn each cage into a list of combinations

The core skill in KenKen is enumerating the number sets a cage could hold. Do it operation by operation.

Addition and multiplication

For a plus or times cage, list every set of numbers (from 1 to N) that reaches the target. A two-cell "6×" on a 4-grid can only be 2 and 3 — because 1 and 6 is impossible (no 6) and 6×1 needs a 6. A three-cell "7+" might be {1,2,4} only, if {1,3,3} or {2,2,3} would force a repeat that the cage's shape cannot allow. Writing these candidate sets in the margin turns a vague cage into two or three concrete options.

Subtraction and division

Minus and divide cages always have exactly two cells, and the operation is order-independent — you take the result regardless of which cell is larger. A "2−" cage is any pair differing by 2: {1,3}, {2,4}, {3,5}, and so on. A "3÷" cage is any pair where one is three times the other: {1,3} or {2,6}. These cages are wonderfully restrictive because there are usually only a couple of qualifying pairs.

Let the grid trim the list

A cage's combinations are only the starting point — the row and column rules do the rest. Suppose a two-cell cage could be {2,3} or {1,4}, but both of its cells sit in a row that already contains a 1. Then {1,4} is dead, and the cage is {2,3}. Every time you place a number, sweep the cages that share its row or column and cross out any combination that now repeats. This back-and-forth between cage lists and Latin eliminations is the whole engine of the solve.

Use the no-repeat shape of a cage

The geometry of a cage tells you when repeats are impossible. If all the cells of a cage lie in one row or one column, no value can appear twice inside it — so a three-cell "6+" stuck in a single row cannot be {1,1,4} or {2,2,2}; it must be {1,2,3}. Conversely, an L-shaped or blocky cage that spans two rows and two columns may legally repeat a number on its diagonal, which keeps options like {1,3,3} alive. Always read a cage's shape before you finalise its combination list.

A practical order of attack

1. Fill every single-cell cage.
2. For each remaining cage, list all combinations that hit the target, pruned by the cage's shape.
3. Place any cage that has only one possible combination and a forced arrangement.
4. Run row/column eliminations from every number you place, trimming the other cages' lists.
5. Repeat — a single placement often collapses a neighbouring cage to one option.

Because each KenKen board has a unique solution, the lists always narrow to one with pure logic. The satisfying part is watching a long cage with five candidate sets shrink to a single answer purely because of digits placed three cells away.