KenKen

Take the free squares

A cage of one square with just a number in it is simply that number. Fill those in first, then clear that digit from the rest of the row and column in your head or in notes — on a small board that alone can settle several squares.

Let the cage type narrow things

Some targets allow only one pair of numbers. On a 6×6 board, a two-square "5−" cage must be 1 and 6, because those are the only digits five apart. A "30×" two-square cage must be 5 and 6. A "3÷" cage on a 6×6 board could be 1 and 3 or 2 and 6, which is two options rather than fifteen. Ask of every cage: which pairs, or triples, hit this target at all?

Multiplication cages are the most revealing, because of prime factors. A cage whose product is a multiple of 5 on a 6×6 board must contain the 5 — there is nowhere else a factor of 5 can come from.

Use the row totals

Every row and column holds each number once, so its total never changes: 10 on a 4×4 board, 15 on a 5×5, 21 on a 6×6. When a row is covered entirely by cages you know the sums of, except for one square, that square is the difference. This "rule of 21" on 6×6 boards finds squares that no single cage would ever give up.

Then treat it like Sudoku

Once cages have narrowed the candidates, the rest is Latin-square logic: a number that can only go in one place in a row goes there, and two squares in a column that can only be 2 or 4 between them use up the 2 and the 4. Pencil the candidates for each cage into its squares and look for these patterns exactly as you would on a Sudoku board.