How to play

Rule one: each number once per row and column

A 4×4 grid uses the numbers 1, 2, 3 and 4. Each row holds all four, and so does each column. A 6×6 uses 1 to 6, a 9×9 uses 1 to 9. If you have solved a sudoku, this part is already familiar — the difference is that there are no boxes, only cages.

That one rule does a lot of work. The moment you know three cells of a row, the fourth is forced, and the same is true down every column. Most of the solving is this rule and the cages taking turns to narrow each other.

Rule two: the cage math

The heavy outlines divide the grid into cages. Each one is labelled with a target and an operator, printed in its top-left cell — 12× means the numbers in that cage multiply to 12; 3− means they subtract to 3.

• Addition and multiplication work in any order, however many cells the cage has.
• Subtraction and division apply to two cells, larger minus smaller. A cage marked 3− could be 4 and 1, and equally 1 and 4 — the order on the grid does not matter.
• A single-cell cage is a gift: its target is the number, with nothing to work out.

A cage may repeat a number, as long as rule one still holds. A six-cell L-shaped cage can hold two 3s if they sit in different rows and different columns. This catches people out more than anything else in the game.

Rule three: one solution, and no guessing

Every grid has exactly one answer, and every step toward it can be justified. If you find yourself picking one of two options to see what happens, there is a deduction available that you have not spotted yet.

That is a promise about the puzzles, not a claim about you. It is also what makes a stuck grid worth staring at rather than restarting: the information is on the board.

A 4×4, solved from scratch

Here is a small grid with its cages written out. Numbers 1 to 4, each once per row and column.

• Top-left two cells, side by side: 3+ (they add to 3).
• Top-right single cell: 4 (a given).
• The cell below that 4, on its own: 2 (another given).

Start with the cage that can be least: 3+ across two cells. The only pair of distinct numbers adding to 3 is 1 and 2, so those two cells hold a 1 and a 2 in some order — you do not know which yet, and you do not need to.

Now use the givens. The top row already contains a 4, and two of its remaining cells are the 1 and the 2. Everything in that row is accounted for except one cell, and the only number left is 3. That cell is a 3, with no guessing involved.

Look down the right-hand column: it holds the 4 and the 2 from the givens, leaving 1 and 3 for the two cells below. Whichever row already carries a 1 elsewhere settles which is which, and the grid unwinds from there.

That is the whole rhythm of the game: find the cage with the fewest possibilities, write down what it forces, then let rule one carry the consequences across the row and down the column.

Where beginners get stuck

• Assuming a cage cannot repeat a number. It can, as long as the repeats are in different rows and columns.
• Reading a subtraction cage as ordered. 3− is satisfied by 4 and 1 in either arrangement.
• Chasing the largest cage first. A five-cell cage has far too many combinations to be useful early; the two-cell cages and the single-cell givens are where a grid opens up.
• Forgetting the columns. Most stuck grids come unstuck by reading down rather than across.

What it is called

Puzzles of this shape are usually called calcudoku, and sometimes mathdoku or cage math. They were popularised by a Japanese mathematics teacher in the mid-2000s as a way of practising arithmetic without drilling it, and the family has grown several commercial names since.

Cages plays the same way: fill the rows and columns, satisfy the cages, one solution, no guessing.