Calcudoku: 4x4

A Latin square with small sums attached

Four by four. Digits 1 to 4, once in each row and once in each column, and the grid divided into bold-outlined cages that each carry a target and an operation.

The 4×4 is the friendliest member of the family by some distance. The arithmetic never gets large, the Latin-square rule takes about thirty seconds to explain, and a practised solver finishes one in under a minute. It is the right first puzzle for a child learning their number bonds and for an adult coming back to logic puzzles after a decade away.

Up to six to a page, four by default.

The rules

• Every row holds 1, 2, 3 and 4, once each.
• Every column does the same.
• Each cage's digits must combine, using the operation printed on it, to make the target.
• A single-cell cage is simply that digit. No working required.

Addition and subtraction only, unless you turn division on. That switch brings multiplication and division into the mix, which raises the arithmetic without changing anything about the logic, and it is the natural second step once the format is familiar.

How a cage actually gets solved

Take a two-cell cage marked 5+. Two digits from 1 to 4 that add to five: either 1 and 4, or 2 and 3. Two possibilities, already.

Now suppose the same row contains a single-cell cage showing a 1. That 1 is spoken for, so the pair cannot include a 1, and the cage resolves to 2 and 3 without any further arithmetic. That interaction between cage and row is the entire puzzle in miniature.

Subtraction works the same way but ignores order: a 2− cage needs two digits whose difference is two, so either 1 and 3, or 2 and 4.

Solving order

• Single-cell cages first. They are free and each one immediately constrains a row and a column.
• Then two-cell cages with extreme targets. On a 4×4, a 7+ has only one legal pair and a 1− has several, so start with the extremes.
• Apply the Latin-square rule after every placement rather than at the end. Each digit placed removes a candidate from seven other cells.
• Pencil candidates inside cages you cannot yet settle, because the resolution usually arrives from a neighbouring row rather than from more arithmetic.

Number bonds in disguise

A child solving a 4×4 Calcudoku works through dozens of small sums and differences without being asked to do a single one. The arithmetic is instrumental: it is how you find out what goes in the box, not the thing being tested.

That matters for the pupils who have decided sums are what they are bad at, because what they are visibly doing here is reasoning. The cage idea also travels: once a child thinks of a cage as a small puzzle with its own target, Killer Sudoku and Kakuro stop looking like new formats and start looking like the same idea on a bigger board.

Six to a page is a car journey, a wet break, or a term of Friday starters from one print run.