How to solve arithmetic cage puzzles

A Latin square where the clues are arithmetic. Each cage tells you what its cells combine to, and you work out which numbers do it.

Fill the grid so each row and column contains every number once — a Latin square. The extra layer is that cells are grouped into cages, each with a target and an operation, and the numbers in a cage must combine to hit it.

Start with the cages that cannot vary

A one-cell cage is a given. Write it in.

Next, look for cages whose target admits only one combination. In a 4×4, a two-cell cage with product 12 must be 3 and 4 — nothing else in range multiplies to it. A two-cell cage summing to 3 must be 1 and 2. These pin down whole regions before you have made a single deduction about rows.

Combine cage and Latin constraints

The real progress comes from using both together. If a cage lies entirely within one row, its numbers are distinct — so a sum of 5 across two cells in a row cannot be 2 and 3 and also 1 and 4 once you know which digits that row still needs.

Conversely, once a cage is solved, its digits are used up in their rows and columns.

The subtraction and division rule

Subtraction and division only ever appear on two-cell cages, and the order is not specified: a cage marked "2−" means the difference is 2, so it could be 3 and 1 or 1 and 3. This is deliberate rather than an omission. Those operations are not order-independent, so a three-cell subtraction cage would have no well-defined meaning — which is why you will never see one here.

Every puzzle has one solution

The generator carves cages out of a completed Latin square, then verifies uniqueness by exhaustive search before offering the puzzle. If more than one filling satisfies the cages, the cages are redrawn. So when you are stuck, there is always a deduction available — you have not been handed an ambiguous grid.

Frequently asked questions

Why are there no three-cell subtraction cages?

Because subtraction and division depend on the order of their operands, and a cage has no order. Only addition and multiplication extend meaningfully to three or more cells, so the tool restricts the other two to pairs.

Does the same number appear twice in a cage?

It can, if the cage bends across rows and columns so the repeated cells are not in the same line. Within a single row or column it cannot, because of the Latin square rule.

Is guessing needed?

No. Every puzzle is verified to have exactly one solution reachable by deduction.