Calcudoku: 6x6

A Latin square driven by arithmetic

Fill the grid so every row and every column holds the digits 1 to 6 once each. That part is a Latin square. The cages are what make it a puzzle: each outlined group carries a target and an operation, and its cells must produce that target.

Addition and multiplication apply to all the cells in the cage in any order. Subtraction and division are only used on two-cell cages, and they work from the larger number — a cage marked 3− holds two digits three apart, and 2÷ holds a pair where one is double the other.

Where a solve starts

With the single cells. A cage of one cell has no operation and simply states its digit, and those are free.

Then the extremes. A two-cell cage marked 11+ has to be 5 and 6, and so does 30×. Marked 5−, it can only be 1 and 6. Anything sitting at the top or bottom of the possible range is fully determined before you have written a single digit, and most boards offer three or four such cages.

After that it is ordinary Latin-square logic: a digit placed rules it out of a row and a column, and the cages narrow what is left.

Turning division off

Division cages are the ones younger solvers stall on, because spotting that 12 and 2 are in a 6÷ relationship is a different mental move from adding.

With the switch off, no cage uses division. The logic is unchanged and the arithmetic becomes far more approachable, which turns this into a reasonable sheet for a Year 4 class rather than a Year 7 one.

Why 6×6 rather than 4×4 or 9×9

Four by four is solvable almost by inspection — there are too few possibilities for the cages to do any work.

Nine by nine takes a long time and the arithmetic gets heavy: a three-cell multiplication cage on a 9×9 grid can reach 504, and factorising that by hand is tedious rather than interesting.

Six is where the cages genuinely constrain and the numbers stay mental. A board here has around 19 cages.

Every board has exactly one answer

Cage layouts are generated over a completed Latin square and then tested with a solver. If a layout does not pin the grid, it is discarded and another is tried; only if several attempts fail does the generator reveal a digit or two, and it never prints a board with more than one solution.

In practice almost nothing is revealed. Measured across sixty boards, the average is under half a digit per puzzle. The cages do all of it.

Sizing it

Puzzles per page from one to six, division on or off, the solutions page, and a seed. A4 or US Letter.

Two to a page leaves room to pencil candidates into the cells, which you will want on anything past the opening moves.

The puzzle was devised by the Japanese teacher Tetsuya Miyamoto as a way of teaching arithmetic without teaching it — the calculation is a side effect of wanting to finish the grid. Print at 100% on A4 or US Letter.