A Latin square with arithmetic cages: each cage hits its target with +, −, × or ÷

Calcudoku

Calcudoku, also called mathdoku, is a Latin square with arithmetic laid over it. Each row and column holds every digit once, and heavy outlines group the cells into cages, each labeled with a target and an operation. Choose a size from the tabs above the board: 20 puzzles each at 4×4, 6×6 and 9×9.

Calcudoku
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Unlike classic sudoku there are no boxes, and the cages do the work that boxes usually do. The arithmetic is times tables and short sums, though a long × cage on a 9×9 can carry a surprisingly large target.

The rules

  1. In a grid of size N, every row and every column contains the digits 1 to N exactly once. A 6×6 uses 1 to 6.
  2. The digits in a + cage add up to its target, and the digits in a × cage multiply to it.
  3. A minus cage has two cells: the larger digit minus the smaller gives the target. A ÷ cage also has two cells: the larger divided by the smaller gives the target.
  4. A cage of a single cell simply shows its digit.
  5. A digit may repeat inside a cage, as long as the repeats are not in the same row or column.

First moves

Fill the single-cell cages, then look for two-cell cages with only one answer. On a 6×6, a minus 5 must be 1 and 6, a 5× must be 1 and 5, and an 11+ must be 5 and 6. On a 4×4, a minus 3 is always 1 and 4. You won't know which digit goes where yet, but you know the pair, and that pair is now off limits for the rest of the row or column it sits in.

Division cages are friendly too. A 3÷ on a 6×6 is 1 and 3 or 2 and 6; a 2÷ is 1 and 2, 2 and 4, or 3 and 6. Write the options in with Pencil and let the row and column cut them down.

Sizes and levels

The 4×4 puzzles are 10 easy and 10 medium. The easy ones use addition cages only, which makes them a gentle place to learn the rules. The 6×6 set is 7 easy, 7 medium and 6 hard, with all four operations, and the 9×9 set skips easy altogether: 7 medium, 7 hard and 6 expert.

Easy puzzles need only cage options checked against the row and column, plus naked singles. Medium adds hidden singles, naked pairs and cage/line interaction: if every way of filling a cage puts a 6 in one row, no other cell of that row can be 6.

Hard grids use line totals. Every row of a 6×6 adds up to 21 and multiplies to 720, so if + cages lying wholly inside a row account for all but two of its cells, those two cells form a hidden cage with a known sum. Hard also brings naked triples and hidden pairs and triples. The 9×9 experts add X-wings on a single digit and cage-pair reasoning, where two cages sharing a line are worked out together.

Help and printables

The calcudoku rules page explains each operation with examples, and calcudoku strategy covers line totals and cage pairs in detail. The calcudoku cheat sheet lists common cage options, so you can stop working out 12× on a 6×6 every time.

To solve on paper, download the printable calcudoku packs, with answers on the last pages; a large print edition is available too. A new 6×6 is posted every day as the daily calcudoku.