Calcudoku

Where it comes from

The puzzle was devised in 2004 by Tetsuya Miyamoto, a Japanese mathematics teacher, as a way of getting students to practise arithmetic without being drilled in it. His stated method was to teach without teaching: hand over a grid, say nothing, and let the reasoning arrive on its own.

That origin still shows in the design. The arithmetic involved is deliberately small — addition, subtraction, multiplication and division on single digits — because the difficulty was never meant to come from the sums. It comes from working out which sums are even possible.

The genre spread under several names. Calcudoku and mathdoku are the generic ones; the best-known commercial version is a registered trademark, which is why puzzle collections and apps describe themselves in the generic terms instead.

How it differs from sudoku

Both are built on a Latin square: each number once per row, once per column. Everything after that is different.

• Sudoku adds boxes — the nine 3×3 regions that must also hold each digit once. Calcudoku has no boxes at all.
• Sudoku starts with numbers already on the board. Calcudoku usually starts empty; the clues are the cages, not the cells.
• Sudoku is always 9×9. Calcudoku is played at any size, and a 4×4 is a genuinely different experience from a 9×9 rather than an easier one.
• A sudoku digit is a symbol — you could play with letters. In calcudoku the numbers have to be numbers, because they are being added and multiplied.

The practical difference is where the information lives. In sudoku you scan for what is already placed. In calcudoku you read the clues and work out what could be placed, which is why the opening moves feel so unlike each other.

And its closer relatives

Killer sudoku is the nearest neighbour: a normal sudoku with cages laid over it, each carrying a sum. It keeps the boxes and uses addition only, so cages never repeat a number within themselves. Calcudoku drops the boxes, adds the other three operations, and allows a repeat inside a cage where the rows and columns permit one — which is the rule newcomers most often get wrong.

Futoshiki is the other cousin: a Latin square constrained by greater-than and less-than signs between neighbouring cells. Same skeleton, no arithmetic.

If you have solved killer sudoku, most of your instincts carry over. Expect to be caught out twice: once by a multiplication cage that factors in only one way, and once by a cage that legally holds two of the same number.

Sizes and difficulty

Grid size sets the vocabulary: a 4×4 uses 1 to 4, a 6×6 uses 1 to 6, a 9×9 uses 1 to 9. Bigger grids are not simply harder — they are more forgiving in one way, because a larger line sum and more cells give you more places to find a foothold.

What actually drives difficulty is the cages. A grid full of two-cell cages is easy whatever its size, because two-cell cages bound tightly and can never repeat. A grid with several four- and five-cell cages is hard, because each one admits many combinations and gives little away until the lines around it close in.

The operations matter too. Multiplication is the most generous clue, since small numbers factor in few ways. Subtraction is the least, since a two-cell difference of 1 in a 6×6 has five possible pairs and tells you almost nothing on its own.

What makes a grid a good one

A well-made calcudoku has exactly one solution, and that solution is reachable by deduction alone. Both halves matter. A puzzle with two solutions is broken; a puzzle with one solution that can only be found by trying a value and backtracking is solvable but not enjoyable, because the satisfying part of the genre is that every step can be justified.

Generators earn their keep here. It is easy to produce a grid with a unique solution and quite hard to produce one that a person can reason through from start to finish — which is why the same target and the same cage shapes can make a pleasant puzzle or a miserable one depending on where the cage boundaries fall.

Every grid in Cages is checked for both. The techniques page walks through the deductions that get you from a blank grid to a finished one.