Calcudoku Puzzles
Fill the grid so each row and column holds every number once, and each outlined cage combines to its target using the operation shown. Every puzzle is checked before you see it: exactly one solution, and solvable by reasoning alone.
Also known as mathdoku, keendoku and arithmetic sudoku. The best-known brand name for this mechanic is trademarked, so this site uses the generic term throughout.
Related Practice
What Is Calcudoku?
Calcudoku is an arithmetic logic puzzle on a square grid. Every row and column must hold each number from 1 up to the grid size exactly once, exactly as in a Latin square. The grid is then divided into outlined blocks called cages, each printed with a target number and an operation — the numbers inside that cage must combine with that operation to reach the target.
Unlike sudoku, a calcudoku usually begins completely empty. The cages are the only clues, and that is what gives the puzzle its character: you are not eliminating from given numbers, you are deducing numbers from arithmetic.
The One Rule People Get Wrong
Numbers may repeat inside a cage, as long as they are not in the same row or column. An L-shaped three-cell cage can legitimately contain two 3s, because two of its cells sit in different rows and different columns. The no-repeat rule belongs to rows and columns; it has never applied to cages.
This matters when listing combinations. A 6+ cage over three cells on a 5×5 includes 1+2+3, but it also includes 1+1+4 and 2+2+2 where the shape allows it — and ruling those out by mistake is the commonest way to convince yourself a solvable puzzle is broken.
How To Solve A Calcudoku
- Fill the single-cell cages. A cage covering one cell has no arithmetic in it — the target is the answer. Each one constrains a full row and column straight away.
- List combinations for the small cages. For a two-cell cage, write every pair that reaches the target. Multiplication is the sharpest clue on the grid: on a 5×5 there is usually only one pair that makes a given product.
- Cross them against the row and column. A combination only survives if its numbers are still available where the cage sits. Striking the impossible ones usually leaves exactly one, which solves the cage outright.
- Alternate, do not exhaust. Swap between cage arithmetic and the Latin square rule. Working either one to a standstill before touching the other is the most common way to stall on a puzzle that is perfectly solvable.
- Leave the big cages for last. A four-cell sum has too many combinations to constrain anything early. Come back once its cells have been narrowed by their rows.
Which Operations Appear Where, And Why
| Operation | Cage size | Why |
|---|---|---|
| Addition | Any | Order does not change a sum, so any number of cells is unambiguous. |
| Multiplication | Any | Same reason — and the tightest clue of the four on small grids. |
| Subtraction | Two only | With three numbers the answer would depend on the order taken, so the clue would not be well defined. |
| Division | Two only | Same, and it appears only where one number divides the other exactly. |
Proved Solvable Without A Single Guess
Every puzzle on this page is checked twice before it reaches you, and it is worth saying what those checks are rather than simply asserting the result.
| Guarantee | How it is checked | If it fails |
|---|---|---|
| Exactly one solution | A backtracking solver counts solutions, stopping at two. | Discarded; a different puzzle is generated. |
| No guessing required | A second solver, restricted to human techniques, must finish unaided. | Discarded, never shown. |
That second solver is what Show me the next move replays. It gives you one deduction with its reason — which cell, which number, and which cage or row forced it — rather than filling the grid in. A solve button answers the puzzle; this is meant to teach it.
Why The Grids Stop At 5×5
This page runs 4×4 and 5×5 grids. It stops there on purpose: at 6×6 the cages frequently leave no route to a solution without guessing at some point, and a puzzle that needs a guess does not meet the guarantee above. Rather than quietly relax the rule for larger grids, the larger grids are simply not offered.
Calcudoku Puzzles Frequently Asked Questions
What is calcudoku?
An arithmetic logic puzzle played on a square grid. Every row and column must contain each number from 1 to the grid size exactly once, and the grid is divided into outlined blocks called cages. Each cage shows a target number and an operation, and the numbers inside it must combine using that operation to reach the target. Unlike sudoku it usually starts completely empty — the cages are the only clues.
Can numbers repeat inside a calcudoku cage?
Yes, provided they are not in the same row or column. A cage shaped like an L can legitimately hold the same number twice, because the two cells sit in different rows and different columns. This trips people up constantly — the no-repeat rule applies to rows and columns, never to cages.
Why is it called calcudoku and not the brand name?
The best-known name for this mechanic is a registered trademark belonging to Nextoy LLC. Calcudoku is the standard generic term, and it is what this site uses in the address, the heading and the text. You may also see the same puzzle called mathdoku or keendoku.
Does every calcudoku have one solution?
A properly made one does. Every puzzle on this page is checked before it is shown: a solver counts the solutions and the puzzle is discarded unless there is exactly one. A second solver, restricted to techniques a person could use, then has to finish it unaided — so nothing here needs a guess.
Which operations are used?
Addition and multiplication can appear on cages of any size. Subtraction and division only appear on two-cell cages, because with three or more numbers the result would depend on the order you took them in, which would make the clue ambiguous. Division cages only appear where one number divides the other exactly.
Is calcudoku harder than sudoku?
It is different rather than uniformly harder. The grids are much smaller — 4×4 or 5×5 here rather than 9×9 — so there is less to track, but you have to do arithmetic as well as logic, and a calcudoku typically starts with no given numbers at all. People who find sudoku mechanical often prefer it for exactly that reason.
Where do you start a calcudoku?
With the cages that can only be filled one way. A single-cell cage is its own answer and goes straight in. After that look for cages whose target is extreme for their size — the largest or smallest total or product available — because those have one combination rather than several. Only once those are placed does row and column elimination have enough to work with.
Can calcudoku be solved without guessing?
Every puzzle here can, and that is checked before it is shown rather than assumed. One solver has to finish the grid using deductions alone, and a second, separate search has to confirm there is exactly one solution; a puzzle failing either is discarded. If you find yourself guessing, there is a cage combination or a line elimination still on the board.