Sudoku Made and maintained by Brian Hamilton and Karan Hamilton.
Brian builds the playable puzzle tools, while Karan helps test clarity, difficulty feel, and the player experience.
Playable puzzles and controls are tested in the browser for rule accuracy, usability, and a clear solving experience. See how we make and test puzzles, our editorial standards, or report a problem.
What is Calcudoku?
Calcudoku is a number puzzle on a square grid, from 4×4 to 9×9 here. Fill it so that every row and every column holds each digit from 1 to the grid’s size exactly once. There are no boxes, as there are in sudoku. Instead the grid is cut into cages, and each cage carries a clue: a target, and the operation that makes it.
- An addition or multiplication cage’s digits add or multiply to the target.
- A subtraction or division cage has exactly two squares: the larger digit minus, or divided by, the smaller gives the target, in either order.
- A one-square cage has no operation: its target is the digit.
The puzzle was devised by the Japanese maths teacher Tetsuya Miyamoto and is sold as KenKen; Calcudoku and Mathdoku are the generic names for the same rules. If you play Killer Sudoku, the cages will look familiar, but here they can subtract, multiply and divide, and there are no boxes to lean on.
Only rows and columns forbid repeats. A cage that bends round a corner, like an L of three squares, can hold the same digit twice when those two squares share no row and no column: a 6+ L can be 1, 4 and 1. Two squares side by side always share a line, so a subtraction or division cage never repeats, and a ÷ clue is never 1.
How do you read a cage?
List the ways its clue can be made from the digits the grid allows. On a 6×6, 11+ over two squares can only be 5 and 6, and 5− only 1 and 6, while 3÷ is 1 and 3 or 2 and 6. A cage that allows just one pair tells you more than where that pair goes:
Longer cages need more listing. On a 6×6, 12× over two squares is 2 and 6 or 3 and 4; over three squares in a straight line it is 1, 2 and 6 or 1, 3 and 4; and when the three squares bend round a corner, 2, 3 and 2 is possible too, with the 2s at the ends. Keep the options in your pencil marks and cross them off as rows and columns fill; the Killer Sudoku combinations tables do the same job for sums.
What does a line’s total tell you?
Every row and column holds each digit once, so on a 6×6 every line adds up to 1 + 2 + … + 6 = 21, and multiplies to 720. When the cages lying wholly inside a line are all additions, their targets and the squares left over make 21 between them:
It is the idea behind the rule of 45 in Sudoku, scaled to the grid: a line adds up to 10 on a 4×4, 15 on a 5×5, 28 on a 7×7, 36 on an 8×8 and 45 on a 9×9. It works for two lines at once as well, and for products when the cages multiply.
Were the boards on this page fair?
Mostly, but not always. The old generator checked each board with a search that stopped after 100,000 steps and called the board unique if it had found one answer by then; when five boards in a row failed that check, it dealt a sixth with no check at all. We dealt 40 boards at every size and level, 960 in all, and counted their answers with a separate solver: 29 had more than one answer. They sat at the big sizes and the hard levels: 13 of the 40 on 9×9 Expert, and 5 on 9×9 Medium.
That matters more than it sounds. The page marks a digit wrong when it differs from the one answer it stored, so on a board with two answers you could place a digit that breaks no rule and still be charged a mistake, and no amount of logic would finish it. The big boards were slow as well: a 9×9 Expert board took a median of 3.7 seconds to appear, and up to 7.3.
The levels also gave too much away. A cage’s size was drawn with equal odds from one square up, so on Easy about half the cages were single squares, each stating its own answer: a median of 16 of the 36 squares on a 6×6, and 35 of the 81 on a 9×9. The generator now makes one-square cages rare, and deals a board only once two plain steps can finish it, which proves it has exactly one answer. Every size and level appears in a few hundredths of a second at most on a desktop computer.
How do the four levels differ?
By the arithmetic. Each level allows larger cages or one more operation than the one before, and hands out fewer one-square cages. We dealt 40 fresh boards at every size and level:
| Level | Cage sizes | Operations | One-square cages, 6×6 | One-square cages, 9×9 |
|---|---|---|---|---|
| Easy | 1–2 squares | + | 6 of 36 | 13 of 81 |
| Medium | 1–3 squares | + − | 6 of 36 | 13 of 81 |
| Hard | 1–4 squares | + − × | 4 of 36 | 10 of 81 |
| Expert | 1–4 squares | + − × ÷ | 4 of 36 | 8 of 81 |
40 boards per size and level from this page’s generator, fixed seed; medians. A cage grows from a random square, so some end up smaller than drawn, and a square nothing else can reach becomes a one-square cage.
Harder here means more arithmetic, not longer chains. Every board at every size and level can be finished with the same two steps, and a solver that knows only those needed about as many passes on Medium as on Expert: a median of 9 passes on a 9×9 Medium board, 8 on Hard and Expert. What Hard and Expert add is the work inside each cage: factoring a product, and four-square cages with more ways to make their clue.
A board is dealt only if two steps, repeated until nothing changes, fill every square: keeping in each square only the digits some way of making its cage’s clue allows, and placing singles: a square with one digit left, or a digit with one square left in its row or column. Both steps only rule out digits that cannot go, so a board they finish has exactly one answer, and you never need to guess. We re-counted all 960 boards with a separate solver that tries every way of filling each cage: every one had exactly one answer, the one the page stored.
Which size and level should you play?
Start on 4×4 or 5×5 Easy if cages are new to you. With addition only and cages of one or two squares, you learn to read clues against rows and columns and nothing else. The page opens on 6×6 Medium, which adds subtraction and three-square cages.
Then move up one step at a time: the size first, which adds squares and more ways to make each clue, then the level, which adds operations. A 9×9 Expert board has 81 squares, cages of up to four and all four operations; it still needs nothing beyond the two steps above, but its cages take real arithmetic.
A solving order that works here
- Take the one-square cages first. Each states its digit, and that digit leaves the rest of its row and column.
- Then the tight cages. The largest and smallest sums for their size, a subtraction one less than the grid size, and on a 6×6 any ÷ clue above 2 leave only one or two ways to make them.
- Use the line totals. When addition cages sit wholly inside a row or column, the squares left over make up the difference.
- Watch cages that bend. Before ruling out a repeated digit in an L-shaped cage, check whether the two squares share a line.
- Sweep one digit at a time. Select a square holding that digit and every copy of it lights up, which shows at a glance the rows and columns that still need it.
What do the buttons do?
Pick a size from 4×4 to 9×9 and a level; the page opens on 6×6 Medium. Select a square, then a digit. Selecting a square highlights its row, its column and its cage, and a wrong digit is marked the moment it goes in. Under the grid:
- Notes turns pencil marks on and off. A correct digit clears itself from the notes of its row and its column; a wrong one leaves the other squares’ notes alone.
- Erase clears the selected square, and pressing the same digit twice does the same.
- Undo takes back the last change to a square, though not the notes a correct digit cleared elsewhere, and not the Mistakes count.
- Hint fills one square with its answer, picked at random rather than the square you could work out next.
- Solve asks first, then fills the grid and stops the timer.
- New Game deals another board at the size and level you are on; the level buttons deal one at theirs, and the size buttons one at the new size.
From a keyboard: the digits from 1 to the grid’s size, the arrow keys, Backspace or Delete and Ctrl+Z. The board is saved in this browser after every move, but the page reopens on 6×6, so that is the board waiting when you come back; choosing a size with its button always deals a fresh one.
Where to go next
Killer Sudoku keeps the cages but only ever adds, and puts Sudoku’s boxes back. If you have a Calcudoku from a book or a newspaper and want to know whether it has one answer, the Calcudoku solver takes its cages and clues and checks. For a warm-up on the same six digits with boxes added, try 6×6 Sudoku.