How to solve Calcudoku
Calcudoku (also known as KenKen, KenDoku and Mathdoku) was invented in 2004 by Japanese maths teacher Tetsuya Miyamoto as a form of brain training. It first appeared outside Japan in 2008, in The Times.
Start here
The rules
Fill the grid so that every row and every column contains each number once, from 1 up to the size of the grid. The grid is divided into outlined areas called cages. The small number in the corner of each cage is its target, and the numbers in the cage must reach that target using the cage's operation (+, −, × or ÷).
- + and × give the same answer in any order.
- For − and ÷, start from the largest number: a two-square "3−" cage could be 4 and 1, or 5 and 2.
- Unlike Killer Sudoku, a number can repeat inside a cage, as long as the copies aren't in the same row or column.
- A cage with a single square simply shows its answer.
Some puzzles use only one operation, shown above the grid. Mixed puzzles show the operation in each cage.
Basic Calcudoku strategies
These eight techniques come from the cages themselves: what a cage's target allows, and how its shape limits it. They're enough for most medium puzzles, and every harder technique builds on them.
A few terms first. A cage is an outlined group of cells, and its target is the number and operation in its corner. A candidate is a number that could still go in one cell. A combination is a set of numbers that makes a cage's target. Keep the two apart: a cage can have a fixed combination while each cell still has two or three candidates.
The diagrams use 5×5 grids so each step is easy to see. The reasoning is the same on the 9×9 grids in our books, where rows and columns hold 1 to 9. Thick lines are cage borders. Green marks the cells the step is about, blue marks a row, column or cage being used, grey numbers are already placed, and red with a cross is a candidate you remove.
After every placement, check the row, the column and any cage that cell touches. In Calcudoku one number often settles a cage, and that cage settles the next.
Starting points
Single-square cages
Seeing it. A cage with one cell just shows its answer. In the diagram the top-left cage is a single 3.
Using it. Write it in straight away, then look along its row and column. Column 1 now has a 3, so the 4+ cage in row 4 (1 and 3) must put its 1 in column 1 and its 3 next to it.
Starting points
List each cage's combinations
Seeing it. For each cage, work out which sets of numbers reach the target with the cage's operation, using only numbers that fit the grid size.
Using it. Note the sets beside the cage before you try to place anything. A three-cell 11+ cage in one row of a 5×5 has only one answer: 2, 4 and 5. Cages with a single combination are the best place to start a puzzle, because you know exactly which numbers they hold.
Starting points
Let rows and columns fix the order
Seeing it. You know a cage's numbers, but not which cell takes which. Check each cell against the numbers already in its row and column.
Using it. In the diagram the 5+ cage could be 1 and 4, or 2 and 3. Column 1 already has 1, 2 and 3, so the left cell can only be 4. That makes the cage 4 and 1, in that order.
Cage shape
Straight cages can't repeat
Seeing it. Calcudoku lets a number repeat inside a cage, but never twice in one row or column. So when a cage's cells all sit in one line, its numbers must all be different.
Using it. Drop any combination with a repeat. A three-cell 6+ cage in a row can't be 2, 2 and 2 or 1, 1 and 4, so it must be 1, 2 and 3.
Cage shape
Bent cages can repeat
Seeing it. In an L-shaped or zigzag cage, cells that share neither a row nor a column may hold the same number. This is the big difference from Killer Sudoku, so it's easy to miss.
Using it. Keep combinations with a repeat when the repeated numbers can go in cells that don't line up. In the diagram, three different numbers from 1 to 5 add to at most 12, so 14+ must use a repeat. Only the two ends of the L can match, and 5 + 5 + 4 is the only way to reach 14: both ends are 5 and the corner is 4.
Cage arithmetic
Difference and quotient pairs
Seeing it. A two-cell − or ÷ cage has only a few possible pairs. In a 5×5, 3− is 1 and 4 or 2 and 5. 2÷ is 1 and 2 or 2 and 4.
Using it. Rule pairs out with the row and column, then decide the order. In the diagram, the 2÷ cage's lower cell could be 1, 2 or 4, but row 2 already has a 1 and a 4, so it is 2. The top cell is then 1 or 4, and column 4 already has a 1, so it is 4.
Cage arithmetic
Smallest and largest totals
Seeing it. Compare a cage's target with the smallest and largest totals (or products) its shape can make. Remember that cells in one line can't repeat.
Using it. A target at or near the limit forces numbers early. The 7+ cage in the diagram has three cells in column 2, which add to at least 1 + 2 + 3 = 6. That leaves at most 1 for the fourth cell, so it is 1, and the three column cells are 1, 2 and 3 in some order.
Cage arithmetic
Work out the last cell
Seeing it. Some of a cage's cells are solved and one isn't.
Using it. Take the known numbers away from a + target, or divide them out of a × target. In the diagram, 11 − 5 − 2 leaves 4. The same works with a partial total you've worked out some other way, such as from a line total on the advanced page.
