How to solve Calcudoku: working out every cage
6 min read ·
Calcudoku combines a Latin square with arithmetic. Every row and column holds the digits 1 to n once, and every outlined cage has a target and an operation that its digits must make. The arithmetic is small; the skill is choosing which cages to work out first.
This guide goes through each kind of cage in the order that pays off fastest, then shows how row and column totals finish the job.
The rules
- On an n×n grid, every row and column contains the digits 1 to n once each.
- Each cage shows a target and an operation: +, −, × or ÷. Its digits must combine to the target.
- − and ÷ cages always have two cells: take the larger digit first (a 3− cage holds two digits that differ by 3).
- A digit may repeat inside a cage, as long as the copies are in different rows and columns.
Single cells first
A one-cell cage simply shows its digit: write it in straight away. Then scan its row and column, because that digit can't appear anywhere else in them.
Two-cell − and ÷ cages
These have the fewest possibilities, especially with large targets:
- On a 6×6 grid, 5− can only be 1 and 6; 4− is 1 and 5 or 2 and 6.
- 3÷ on a 6×6 grid is 1 and 3 or 2 and 6; 2÷ is 1 and 2, 2 and 4, or 3 and 6.
- A large ÷ target like 6÷ needs the digits 1 and 6, so it only appears on grids of 6×6 and up.
Multiplication cages: think in factors
Break the target into its prime factors and share them out between the cells. On a 6×6 grid, 15× in two cells must be 3 and 5, and 60× in three cells is 3, 4 and 5 or 2, 5 and 6. A factor of 5 always means a 5 in the cage (on grids up to 9, only 5 itself is a multiple of 5).
Large products are the easiest. On a 6×6 grid, 120× in three cells must be 4, 5 and 6, because no smaller set reaches it.
Addition cages
Addition cages work like Killer Sudoku cages, except that digits can repeat when they sit in different rows and columns. Very small and very large sums have few combinations: a two-cell 3+ is 1 and 2, and a two-cell 11+ on a 6×6 grid is 5 and 6.
Use the row and column totals
Every row and column contains 1 to n, so each adds up to the same total: 10 on a 4×4 grid, 15 on 5×5, 21 on 6×6 and 28 on 7×7. When the cages of a row are known except for one cell, that cell is the total minus the rest.
The same holds for products: every row of a 4×4 grid multiplies to 24, and every row of a 6×6 grid to 720.
A solving routine
- Fill single-cell cages and cross their digits off their rows and columns.
- List the pairs for every − and ÷ cage.
- Factor the × cages and find the one-way + cages.
- Look for rows and columns where only one or two cells are still open, and use the totals.
- Write notes and treat it like a Sudoku without boxes: singles and pairs work in every row and column.
Practise
Puzzimo's Calcudoku has four sizes, 4×4, 5×5, 6×6 and 7×7; the daily challenge is 5×5, and every puzzle has exactly one solution. Notes, undo and hints are built in (a hint adds 30 seconds to a ranked time), and a wrong digit counts as a mistake; the third ends the round.
Frequently asked questions
Which Calcudoku cages should I solve first?
Single cells, then two-cell − and ÷ cages with large targets, then multiplication cages with big or unusual products. They have the fewest possible digit pairs.
How do I work out a multiplication cage?
Split the target into prime factors and share them between the cells, keeping every digit between 1 and the grid size. A factor of 5 or 7 means that digit itself is in the cage.
Is Calcudoku the same as Sudoku?
It shares the rule that each row and column holds every digit once, but it has no boxes and no starting digits. The cages and their arithmetic take their place.