How to solve Mathdoku: 5 beginner techniques
Mathdoku looks as if it needs a lot of arithmetic, but most puzzles are solved with a handful of simple techniques used in the right order. Here are five that will take a beginner through an easy grid without guessing.
The two rules to keep in mind
Every technique comes from two rules. No digit repeats in a row or column. The digits in each cage must reach the cage’s target using its operation.
The examples below use a 6×6 grid, where the digits are 1 to 6. The same ideas work on any size.
1. Fill the single-cell cages
A cage with one cell has no sum to do. The target is the digit, so fill these first.
Each one removes a digit from its row and its column, and that makes the cages next to it easier.
2. Find the cages with only one answer
Some targets can be made in only one way. Learn to spot these:
- Two cells that add to 3 must be 1 and 2.
- Two cells that add to 11 must be 5 and 6.
- Two cells that subtract to 5 must be 1 and 6.
- Two cells that multiply to 30 must be 5 and 6.
- Three cells in a straight line that add to 6 must be 1, 2 and 3.
Two cells that add to 4 must be 1 and 3. They cannot both be 2, because two cells side by side share a row or a column.
You may not know yet which cell takes which digit. That is fine. Note both digits in both cells and move on.
3. Let the rows and columns decide the order
Once you know which digits a cage holds, the row and column rule usually tells you where each one goes.
Say a cage holds 1 and 2 in some order. If the column of the left cell already has a 1, the left cell must be 2 and the right cell must be 1.
Each time you place a digit, look along its row and its column for a pair it settles.
4. Use the row and column total
Every row and column on a 6×6 grid holds the digits 1 to 6 once, so it always adds up to 21. This helps when cages sit entirely inside one row or column.
Say a row holds a three-cell cage marked 12+ and a two-cell cage marked 7+. Those five cells add up to 19, so the sixth cell must be 2, whatever cage it belongs to.
On a 4×4 grid the total is 10, and on a 5×5 grid it is 15.
5. Break multiplication targets into factors
For a multiplication cage, ask which digits divide the target. On a 6×6 grid:
- If the target divides by 5, the cage must contain a 5. A two-cell cage marked 20× is 4 and 5.
- If the target does not divide by 5, the cage has no 5 in it. In the same way, an odd target means no 2, 4 or 6.
- Then list what is left. A two-cell cage marked 12× is 2 and 6, or 3 and 4. The row and column rule picks between them.
Division works the same way in reverse. A two-cell cage marked 3÷ can only be 1 and 3, or 2 and 6.
When you get stuck
Go back through the five techniques in order. A digit you placed later often unlocks an earlier step.
Turn on Notes and mark the digits each cell could still hold. It works the same way as Sudoku pencil marks.
Avoid guessing. A wrong digit can stay hidden for many moves, and by then everything placed after it is wrong too.
Practise in our Mathdoku
Try these techniques in our online Mathdoku. Start on Easy, which uses only addition and multiplication. Move to Medium when you are comfortable with all four operations.