Puzzle guide

How to solve Kakuro

Kakuro is a crossword made of numbers. Instead of words you write digits, and instead of clues you get sums. It looks like arithmetic, but most of the work is noticing which digits can still fit. This guide shows you the ideas that solve a board, one at a time.

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The rules

Fill every white cell with a digit from 1 to 9.
A number on a dark tile is a sum. The number at the top right is for the white cells going right; the number at the bottom left is for the white cells going down.
The digits of one run (the white cells that belong to one sum) add up to that sum.
A digit never appears twice in the same run.

A solved puzzle

A solved puzzle. The 3 on the left: 1 + 2. The 4 at the top: 1 + 3. Every run adds up and no run repeats a digit.

Your first moves

Look for small and big sums in short runs. A 3 in two cells can only be 1 + 2, and a 17 in two cells can only be 8 + 9.
Find the cell where two such sums meet. The digit it holds has to fit both of them.
Whenever a run has one cell left, take the sum minus the digits already there.
Write small notes in a cell when two or three digits are still possible. They show you pairs and patterns later.

Techniques, step by step

Take the sum minus the digits already there

This run adds up to 12, and two of its three cells already hold 5 and 4. What is left is 12 − 5 − 4 = 3, so the last cell is a 3. This is the move you will make most often: every time a run has a single empty cell, it is decided.

12 − 5 − 4 = 3.

Sums with only one way

Some sums can be made in only one way. 4 in two cells is always 1 + 3 (2 + 2 is not allowed, because a run never repeats a digit). You do not know yet which cell is the 1 and which is the 3, but both cells can only be 1 or 3. Learn the most useful ones: in two cells 3 = 1+2, 4 = 1+3, 16 = 7+9, 17 = 8+9; in three cells 6 = 1+2+3, 7 = 1+2+4, 23 = 6+8+9, 24 = 7+8+9.

4 in two cells: only 1 and 3.

Where two sums meet

The 3 on the left and the 4 at the top meet in the framed cell. The 3 can only be 1 + 2, and the 4 only 1 + 3. The only digit in both is 1, so the framed cell is a 1. After that the rest follows by subtraction: 3 − 1 = 2 to its right, 4 − 1 = 3 below it.

1 or 2 from the left, 1 or 3 from the top: only 1 fits both.

Big sums work the same

At the other end of the range the same idea works. 16 in two cells is only 7 + 9, and 17 only 8 + 9. Where they meet, the shared digit is 9. Big sums are just as helpful as small ones, because they leave just as few choices.

7 or 9 from the left, 8 or 9 from the top: the cell is a 9.

Too big or too small

The 4 on the left makes the framed cell 1 or 3. Now look at the 11 at the top: the framed cell and the cell below it add up to 11. If the framed cell were 1, the cell below would have to be 10, and no digit is bigger than 9. So the framed cell is 3. Use this whenever a run is short: the other cells can only add up to between their smallest and their largest possible digits.

With a 1 on top, the 11 would need a 10 below it. So it is 3.

Pairs lock their digits

This run adds up to 6 in three cells, which is always 1 + 2 + 3. Two cells can only be 1 or 3, and they have to share those two digits between them. So no other cell of the run can be a 1 or a 3, and the last cell is a 2. When two cells in a run have the same two possible digits, cross those digits off everywhere else in the run.

1 and 3 are taken by the pair, so the third cell is 2.

Common mistakes

Repeating a digit in a run. 4 in two cells is never 2 + 2, and 6 in three cells is never 2 + 2 + 2.
Reading the tile the wrong way round. The top-right number belongs to the cells going right, the bottom-left number to the cells going down.
Playing it like Sudoku. A run does not have to contain every digit. A run of two cells holds only two of the nine digits.
Forgetting the digits already there. When a run is half full, work with what is left of its sum, not with the whole sum.

Questions and answers

What is Kakuro?

A number puzzle like a crossword: you fill white cells with digits 1 to 9 so that each run adds up to the sum on its dark tile, without repeating a digit in a run. It is also called Cross Sums.

Do I need to be good at maths?

No. You only add and subtract small numbers. The real skill is knowing which sums have only a few ways to be made, and the list in this guide covers the useful ones.

Where should I start?

With short runs whose sums are very small or very large, like 3, 4, 16 or 17 in two cells. Then look at the cells where two of them cross.

Can a run contain a 0?

No. Every white cell holds a digit from 1 to 9.

What should I do when I am stuck?

Write the possible digits as notes into the cells of one run, then check where it crosses other runs. A pair of cells with the same two digits, or a digit that would need a number bigger than 9, usually opens the next step.

Ready to play?

Subtract, look for sums with only one way, and check where runs cross. That is how every Kakuro is solved, from the first easy board to the expert ones.