Kakuro

Rules

1.Write a digit from 1 to 9 into every white cell. Zero is never used.

2.A gray clue cell is split by a diagonal. The number in the top right is the sum of the white cells running to the right of it. The number in the bottom left is the sum of the white cells running down from it. A clue cell may carry one number or two.

3.A run is the white cells between two clue cells. Within one run, all digits are different: even where the sum alone would allow a repeat.

4.Runs are at least two cells long. In this book they are never longer than six.

5.The puzzle is solved when every white cell holds a digit and every run adds up to its clue.

The combination table

For many combinations of sum and run length, only one set of digits is possible. Those are your entry points. This table is complete for the run lengths that occur in this book.

Cells	Sum	The only possible digits
2	3	1 2
2	4	1 3
2	16	7 9
2	17	8 9
3	6	1 2 3
3	7	1 2 4
3	23	6 8 9
3	24	7 8 9
4	10	1 2 3 4
4	11	1 2 3 5
4	29	5 7 8 9
4	30	6 7 8 9
5	15	1 2 3 4 5
5	16	1 2 3 4 6
5	34	4 6 7 8 9
5	35	5 6 7 8 9
6	21	1 2 3 4 5 6
6	22	1 2 3 4 5 7
6	38	3 5 6 7 8 9
6	39	4 5 6 7 8 9

Almost as useful are sums with exactly two possible sets, where one crossing decides it:

Cells	Sum	Either	or
2	5	1 4	2 3
2	6	1 5	2 4
2	14	5 9	6 8
2	15	6 9	7 8
3	8	1 2 5	1 3 4
3	22	5 8 9	6 7 9

You need not memorize this. For a run of length L the lowest sum is 1+2+…+L and the highest is 9+8+…, and the two sums nearest each end are always forced.

How to solve

1.Find the forced runs. Go over the grid and mark every run whose sum and length appear in the first table. Pencil the digit set into its cells.

2.Cross the runs. Every white cell belongs to one horizontal and one vertical run. Write down the digits allowed by its horizontal run and those allowed by its vertical run, then keep only the digits that appear in both. Very often that leaves one digit.

3.Clear up inside a run. Once one digit of a forced set is placed, the remaining digits of that set must fill the remaining cells of the run, which usually settles a second cell by crossing.

Worked example

A Medium grid. Each clue cell carries the sum of the run to its right (upper number) or below it (lower number). A run never repeats a digit, and there is no zero. The small digits are pencil marks.

The full solution takes 46 of these run-by-run steps. Here are the two that matter: the way in, and the move that only Kakuro has.

The way in. A run of 2 cells that has to add up to 16 can only be made of 7 + 9: there is no other set of 2 different digits with that sum. So r6c2 and r6c3 carry those digits in some order, and nothing else.

Where two runs cross. The run r5c2, r6c2, r7c2 and r8c2 has to add up to 13. Checking every set of 4 different digits that adds up to 13 leaves only part of the digits possible in each cell. Combined with what the crossing runs already allow, that settles r6c2.

The finished grid: