Kakuro combinations and techniques

These are the techniques our solver uses, from easiest to hardest. The hint button uses the same ideas and explains each step in the same words.

1. Learn the unique sums

Most totals can be made in several ways. 10 in two cells could be 1 + 9, 2 + 8, 3 + 7 or 4 + 6. But the smallest and largest totals for each run length have only one combination. These are the anchors of every kakuro.

A pattern makes the table easy to remember. For any length, the lowest total uses the smallest digits, and the next one up swaps the biggest of them for the next digit. The high totals mirror the low ones: swap each digit d for 10 minus d. So 1 + 2 = 3 mirrors 9 + 8 = 17, and 1 + 2 + 4 = 7 mirrors 9 + 8 + 6 = 23.

Some near-unique totals are worth knowing too. They have just two combinations:

• 2 cells: 5 (1 + 4, 2 + 3), 6 (1 + 5, 2 + 4), 14 (5 + 9, 6 + 8), 15 (6 + 9, 7 + 8).
• 3 cells: 8 (1 + 2 + 5, 1 + 3 + 4) and 22 (5 + 8 + 9, 6 + 7 + 9).
• 4 cells: 12 (1 + 2 + 3 + 6, 1 + 2 + 4 + 5) and 28 (4 + 7 + 8 + 9, 5 + 6 + 8 + 9).

Notice the useful bits. Both ways of making 8 in three cells use 1. Both ways of making 12 in four cells use 1 and 2. Digits that every combination shares are gold, as you'll see in technique 4.

2. Cross the runs

Every white cell sits in one across run and one down run. Its digit has to be allowed by both. Write down what each run allows, then keep only the digits that appear in both lists.

Example: an across run of 4 in two cells needs 1 and 3. A down run of 3 in two cells crosses its first cell, and that run needs 1 and 2. The only digit in both lists is 1, so the crossing cell is 1. The other cell of the across run must then be 3.

3. High and low digits

Even when a total has many combinations, it limits the digits. Two cells adding up to 6 can't hold anything above 5. Three cells adding up to 22 can't hold anything below 5, because the other two cells can add up to 17 at most.

A quick way to check a digit: fill the rest of the run with the biggest digits you can. If that still doesn't reach the total, the digit is too small. Fill them with the smallest digits. If that already goes over, the digit is too big.

The same trick finishes runs. Once a run has one empty cell, subtract the placed digits from the clue. Once it has two, the remaining total in two cells often has a unique answer, even if the full clue didn't.

4. Digits a run must contain

Look at every combination that still works for a run. If a digit is in all of them, the run must contain it somewhere. Now check which cells of the run can still take that digit. If only one can, that's where it goes.

Example: 10 in four cells must be 1 + 2 + 3 + 4. If the crossing runs rule out 4 in three of those cells, the fourth cell is 4. This is the kakuro version of a "hidden single" in sudoku.

5. Rule out combinations that don't fit

For a harder run, list its combinations and try each one against the cells. A combination only works if every one of its digits has a cell that can take it, all at once.

Example: 9 in two cells could be 1 + 8, 2 + 7, 3 + 6 or 4 + 5. If the crossing runs only let the first cell be 3 or 6 and the second be 2, 3 or 6, then 1 + 8, 2 + 7 and 4 + 5 are out. The run is 3 + 6, and both cells are now down to two digits each.

This is where pencil marks earn their keep. Turn on Notes and write the candidates into tricky cells. Then cross out combinations one by one.

6. Where to start a big grid

• Start with unique sums, especially short runs in corners.
• Next, look at short runs that cross unique ones.
• Big totals in short runs (like 16 or 17 in two cells) and small totals in long runs are your friends.
• After each digit, check both runs through it. Most progress comes from the cells right next to what you just filled.