How to Play Kakuro

Kakuro looks like a crossword crossed with arithmetic — but there's no English in it. You fill blank cells with digits 1 through 9 so that each run of cells sums to the clue at the start of the run, with the catch that no digit may repeat within a single run.

Once you internalize one trick — “magic combinations” — easy Kakuro puzzles unlock quickly. Let's walk through one from start to finish.

The Puzzle

Here is a 4×4 puzzle with seven blanks to fill. The dark cells with two numbers are clue cells. The number above the diagonal applies to the row going right; the number below the diagonal applies to the column going down. Solid navy squares are walls — they're not part of any run.

The Three Rules

Rule 1: Each run sums to its clue

A “run” is a horizontal or vertical sequence of blank cells, bounded by a clue cell or a wall. The number on the clue cell tells you what the digits in that run must add up to.

Rule 2: No digit may repeat within a run

Even if the digits add up correctly, you cannot use the same digit twice in the same run. This is the rule beginners miss most often.

Important: digits can repeat across different runs. Two cells in the same row that belong to separate runs (separated by a wall or clue cell) are free to hold the same digit.

Rule 3: Only digits 1-9

No zeros, no double digits. Just 1, 2, 3, 4, 5, 6, 7, 8, or 9 in each blank cell.

Magic Combinations

For a given run length and a given clue, only certain sets of distinct 1-9 digits can possibly fit. Many sums have multiple valid sets — but a few have exactly one. Memorizing these unlocks the puzzle.

The most useful unique combinations:

Notice the pattern: very low sums and very high sums for a given length are most constrained. These are the cells you attack first.

Walkthrough

Step 1 — Find the magic combination on column 4

The clue cell at row 2, column 4 says the column going down sums to 17, across two cells (rows 3 and 4).

From the magic combinations table: 17 in 2 cells = {8, 9}, the only option. So the cells at row 3, column 4 and row 4, column 4 contain 8 and 9 in some order — we don't know which yet.

Step 2 — Find another magic combination on row 2

The clue cell at row 2, column 1 says the row going right sums to 16, across two cells (columns 2 and 3).

From the table: 16 in 2 cells = {7, 9}, the only option. So the cells at row 2, column 2 and row 2, column 3 contain 7 and 9 in some order.

Step 3 — Try one possibility, find a contradiction

We know row 2, column 2 is either 7 or 9. Let's try assuming it is 9 and see what happens.

• If row 2, column 2 = 9, then row 2, column 3 = 7 (to make the across=16 work).
• Now look at the down=15 clue at row 1, column 2. Its column sums to 15 over two cells: row 2, column 2 + row 3, column 2. Since the top is 9, the bottom must be 15 − 9 = 6. So row 3, column 2 = 6.
• Now look at row 3 (across=22, three cells). It already has row 3, column 2 = 6. The other two cells must sum to 16.
• Row 3, column 4 is 8 or 9 (from Step 1). Try both:
• If row 3, column 4 = 9, then row 3, column 3 = 7. But column 3 already has a 7 at row 2 — that would duplicate 7 in column 3's down=16 run. Contradiction.
• If row 3, column 4 = 8, then row 3, column 3 = 8. But row 3 would now have two 8s. Contradiction.
• Both branches fail. So our assumption (row 2, column 2 = 9) is wrong.

Therefore row 2, column 2 = 7, which forces row 2, column 3 = 9.

Step 4 — The rest cascades by simple subtraction

Now everything else falls out by subtracting from the clues:

• Down=15 column 2: top is 7, so bottom (row 3, column 2) = 15 − 7 = 8.
• Across=22 row 3: 8 + ? + ? = 22, so the remaining two cells sum to 14. Combined with column 4's constraint (row 3, column 4 ∈ {8, 9}):
• If row 3, column 4 = 8, then row 3, column 3 = 6. But row 3 would have two 8s. ✗
• If row 3, column 4 = 9, then row 3, column 3 = 5. ✓
• So row 3, column 3 = 5 and row 3, column 4 = 9.
• Down=17 column 4: 9 + ? = 17, so row 4, column 4 = 8.
• Across=10 row 4: 2 cells sum to 10, and we just placed 8 at row 4, column 4. So row 4, column 3 = 10 − 8 = 2.

Tips for Beginners

• Start with extreme sums. Tiny sums like 3, 4 and giant sums like 24, 23 collapse to a single magic combination — they're your foothold.
• Find the intersection of two runs. A blank cell sits at the intersection of one across-run and one down-run. The cell's value must appear in both runs' valid candidate sets — often that narrows it to one digit.
• Pencil candidates in. When you've identified a magic combo but don't yet know the order, jot the candidate set in the cell corner. Other constraints will eliminate one as you go.
• Don't forget the no-repeat rule. When you're tempted to place a digit, scan the whole run first to make sure it doesn't already appear there.
• Try both branches when stuck. Like in Step 3 above, picking one option and chasing constraints often surfaces a contradiction quickly — which tells you the other option is correct.

Ready to try one yourself? Hit the button below to play your first Kakuro.