Kakurasu is a logic puzzle where you shade squares in a grid so that every row and column adds up to its target. A shaded square is worth its column number to its row, and its row number to its column. There are no digits to write in: the whole game is deciding which squares to shade.
The rules
The columns are numbered 1, 2, 3 and so on from the left, and the rows from the top. Every row and column has a target at its end.
- Shade some squares. You can also cross out squares you know are empty, as a reminder.
- Each row adds up its shaded squares’ column numbers. The total must equal the target at the end of the row.
- Each column adds up its shaded squares’ row numbers. The total must equal the target below the column.
That’s all. When every row and column meets its target, the puzzle is solved.
How a row adds up
A square’s value comes from where it sits, not from how many squares you shade. In this row, squares 1 and 4 are shaded, so the row adds up to 1 + 4 = 5.
Most targets can be made more than one way. A target of 5 in a five-square row could also be square 5 on its own, or squares 2 and 3. The puzzle is working out which way is right, using the crossing columns.
A small example, solved
Here is a 4×4 puzzle. The row targets are on the right and the column targets are underneath.
Step 1: start with the targets that have only one way
- Row 4 needs 2. The only way to make 2 is square 2 on its own. (Squares 1 and 1 would be the same square twice.)
- Column 4 needs 1. Only row 1 is worth 1, so that’s the only shaded square in the column.
- Column 2 needs 8. All four rows together make 1 + 2 + 3 + 4 = 10. A target of 8 is 2 short, so exactly the square worth 2 (row 2) is empty, and rows 1, 3 and 4 are shaded.
Cross out every other square in those three lines.
Step 2: finish the rows
- Row 1 needs 7. It already has squares 2 and 4, which make 6. It needs 1 more, so square 1 is shaded and square 3 is empty.
- Row 3 needs 5. It has square 2 already, and square 4 is crossed out. It needs 3 more, and the only square left that is worth 3 is square 3. Square 1 is empty.
- Row 2 needs 4. Squares 2 and 4 are crossed out, so the only way left is 1 + 3: shade squares 1 and 3.
Check the columns you haven’t used yet: column 1 now has rows 1 and 2 (1 + 2 = 3) and column 3 has rows 2 and 3 (2 + 3 = 5). Every target is met.
Where to start in any puzzle
Two ideas get every Kakurasu going.
Targets with only one way. At every grid size, exactly five targets can only be made one way: 1, 2, and the three largest. Look for these first.
| Grid | Largest target | One-way targets |
|---|---|---|
| 4×4 | 10 | 1, 2, 8, 9, 10 |
| 5×5 | 15 | 1, 2, 13, 14, 15 |
| 7×7 | 28 | 1, 2, 26, 27, 28 |
| 9×9 | 45 | 1, 2, 43, 44, 45 |
Count the empty squares instead. A line’s shaded and empty squares always add up to the largest target. So a big target is really a small target for the empty squares. In a 5×5 grid, a target of 13 means the empty squares add up to 15 − 13 = 2, so square 2 is the only empty one. That’s how column 2 fell in step 1 above.
Common mix-ups
- Using the wrong number. A square counts its column number towards its row, and its row number towards its column. The square in row 1, column 4 is worth 4 to its row but only 1 to its column.
- Counting squares instead of adding them. A target of 4 doesn’t mean four shaded squares. It could be one square (square 4) or two (squares 1 and 3).
- Confusing it with Kakuro. The names are close, but Kakuro has you write digits. See the questions below.