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Kakurasu One-Way Targets: Where to Start

A one-way target is a row or column target that only one set of squares can make. At every grid size there are exactly five: 1, 2 and the three largest. Each settles its whole line straight away, so fill those lines in before anything else.

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This is the first of the Kakurasu techniques.

Why there are exactly five

At the small end, 1 can only be square 1, and 2 can only be square 2 (you can’t shade square 1 twice). From 3 upwards there are always at least two ways: 3 is square 3 or squares 1 + 2, 4 is square 4 or squares 1 + 3, and it only gets more crowded from there.

At the large end, think about the empty squares instead. A line’s shaded and empty squares always add up to the largest target, so:

  • the largest target means no empty squares: every square is shaded;
  • one less means the empty squares add up to 1: only square 1 is empty;
  • two less means they add up to 2: only square 2 is empty.

Three less would mean the empty squares add up to 3, which is square 3 or squares 1 + 2: two ways again.

The five at each size

Grid Small targets Large targets
5×5 1 (square 1), 2 (square 2) 13 (all but 2), 14 (all but 1), 15 (all)
7×7 1 (square 1), 2 (square 2) 26 (all but 2), 27 (all but 1), 28 (all)
9×9 1 (square 1), 2 (square 2) 43 (all but 2), 44 (all but 1), 45 (all)
One-way target: a 9-square row with target 44 A row of nine squares with target 44, one short of the largest, 45. Only square 1 is empty; squares 2 to 9 are shaded. 1 2 3 4 5 6 7 8 9 44
44 in a 9×9 is one short of 45, so only square 1 is empty.
One-way target: a 7-square row with target 1 A row of seven squares with target 1. Only square 1 is worth 1, so it is shaded and squares 2 to 7 are crossed out. 1 2 3 4 5 6 7 1
1 can only be square 1. Everything else in the line is crossed out.

In a real puzzle

Here is the 7×7 puzzle used on every technique page.

A 7 by 7 Kakurasu puzzle An empty 7 by 7 grid. Row targets from top to bottom: 12, 4, 11, 23, 25, 17, 2. Column targets from left to right: 9, 25, 13, 7, 19, 10, 15. 1 2 3 4 5 6 7 1 2 3 4 5 6 7 12 4 11 23 25 17 2 9 25 13 7 19 10 15
Rows need 12, 4, 11, 23, 25, 17 and 2. Columns need 9, 25, 13, 7, 19, 10 and 15.

Only one line has a one-way target: row 7, which needs 2. That fills the whole row: square 2 is shaded and the other six squares are crossed out.

One-way targets at the start of a 7 by 7 puzzle Only row 7 has a one-way target: 2, which is square 2 alone. Square 2 is shaded and the rest of row 7 is crossed out. 1 2 3 4 5 6 7 1 2 3 4 5 6 7 12 4 11 23 25 17 2 9 25 13 7 19 10 15
Row 7 (target 2) is settled in one move.

Seven squares for one glance is a good start, but it’s only one line. The next technique finds squares in lines that have more than one way.

How to use it

  • Scan every target first. Look for 1, 2 and the top three for the grid size before doing anything else.
  • Fill the line completely, crossing out as well as shading. The crosses matter: they cut the combinations of every column they sit in (see crossing lines).
  • Know your top three. On a 7×7 that’s 26, 27 and 28; on a 9×9, 43, 44 and 45. They’re easy to miss because they look like any other big number.

Next steps

Questions

Does every Kakurasu have a one-way target?

No. Across 400 of our puzzles at each size, 14% of 5×5 puzzles, 40% of 7×7 and 21% of 9×9 had none at all. When there isn’t one, open with too big to fit and counting the empty squares instead.

Why isn't 3 a one-way target?

Because it can be made two ways: square 3 on its own, or squares 1 and 2. That is true at every grid size, so from 3 upwards every small target has at least two combinations.