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Kakurasu Big Targets: Count the Empty Squares

Big Kakurasu targets are easier as small ones. A line’s shaded and empty squares always add up to the largest target, so the empty squares add up to the largest target minus the line’s target. Any square worth more than that can’t be empty, so it’s shaded. A 9-square row needing 40 has empty squares adding up to just 5, so squares 6, 7, 8 and 9 are all shaded.

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This is the third of the Kakurasu techniques.

The rule

  1. Find the largest target for the grid: 15 for a 5×5, 28 for a 7×7, 45 for a 9×9.
  2. Subtract the line’s target. That’s what the empty squares add up to.
  3. Subtract any squares already crossed out. What’s left is the room for more empty squares.
  4. Every open square worth more than that room must be shaded.
Counting empty squares: a 9-square row with target 40 A row of nine squares with target 40. The empty squares add up to 45 minus 40, which is 5, so squares 6 to 9 can't be empty and are shaded. 1 2 3 4 5 6 7 8 9 40
45 − 40 = 5, so the empty squares add up to 5. Squares 6 to 9 are too big to be empty, so they’re shaded.

Crossed-out squares use up some of the room, which makes the rule stronger as a puzzle goes on.

Counting empty squares with a cross: a 7-square row with target 20 and square 3 crossed out A row of seven squares with target 20 and square 3 crossed out. The empty squares add up to 28 minus 20, which is 8, and square 3 already uses 3 of that, leaving 5. Squares 6 and 7 are worth more than 5, so they are shaded. 1 2 3 4 5 6 7 20
28 − 20 = 8 for the empty squares, and crossed-out square 3 uses 3 of it, leaving 5. Squares 6 and 7 are shaded.

The same as “can’t do without it”

You may also see this put the other way round: if the rest of the line can’t reach the target without a square, that square must be shaded. In the row above, without square 7 the other open squares and the shaded ones reach at most 1 + 2 + 4 + 5 + 6 = 18, short of 20, so square 7 is needed.

That is the same rule in different words, and it always shades exactly the same squares. We checked it on every possible state of a 5-square and a 7-square line, for every target. Use whichever way of thinking comes more easily to you.

In a real puzzle

On the 7×7 used on every technique page, three lines have big targets.

Counting empty squares at the start of a 7 by 7 puzzle Row 4 needs 23, so its empty squares add up to 5 and squares 6 and 7 are shaded. Row 5 needs 25, leaving 3 for the empty squares, so squares 4 to 7 are shaded. Column 2 needs 25 as well, so rows 4 to 7 are shaded. 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 4 (23): empty squares add up to 5, so squares 6 and 7 are shaded. Row 5 (25): 3, so squares 4 to 7. Column 2 (25): rows 4 to 7.

That’s ten shaded squares from three subtractions. Together with too big to fit, which crossed out eight squares in the same puzzle, it gets a lot of the grid started without listing a single combination.

Its mirror

This is too big to fit turned inside out. Too big to fit crosses out squares worth more than the shaded squares still need; counting empty squares shades squares worth more than the empty squares can take. The mirror tip is the same idea applied to whole combinations.

Watch out for

  • The wrong largest target. 15 on a 7×7 makes every count 13 too small. See using the wrong total.
  • Forgetting the crosses. Crossed-out squares are empty squares too, and they count towards the total.

Next steps

Questions

What's the largest target for each grid size?

It’s every square shaded: 15 in a 5×5, 28 in a 7×7 and 45 in a 9×9. In general, the grid size times one more than itself, halved.

When should I count empty squares instead of shaded ones?

Whenever the target is more than half the largest. The empty squares then add up to less than the shaded ones, so they’re the easier thing to think about.