Push each run to both ends
Push every run in a line as far left as it fits, then as far right. A square that the same run covers both times is filled, wherever the runs end up. A square that no run can reach is empty.
The drawing shows a row of 10 squares with the clue 4 3. Pushed left, the 4 covers squares 1 to 4 and the 3 covers squares 6 to 8. Pushed right, the 4 covers squares 3 to 6 and the 3 covers squares 8 to 10.
The 4 covers squares 3 and 4 both times, and the 3 covers square 8 both times. Those three squares are filled in every arrangement of the row, so you can fill them now.
A quick count gives the same squares. Add the runs and one gap between each pair: 4 + 1 + 3 is 8, so the row has 2 squares to spare. A run longer than the spare fills its length take away the spare. The 4 fills 2 squares and the 3 fills 1.
Empty squares come once a square is filled. A run must then cover that square, so it can only reach so far from it. Push it both ways again, and every square it cannot reach is empty. A clue of 0 empties the whole line.
Only the same run counts
The common slip is to lay the two pushes over each other and fill every square that is dark in both, without asking which run is there.
This row has 6 squares and the clue 2 1. Pushed left, the 1 sits on square 4. Pushed right, the 2 covers squares 3 and 4. Square 4 is filled both times, but by different runs.
That proves nothing. The third row puts the 2 on squares 1 and 2 and the 1 on square 5. Both clues still fit, and square 4 is empty. In this row no square is certain yet, so leave it and work on a crossing column.
Count the runs from the left, first, second, and compare each run only with itself.
Which squares can you mark empty?
This row has 9 squares and one run, a 4. Its crossing column has already filled square 3.
Push the 4 as far left and as far right as it can go while it still covers square 3.
Try it
Square 3 is filled. Which squares can you mark empty?
- Squares 7, 8 and 9
- Squares 1 and 2
Show the answer
Squares 7, 8 and 9
Correct. The 4 must cover square 3, so it starts at square 3 at the latest and ends at square 6 at the latest. Squares 7, 8 and 9 are out of its reach. Square 4 is filled too: every way of placing the 4 covers it.
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