Follow one number around the board, alternating two kinds of link: a line where it has only two places, so one of them is it, and two squares that see each other, so they can't both be it. Where the chain's ends meet a square that sees both, the number goes.
123456789
123456789
125
2
8
7
6
4
4
3
2
5
8
8
1
9
4
2
125
2
4
27
6
8
3
8
3
5
6
1
4
7
7
6
4
2
8
3
4
27
8
27
1
9
3
5
6
3
6
5
8
4
9
5
6
4
3
27
8
1
The chainLoses a pencil mark
Strong and weak links
A line with two places gives a strong link: if one isn't the number, the other is. Two squares that see each other give a weak link: they can't both be it. The chain alternates them.
How to spot one
Pick one number and mark its two-place lines. Join them where a square of one sees a square of another, and look for ends that share a square.
What it gives you
Removals where the ends see a common square. It's colouring taken further, on one number.
Practise X-Cycles
Boards that need it, with the owl's hints when you're stuck.