The rule: draw any line that cuts the grid into two parts. A single closed loop crosses that cut an even number of times — every time it goes out, it has to come back. So if all but one of the crossing edges are known, the last one is decided by whether the known crossings are odd or even.
Why crossings are always even
Walk around the loop from any starting cell. Each time you cross the cut you switch sides; to arrive back where you started you must have switched an even number of times. Mathematicians recognise this as a consequence of the Jordan curve theorem — a simple closed curve has a well-defined inside and outside — but for Masyu the walking argument is all you need. More on the maths in Masyu and mathematics.
Using a cut
- Pick a straight cut between two rows or columns, or around a region.
- Count the crossing edges that are already lines.
- If exactly one crossing edge is still undecided, it must make the total even.
Sealed regions
A special case of parity: if a region contains a pearl, the loop must cross into it at least once — so at least two crossings. When exactly two entrances remain, both are lines.
And if a move would leave a pearl’s region with only one entrance, that move is impossible. This is one of the most common contradictions in what-if chains.
Checkerboard colouring
Colour the grid like a chessboard. Every step of the loop moves to a cell of the other colour, so a loop alternates colours and always has an even length, visiting equally many light and dark cells. In standard Masyu this is rarely decisive on its own, but it’s a useful sanity check, and it powers parity arguments in related puzzles and in some variants.
When to reach for parity
- Large empty regions with few or no pearls — common on 20×20 and 25×25 boards.
- Long walls of ✕ marks that almost seal off part of the grid.
- Rows or columns where every crossing but one is decided.
Before reaching for parity, make sure you’ve exhausted the loop logic sweep — parity is powerful but slower to apply.