Draw a single loop through the centres of cells. Segments run horizontally or vertically between neighbouring cells.
The loop is one closed circuit: it never crosses itself, never branches, and never enters a cell twice. Every cell it uses has exactly two segments — it either passes straight through or makes a right-angled turn.
The loop must pass through every white and black circle. Cells without a circle may be used or left empty; the loop does not have to fill the grid.
A white circle must be passed straight through, and the loop must turn in at least one of the two cells immediately before or after it.
A black circle must be turned on, and the loop must run straight through the next cell on both sides of it.
The puzzle is solved when one closed loop satisfies every circle rule. Every published Masyu puzzle has exactly one valid solution.
A white circle in the top or bottom row has to run left–right: the up–down straight it would otherwise need would leave the board. One in the leftmost or rightmost column has to run up–down for the same reason. Draw both of its segments straight away — no deduction needed, just geometry.
Example: A white circle sits in the top row. Draw the segments joining it to the cell on its left and the cell on its right.
From a black circle the loop leaves in two perpendicular directions and must then run straight through the next cell, so it needs two cells beyond it in each direction it takes. A black circle in the first two rows can therefore only travel down, and one in the first two columns only right — with the mirror at the far edges.
Example: A black circle sits in row 2. It cannot go up, so it goes down, and the loop runs straight through the two cells below it.
Combine the two edge rules and a black circle inside the 2×2 block at any corner has only one legal pair of directions left. Everything about it — both segments and both straight runs beyond them — falls out at once.
Example: A black circle sits one cell in from the top-left corner. It must go right and down, with straight runs of two cells each way.
This is the single most productive first pass. Once you know which two directions a black circle takes, you get four segments, not two: the loop must continue straight through the first cell in each direction. Those extra cells are then straight-through cells with known neighbours, which usually decides something next to them.
Example: A black circle turns right and down. Mark the segment to its right and onward to the next cell, and the segment below it and onward to the next cell.
A white circle needs a turn in one of the two cells beside it. If three white circles sit in an unbroken row or column and all run along that line, the middle one has straight-through white circles on both sides and no turn anywhere near it. So at least one of the three runs across the line instead.
Example: Three white circles sit side by side in a row. They cannot all run left–right, so look for which of them is forced to run up–down.
If two black circles are orthogonally adjacent, the loop cannot run between them: the segment joining them would have to continue straight through the second circle, and a black circle always turns. Cross that join off before you do anything else with either of them.
Example: Two black circles sit next to each other in a row. Put a cross on the join between them; each must take its segments elsewhere.
Where a white circle sits directly in line with a black one, the two rules fit together: the black supplies the turn the white needs, and the white is the straight-through cell the black requires. Reading the pair together often settles both at once.
Example: A white circle sits directly to the left of a black circle. The loop runs straight through the white, into the black, and turns there.
Every cell carries exactly two segments or none. As soon as a cell has two, cross off both of its remaining joins — a third would be a branch, which is never legal. Doing this after every move is what keeps the board honest and turns up the next forced line.
Example: A cell already has a segment above it and one to its right. Cross the joins to its left and below it.
The other half of the same rule. A cell with a single segment must find a second one, so if three of its four joins are crossed, the fourth is a line. Chase these all over the board: each one completes a cell, which then crosses off more joins elsewhere.
Example: A cell has one segment coming in from above and crosses on its left, right and below — an impossible cell, so go back and check what boxed it in.
A closed circuit that leaves any segment outside it, or misses any circle, can never grow into the answer — there is only one loop in a finished Masyu. So a join whose two cells are already linked by a chain of segments is a cross, unless closing it would finish the whole puzzle.
Example: Two ends of the same chain come back next to each other with circles still unvisited elsewhere. Cross the join between them and route the chain the long way round.
The loop does not have to visit every cell. A cell with no circle that ends up fully crossed off is a perfectly good deduction, not a mistake — do not force the loop to fill the grid.
Example: A cell in the corner has crosses on both of its joins. Leave it empty and carry on.