Masyu Rules: How to Play, Strategies and Variations

What is Masyu?

Masyu is a loop-drawing puzzle. You get a grid of empty cells, some of which contain a white or black circle, often called pearls. Your job is to draw a single closed loop that passes through the centers of cells, moving only horizontally and vertically, and visits every pearl.

The two colors tell you how the loop behaves. At a white pearl the loop goes straight through, and turns just before or just after. At a black pearl the loop turns a corner, and goes straight on either side. That is the entire rule set, yet those two small instructions create long, elegant chains of logic.

Masyu belongs to the same family as Slitherlink, which also asks for one closed loop. The difference is that Slitherlink draws along the grid lines using number clues, while Masyu draws through the cells and uses no numbers or letters at all.

A short history

Masyu was designed and published by the Japanese puzzle company Nikoli, best known outside Japan for popularizing Sudoku. The goal was a puzzle that used no numbers or letters and yet kept real depth and a clean look on the page.

An early version appeared in issue 84 of Puzzle Communication Nikoli under the title Shinju no Kubikazari, meaning "pearl necklace". That first form used only white circles. Black circles arrived in issue 90, when the puzzle was renamed Shiroshinju Kuroshinju ("white pearls and black pearls"), and the extra circle type made the logic much richer. The short name Masyu was adopted in issue 103.

Because of that history, some English-language collections simply call it a pearl puzzle. You can read more in Wikipedia's Masyu article.

The core rules

A standard Masyu follows five rules. The first three describe the loop, the last two describe the pearls.

1. 1Draw one closed loopThe solution is a single loop with no loose ends. It moves from cell center to cell center, horizontally or vertically, so every turn is a right angle.
2. 2No crossings or branchesThe loop visits each cell at most once. It never crosses itself, touches itself or splits. Cells without pearls may be left empty.
3. 3Visit every pearlThe loop must pass through every white and every black circle in the grid.
4. 4White pearls: straight throughThe loop goes straight through a white pearl, and must turn in at least one of the two cells directly before or after it.
5. 5Black pearls: turnThe loop turns 90 degrees on a black pearl, and must go straight through the next cell on both sides. So each black pearl sends out two arms at least two cells long.

The two pearl rules mirror each other: a white pearl is straight with a turn nearby, a black pearl is a turn with straight lines nearby. Keeping that symmetry in mind makes the deductions much easier to remember.

A solved example

Here is a 6×6 Masyu with three black and three white pearls. It has exactly one solution. Try it before looking at the loop on the right.

The black pearls near the edges are the way in. The pearl on the bottom row cannot send an arm downward, so it must go up two cells. The pearl in the top row cannot go up, so it goes down two cells, and because it is only one cell from the left edge it cannot run two cells left either, so its other arm goes right. With those arms fixed, the remaining pearls link up around them, and there is only one way to close the loop.

How to play: a step-by-step method

This routine gets you started on any Masyu, and most small grids fall to it completely.

1. Start with pearls on or near the edges. A black pearl on the edge, or one cell in from it, must send its arm away from that edge. A white pearl on the edge must run parallel to it.
2. Draw every forced black-pearl arm. Each black pearl needs two arms of at least two cells, at right angles. If one direction is blocked, the opposite direction is forced.
3. Mark lines you know are impossible. Use a small cross on the edge between two cells when the loop cannot pass that way. Crosses are as useful as lines.
4. Extend every loop end. Every cell on the loop has exactly two connections. If a line end has only one open direction, extend it.
5. Check the white pearls' turns. When a line runs straight through a white pearl and one side is already straight, the other side must turn.
6. Avoid closing small loops. The solution is one loop. Never join two line ends if doing so would close a loop before every pearl has been visited.

Strategies experienced solvers use

These patterns come up again and again. Learning to spot them at a glance is what turns a slow solve into a fast one.

Most hard Masyu puzzles are solved by alternating between local pearl rules and global loop rules: the pearls fix short pieces of line, and the need for a single closed loop tells you how those pieces join. If you enjoy the global side, Nurikabe and Hashi use similar connectivity thinking.

Popular variations

Standard Masyu is by far the most common, but puzzle collections and competitions use a few twists.

• Gray pearls. Some or all circles are gray, and you must work out whether each behaves as white or black.
• Wrap-around grids. The grid is treated as a torus, so the loop can leave one edge and reappear on the opposite side.
• Region Masyu. The grid is split into regions, and the loop must turn at least once in every region.
• Hexagonal Masyu. Played on a hexagonal grid with 60 and 120 degree turns in place of right angles.
• White-only puzzles. A throwback to the original pearl necklace form, where every circle is white. Often used as a gentle introduction.

Common mistakes

The most frequent mistake is forgetting the second half of a pearl rule. It is easy to remember that a white pearl goes straight and a black pearl turns, and to forget the conditions on the neighboring cells. A white pearl with straight lines on both sides, or a black pearl with a turn right next to it, breaks the rules.

Second, beginners often close a small loop by accident. It looks tidy, but the answer is one loop through every pearl, so a closed section that leaves pearls outside is always wrong.

Third, solvers sometimes assume every cell is visited. Unlike some loop puzzles, Masyu allows empty cells, and many solutions leave a good share of the grid untouched.

Finally, watch out for branching. A cell on the loop has exactly two connections. If you find yourself drawing a third line into a cell, one of your earlier lines is wrong.

How Masyu compares to other puzzles

Masyu is one of the most approachable loop puzzles. Slitherlink asks you to count edges around numbered cells, which takes more bookkeeping, while Masyu's pearls give you short pieces of line almost immediately.

It also contrasts nicely with placement puzzles such as Star Battle: both are number-free, but Star Battle is about where objects go, while Masyu is about how one continuous path flows. Solvers who like one usually enjoy the other.

Frequently asked questions