How to Solve Masyu Puzzles
Masyu first appeared around the year 2000 in the Japanese puzzle magazine Nikoli. It was originally invented by Yano Ryuoh, and called Shinju-no-Kubikazari, meaning “Pearl Necklace.” This is fitting, given that this is a loop puzzle. At the time, it had only white circles.
Later, to add a little more flavor to the puzzle, a reader calling themself Acetonitrile added the idea of black circles, which followed different rules. The two authors renamed the game Shiro-Shinju Kuro-Shinju, which meant “White Pearl, Black Pearl.”
However, many kanji have multiple readings, and the then-CEO of Nikoli, Maki Kaji, misread the title as Shiro-Masyu Kuro-Masyu. “Masyu” means “evil influence,” which the staff thought was hilarious, and it quickly became an inside joke. Eventually, the nickname stuck, and Masyu is how the puzzle is known today..
Rules
Draw a single loop which passes through all circles without branching or crossing itself.
- Line segments travel from the center of one cell to the center of an orthogonally adjacent cell.
- White pearls force the line to travel straight through them. It must turn 90 degrees in at least one of the cells before or after the pearl.
- Black pearls require a 90-degree turn within them. Line segments each must then travel straight through at least one cell after exiting.
- You are not required to use every empty cell.
Pass straight through white pearls. You must turn at least once before or after the pearl. 
You may also turn both before AND after the white pearl. 
Turn in a black pearl. Both segments must travel straight for at least one cell before they may turn again.
Basic Techniques to Solve
- Start at the border.
- Look at black pearls one cell away from the edge.
- Some groupings of pearls force patterns.
- Watch for new borders.
- Avoid trapping lines.
Start at the Border
As with many loop puzzles, your first lines will usually be around the edge. Here, the pearl restrictions cause two effects:
- A white pearl on the edge must have a line running parallel to the border.
- A black pearl on the edge will have one line perpendicular to the border.
A white pearl on the border always produces a parallel line. This is because a line must always pass all the way through a white pearl. 
The loop always turns in a black pearl and extends for two cells. So one on the border will always produce one of these options. 
Therefore, a black pearl on the border must have one line perpendicular to the edge.
Black Pearls Near the Edge
This deduction is similar to the one for a black pearl on the border. Remember that a line exiting a black pearl must extend for two cells before it may turn. If the black pearl is only one cell away from the border, the loop will turn again immediately. This is against the rules.
Therefore, a black pearl one cell away from the edge will act exactly like a black pearl on the edge. As we saw above, that means one of two possible lines parallel to the border, and a definite one perpendicular to the border.
A black pearl only one cell away from the border also forces a perpendicular line.
Pearl Group Patterns
Learning pearl patterns helps to solve Masyu puzzles. Here are a few common ones to get you started.
- Black Pearls – The loop must travel two cells before turning again after turning inside a black pearl. So any time you can’t draw a 2-cell straight line in one direction, there must be one pointing the opposite way.
- An adjacent pair of black pearls must each have one line pointing away from the other one. This is because the line has to turn, so there can’t be a line between the two pearls.
- Similarly, when a pair of black pearls appears on the border, or one cell away, they form a back-to-back pair of L shapes.
- White Pearls – Loop lines must pass through white cells, and turn at either or both cells around it. This results in some interesting effects with adjacent white pearls.
- Along the border, a pair of white pearls that are directly adjacent, or that have only one cell between them must have a line parallel to the border that turns at both ends. This is because the line can’t turn between the pearls, and they each require the line to turn at one end, minimum.
- Three or more white pearls in a row always contain parallel lines. A single line can’t pass through all of them without breaking the turning rule for at least one pearl.
- Combination – Look for a black pearl with two white pearls diagonally adjacent on the same side. If one leg from the black pearl runs between the white pearls, you will force a corner for at least one of them. This would force a branch by the white pearl, which is invalid. Therefore, one leg from the black pearl must always point away from the white pearls.
New Borders
Every time a line passes completely through a cell, whether or not it contains a pearl, you are creating a new “edge.” Look at pearls next to that edge, and apply the same border techniques we’ve already covered.
Don’t Trap Your Lines
Put simply, always leave an exit. Since you can’t create small loops or branches, look carefully at paths you create. Every line segment must have an exit point from any cell it enters. So when deciding which way a line has to cross a cell, be sure you’re not blocking a different line segment in the process.
Solving the Puzzle
Now that we’ve learned the basic techniques, we’re ready to solve our example Masyu puzzle.
At the Border
We start with an easy one. The black pearl in the corner must form – a corner! 
The rest of the black pearls on the edge must have perpendicular lines. 
We can draw parallel lines through these white pearls. Obviously, a couple will turn in the adjacent corners.
This line can’t continue forward. If it did, the white pearl would either have a vertical line with no exit, or a horizontal line that closes a loop. 
So we draw down through the white pearl. Now, the line on the border must move down at least two cells to avoid a loop. Below the white pearl, it must turn.. 
Now we know the direction of the other leg for this black pearl. It can’t go left, so it must go right.
Moving Inward
Because they’re one cell inward, these black pearls must have lines pointing away from the border. 
A black pearl can’t have a leg turn in an adjacent cell. Therefore, at least one line must point away from the perpendicular line segments. 
This black pearl is next to a new “edge,” so one leg must point downward through both white pearls.
After passing through two white pearls, this line must turn. As a result, the line below it will continue straight into the black pearl. 
Speaking of white pearls, this one is next to a new edge. That means a vertical line passes through both pearls, and turns leftward at the top. 
This line has to turn to avoid creating a branch at the black pearl.
Completing Black Pearls
Because they’re next to new edges, we know the second legs for these black pearls. 
Once again, the second leg can’t go up, because you can’t turn next to a black pearl. Instead, the leg moves down through the white pearl. 
Notice this closed-in blue area. If we extend the line in the yellow cell forward, any line through the white pearl will be trapped. Therefore, it must turn.
This black pearl is against another edge, so draw a line to the left. 
Look below this black pearl. Because there isn’t a clear 2-cell path to draw a leg, the second line must go upward. 
The same applies to this black pearl. It can’t point right, so it must have a leg pointing left.
Making Connections
Let’s start connecting some of these segments together. First, each of these cells has only one valid exit point left. 
A horizontal line through this white pearl forces a loop. Instead, it must be vertical, and turn left at the top. 
The segment on the left can’t go forward, so it has to connect downward. Meanwhile, the segment on the right must extend down at least four cells.
These tips can never connect to each other, or they’d complete a loop. So they both extend downward. 
I think it’s about time we completed this last black pearl. Draw the downward leg. 
Once again, we have two cells with only one remaining exit point.
Completing the Loop
This tip must extend forward, because an upward turn would trap it. 
Doing so forces this line to turn downward. 
And now we connect this line to the left.
Once again, these tips can’t connect without creating a premature loop. So they both extend all the way down. 
The final connection of these two lines is obvious now. 
Completing the entire loop solves our Masyu puzzle!
