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Daily Masyu

Monday 30th October 2017

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Completed battleships puzzle
This is an example of a completed Masyu puzzle. The aim is to draw a path round the grid so that it passes through every black and every white node. These are the rules,
  1. The path must turn on a black node. However, it must pass straight through the nodes before and after.
  2. The path must pass straight through a white node. However, it must turn on one (or both) of the nodes before and after.
This is an example of a starting Masyu grid. The first thing we do with a new puzzle is to look at the black nodes and the white nodes that are on the outside of the puzzle.
We will first look at the black nodes. We know that the path must form an 'L' shape as it passes through a black node. This means that one of the 'legs' from a black node must be horizontal, and the other vertical. For the four black nodes in this starting grid, there is only one vertical direction available to us, i.e. the direction away from the edge of the puzzle.
Since the path through a black node must be straight for two nodes in both directions, we can also apply the same logic for black nodes that are one cell away from the edge.
There's still a little bit more we can do with the black nodes, before looking at the easy white nodes. We know the path must turn on black circles, and then continue straight over the next node. Looking at the left black node on the bottom row, the path can't turn to the right here as that would mean going straight through the black node (and also forming a T-junction, which also isn't allowed). We can apply the same reasoning to the other black node on the bottom row.
We can also look at the black node on the second row. If the path turned to the left here, then it would have to carry on straight over the next node and form a T-junction with the existing path. That isn't allowed, so the path must continue to the right here.
For the black node on the top row of the puzzle, we need to fill in a little more of the puzzle before we can look at this node.
We can now look at the white nodes on the edge of the puzzle. We know that the path must pass straight through white nodes. Since the path can't leave the puzzle, this means that the path must move along the edge of the puzzle in this situation
Next, we will look at where we have 3 or more white nodes in a straight live. We know that when the path moves through a white node, one (or both) of the nodes before or after must turn. If we have only two white nodes in a line, then it is possible for the path to move through both white nodes and turn at the next node in both directions.
This isn't possible with 3 (or more) nodes, hence the path must cross through these white nodes in a perpendicular direction.
In the example in the bottom left, there is also 2 white nodes in a line as well, so the path must continue straight through both of these.
We can now go through the puzzle and fill in the path where there is only one option for it. We have made good progress on this puzzle by simply applying a basic set of rules that we would apply at the start of any puzzle. It does start to get a little trickier now though.
There are still a couple of path sections we can fill in for the bottom half of this puzzle since there is only one way the path could go here.
We can also look at the 2 white nodes in the top half of this puzzle. We know the path must pass straight through both of these white nodes. For the white node on the left, we know this can't be vertical as this would create a T-junction on the top row, i.e. it must be horizontal. For the white node in the middle, we know this can't be horizontally as that would create a T-junction to it's right, i.e. it must be vertical.
At this point in the puzzle it can be helpful to insert a 'X' where we know we can't have a link between two nodes. This makes it clearer where our path can and cannot go in this puzzle.
There are now more links we can insert where there is only one available path.
We will now look at this highlighted link. Just above this is a white node. We know that the path must pass straight through a white node, but it must turn at one (or both) of the nodes before or after. Looking at the path just above the white node, we can see that it carries on straight through the next node.
This means the path can't carry on straight as I've drawn the highlighted path here - it must turn to the left or to the right. We can then insert a 'X' for the highlighted node.
We have now arrived at this position. There are some obvious links we can insert in the top half of this Masyu puzzle - situations where there is only one way the path could go.
The interesting part of the puzzle here though is the two white nodes in the center of the puzzle. Due to the 'X' we just inserted, we know the path must pass straight through the two white nodes horizontally. Further, we also know the path must turn up or down either side either side of the new path drawn in - otherwise we'd be left with a path that passes straight through a white node and both nodes before and after, i.e. similar situation to previous step.
We have now almost solved the top half of this puzzle - this stage has 3 links that we can insert as there is no other options for the path to take.
Another two more links we can insert here.
Another two links we can insert here since there is only option for the path to take without creating a closed path and/or T-junctions.
The bottom section of the puzzle is becoming quite crowded now, with not many links left to insert. We can insert the highlighted links here since these are the only options we have available.
Again, by looking at where the path can go without creating closed loops, we can insert these links.
There is only one route for the path to take here, and that is the highlighted route. We have finished this Masyu puzzle!
We arrived at this solution by taking small logical steps along the way. Sometimes it is easier (and quicker) to look at the possible combinations of where the path can go, and work from there. That is always an option!

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