Edges and corners

Forced moves look at path ends. The squeeze looks at empty cells, and it finds moves that the forced-move scan misses. Corners and wall cells are where it applies most often, because the border has already taken some of their neighbors away.

The squeeze rule

Every empty cell must end up on some path. A cell without a dot can't be where a path starts or stops, so the path through it enters from one neighbor and leaves through another.

Count a cell's usable neighbors: empty cells, untouched dots, and path tips. Everything else is unusable. That means the wall, holes, the middle of any drawn path, a dot whose path has already left it, and any finished color.

If an empty cell has exactly two usable neighbors, the path through it uses both. What that tells you depends on what those two neighbors are:

• One is a path end. That path must enter the cell, and you can draw it now.
• Both are ends of the same color. That color runs through the cell and connects.
• Both are ends of different colors. The cell can never be filled, because no single path can come from both. An earlier move was wrong. This is one of the dead ends.
• Both are empty. You know the shape the path makes there but not its color yet. Remember the cell for later.

A cell with fewer than two usable neighbors is already lost. That is also a dead end.

Corners

A corner cell has only two neighbors to begin with, so every empty corner is a squeeze from the first move.

• Row 1, column 1. Its neighbors are Orange's dot at row 2, column 1 and the empty cell to its right. Orange must come up into the corner and turn right.
• Row 1, column 5. Its neighbors are Blue's dot at row 1, column 4 and the empty cell below. Blue must enter it.
• Row 5, column 1. Its neighbors are Green's dot at row 5, column 2 and the empty cell above. Green must enter it.
• Row 5, column 5. Both neighbors are empty. Some path turns through this corner, using the cells above and to the left of it. The color is still unknown.

Orange's dot has two open sides, so a scan for forced moves skips it. The corner settles its first move anyway.

From the corner, the rest of Orange is forced. At row 1, column 2, Green's dot is below, so Orange goes right. At row 1, column 3, Blue's dot is to the right, so Orange turns down to row 2, column 3. That cell touches Orange's other dot at row 2, column 4, and its other neighbors are Green's dot and Red's dot. Orange joins. One corner settled an entire color.

Wall cells next to a path

A cell on the wall that isn't a corner has three neighbors. As soon as a path passes through one of them, it becomes a squeeze just like a corner.

Look at row 3, column 1. The wall is on its left. Above it is Orange's dot, which is no longer usable because Orange has already left it. Only the cells to its right and below are usable, so whatever fills it turns between those two.

Green's dot at row 2, column 2 is now boxed in by Orange on two sides and Orange's dot on the third, so Green is forced down to row 3, column 2, beside the squeezed cell. Green enters row 3, column 1 and turns down to row 4, column 1. Blue's dot is to the right of that cell, so Green continues into the corner at row 5, column 1, the one we already knew Green would use, and joins its dot at row 5, column 2.

Blue fills its top-right corner and runs down the right wall. Its other end drops from row 4, column 2 to the bottom wall, follows it through the bottom-right corner and climbs to meet the first half. Red takes the last cell.

Why following the wall works so often

On the finished board above, each of the three paths that reach the border runs along it for several cells. That is common, and the squeeze rule explains it.

Suppose a path runs parallel to a wall with one row of empty cells between them. Each cell in that strip has the wall on one side and the middle of a path on the other, so its only usable neighbors are the strip cells on either side. Every cell in the strip is a squeeze, so the strip can only be filled by one path that enters at one end and runs along it to the other. Usually no color can do that, and the path beside the strip was in the wrong place.

Corners add to the effect. The path through a corner uses the wall cells on both sides of it, so a path that follows one wall into a corner is carried around onto the next wall.

So a path that reaches the border tends to stay on it, and "this path follows the wall" is often the best first guess on a stuck board. It is still a guess, so check it.

When following the wall fails

The most common warning sign is another color's dot one cell in from the wall, along the stretch you are about to cover. A path along the wall wraps around that dot and can seal it in.

On this board, Blue's dot sits in the top-left corner with Orange's dot below it, so its first move to row 1, column 2 is forced. There it can continue right along the wall or turn down.

Follow the wall. From row 1, column 3, Green's dot is below, so Blue must go on to row 1, column 4. Red's dot is below that, so Blue goes to the corner and turns down to row 2, column 5. From there, with Red's dot to the left, Blue drops to row 3, column 5. Green's dot at row 2, column 3 now has only one open side, so Green moves left to row 2, column 2 and then down to row 3, column 2, beside its partner. That leaves Blue's other dot at row 3, column 3 with a single open side, row 3, column 4. Blue takes it, and Red's dot at row 2, column 4 is left with Blue on three sides and Green's dot on the fourth. Red has nowhere to go.

A path end with no open side is a bottleneck. Every step after Blue's choice was forced, so the failure is certain, and Blue must turn down at row 1, column 2. Testing a choice this way is the subject of thinking ahead.

The wall still gets followed. Green does it instead of Blue, running from its dot at row 2, column 3 all the way around to its partner at row 4, column 2. Before you send a path along a wall, ask which color can take that stretch without trapping a dot behind it.

To practice, try the edges-and-corners puzzles, listed from easiest to hardest.