Dead ends and stranded cells

Every cell on the board has to end up inside a path. A path enters an empty cell from one side and leaves through another, so each empty cell needs two neighbors that a path could actually use. A move that takes that away from even one cell is wrong, however good it looks.

This lesson covers the three ways a move can strand a cell, how to check for them quickly, and a worked example where the obvious move fails.

What counts as a usable neighbor

When you look at an empty cell, count the neighbors a path could still come from:

• other empty cells,
• the tip of a partly drawn path,
• a dot whose path hasn't left it yet.

A cell that is already in the middle of a path doesn't count, and neither does a dot whose path has already set off in another direction. Both are spent. That second point is easy to miss, and it causes the first kind of dead end.

A cell with only one way in

If an empty cell has fewer than two usable neighbors, nothing can ever pass through it. Corners are the usual victims, because they start with only two neighbors.

Green's dot at row 5, column 4 sits next to the corner. Green's partner is up at row 4, column 1, so heading up to row 4, column 4 feels natural. But once Green leaves its dot that way, the dot is spent. The corner then touches only the empty cell above it, at row 4, column 5. A path could come into the corner from there and would have nowhere to go next. Green has to go right into the corner instead, and that is the move the puzzle's solution makes.

A cell trapped between two colors

The second shape is a cell with exactly two usable neighbors that belong to different colors. A path through that cell would have to join those two neighbors, and a path can't change color halfway.

Blue has come down the left edge from its dot at row 1, column 1, and its partner is at row 4, column 2, diagonally below. Cutting across to row 3, column 2 puts Blue right next to that dot. Now look at row 4, column 1. Above it is Blue's path, now spent. Its only usable neighbors are Blue's dot to the right and Purple's dot below. Any path through the cell would have to run from one of those dots to the other, which would join Blue to Purple. So Blue goes straight down through row 4, column 1 and turns right into its dot.

This trap only needs the two neighbors to be path ends of different colors. If both were ends of the same color, that color would simply connect through the cell.

A pocket nobody can finish

The third shape involves a whole group of empty cells. Once a move walls off a pocket, any color that goes inside can't come back out, so the pocket can only be filled by a color whose two ends both border it. If no color has both ends touching the pocket, it's dead.

Blue's dot at row 1, column 5 has already been forced one step left. Its partner is at row 3, column 4. Continuing along the edge to row 1, column 3 looks harmless, but it closes off the cells at row 1, columns 1 and 2, and row 2, column 1. The only things touching that pocket are Blue's tip, one Orange dot (row 2, column 2) and one Red dot (row 3, column 1). Orange's other dot is at row 4, column 2 and Red's is at row 4, column 4, so neither can enter the pocket and get back out. Blue would be in the same position. So Blue turns down at row 1, column 4 instead.

How to scan for them

You don't need to recount the whole board after every move. A move changes very little. The cell you moved into is filled, and the cell you moved from has just gone from a usable tip to a spent middle. Only the cells around those two can have lost a usable neighbor. Check them in this order:

1. Neighbors of the cell you just left. This is where dead cells appear, as with Green's corner above.
2. Neighbors of your new tip. Count each one's usable sides. A cell with exactly two, both of them path ends of different colors, is trapped.
3. Any wall you just touched. If your tip is now next to the edge, a hole, or another path, check whether you've closed a pocket. If so, list the colors that touch it and ask whether any of them touches it with both ends.

Corners and edge cells deserve a look first, because they start with fewer neighbors. A cell next to two or three dots is also fragile, because each dot stops counting the moment its path leaves.

Worked example: the direct route that fails

Here is a full puzzle. Try it first if you like.

The opening moves are quick. Orange's top dot can only go down, to row 2, column 6. The corner at row 6, column 1 has only two usable neighbors, and one of them is Blue's lower dot, so Blue runs into the corner and up the left edge to row 4, column 1.

Blue's top dot at row 1, column 5 now has two choices, left or down. Its partner is down at row 4, column 1, so down looks like the shortcut. Follow it one move further. At row 2, column 5, Blue's right side is Orange's path and its lower side is Red's dot, so it must step left to row 2, column 4.

Now look at row 3, column 4. Above it is Blue's tip. To its left is an Orange dot, to its right a Red dot, and below it a Green dot. Four neighbors, four different colors, and each of those colors has its other end somewhere else. The cell is a pocket of one, and nothing can fill it. Going down was a dead end.

So Blue goes left along the top edge. As it continues, the same habit keeps it right. Turning down at row 1, column 4 would cut Blue off from its partner, which is the subject of bottlenecks. Once Blue reaches row 2, column 1, turning right would strand a single cell again. Each time, the check takes a couple of seconds and saves you from untangling a broken board later.

When the failure only shows up several forced moves later, you need the habit described in thinking ahead. For practice, the dead ends practice set collects puzzles whose walkthroughs use these checks.