Forced moves

A path end with exactly one open side must move there. That single rule solves most beginner boards on its own and opens almost every harder one. This lesson shows the rule on one 5×5 puzzle that falls to forced moves from the first cell to the last.

The one-open-side rule

A path end is either a dot you haven't drawn from yet or the tip of a path you've started. Look at its four neighbors. A neighbor is closed to that path when it is:

• outside the board (the wall) or a hole,
• already part of any path, including this path's own earlier cells,
• a dot of another color.

If exactly one neighbor is open, the path goes there. No guessing is involved, and the rule doesn't depend on the puzzle having a unique answer. If the puzzle can be solved at all, that path has to leave through its only open side.

Green's dot at row 3, column 4 has Blue's dot above it and Orange's dots to its left and below. The only open cell is to its right, row 3, column 5. Green's other dot at row 5, column 2 has the wall below, Blue's dot above and Red's dot to its right. Its only open cell is to its left, row 5, column 1.

Notice that neither Green dot needed any drawing first. Untouched dots are path ends too, and players who only extend paths they have already started miss these.

Chains of forced moves

A forced move creates a new tip, and that tip often has one open side as well. Keep going until the path reaches a real choice.

Follow Green from row 3, column 5. The cell below is Red's dot, so Green goes up to row 2, column 5. Blue's dot is to the left of that cell, so Green goes up again to the top-right corner. A corner has two neighbors and Green came from one of them, so it turns left to row 1, column 4. Blue's dot is below, so Green goes left again to row 1, column 3. There it finally has two open sides, row 1, column 2 and row 2, column 3, and the chain stops.

The other end runs the same way. From row 5, column 1 the only way on is up to row 4, column 1. Blue's dot is to the right of that cell, so Green goes up to row 3, column 1, where it has two open sides and stops.

Eight cells are filled, and every one of them was certain.

Rescan after every move

At the start, Blue's dot at row 2, column 4 had three open sides. Green's chain has since taken two of them, row 1, column 4 above and row 2, column 5 to the right. Blue now has one open side, to its left. A move that looked impossible to judge a minute ago is now forced.

This is why you rescan after each move. Every cell a path takes closes a side for each of its neighbors. The next forced move is usually right beside the last one.

Blue steps left to row 2, column 3. Green's tip is above that cell and Orange's dot is below, so Blue goes left again to row 2, column 2. Blue's first step took one of the two open sides of Green's tip at row 1, column 3, so that tip now has only row 1, column 2 open. Green runs left along the top row, turns down at the corner to row 2, column 1 and meets its other end at row 3, column 1. Blue, now walled in by Green above and to the left, drops to row 3, column 2 and joins its dot at row 4, column 2.

The joining case

Orange's dot at row 3, column 3 has Blue above and to its left and Green's dot to its right. Its one open side is down, row 4, column 3. That is an ordinary forced move.

Now look at Orange's new tip. Its partner dot is to its right at row 4, column 4. Blue's dot is to its left and Red's dot is below. Joining its partner is the only thing Orange can do, so it joins. A path end whose only option is its partner is as forced as one with a single empty neighbor, and the rule treats both the same way.

The condition matters. Two ends of one color can sit side by side while either still has other open sides. Then joining is just one option, and it is often wrong. How to play shows a board where two touching Green dots must not join directly.

Red finishes the board. Its dot at row 4, column 5 has Green above and Orange's dot to its left, so it moves down into the corner, turns left to row 5, column 4 and joins its dot at row 5, column 3.

How to spot forced moves quickly

On a fresh board, scan in this order.

• Corners and walls. A dot in a corner starts with two neighbors and a dot on the edge with three. One more blocked side and it's forced.
• Clusters of dots. Each dot of another color next to a dot closes one of its sides. Both Green dots above were forced only because other dots, and in one case the wall, surrounded them.
• Cells next to your last move. After each move, check the path ends that touch the cell you just filled. This is the rescan, and it's where chains continue.
• Both ends of every color. A path often stalls at one end while its other dot is forced. Green made progress from both ends before either chain stopped.
• Partners side by side. When a tip arrives next to its partner, check whether anything else is open. If nothing is, join.

Mark finished colors in your head as done. A completed path can't take part in anything further, so ignore it when you scan.

When nothing is forced

On harder boards the forced moves run out. Then switch from path ends to empty cells. A cell with only two usable neighbors tells you which path must come through it, which is the squeeze in Edges and corners. If that fails too, test a move and see whether it leaves a cell nobody can fill, as in Dead ends and stranded cells. Every new cell those techniques settle can start a fresh chain of forced moves, so return to this rule after each one.

One warning sign belongs here. If a path end ever has no open side at all, a move you made earlier was wrong. Undo back to your last guess or recheck your last few moves. That situation is the starting point of Bottlenecks.

To practice, try the forced-move puzzles, listed from easiest to hardest.