The uniqueness trick

When you know a puzzle has exactly one solution, that fact is a clue. Any move that would leave room for a second solution has to be wrong, so you can rule it out without working out where it fails. Experienced players use this all the time, usually as "a path never makes a 2×2 block". That rule is mostly right, and the exceptions matter.

Why good puzzles have one answer

A puzzle with two answers can't be solved by pure reasoning. At some point both options survive every check, and you have to pick one. So puzzle makers aim for exactly one answer, and players come to rely on it.

Every puzzle on this site is checked by the exact solver before it is published. The solver finds one solution, then searches for any different set of paths that also fills the board. The puzzle is only kept if that second search comes back empty. How the solver works explains the method.

How a second solution hides

Here is a small board with exactly two solutions.

Compare the two. Orange is identical. The only change is who owns the two cells at row 2, columns 2 and 3. In the first drawing Blue takes them with a short detour. In the second, Green takes them with a bulge of its own. Both drawings fill every cell, so both are legal.

That is the simplest shape of a hidden second solution. One path makes a shallow detour. It steps aside by one cell, runs for two cells and steps back. Another path runs straight past the far side of those two cells. The two cells can then belong to either path, and nothing else on the board has to change. A puzzle with one answer can never contain this shape, in either version.

The 2×2 rule, stated carefully

In the first drawing, Blue fills a 2×2 block, rows 1 and 2, columns 2 and 3. Any 2×2 block of one color means the path touches itself. Two of its cells sit next to each other without being consecutive along the path.

The rule "never a 2×2 block" is airtight in Numberlink variants where paths don't have to fill every cell. There, a path that touches itself can always cut the corner and leave the skipped cells empty, which gives a second answer. In Flow-style puzzles every cell must be filled, so cutting the corner only works if another path can take over the cells that were given up. When one can, as Green can above, the block proves a second solution. When nothing can, the block may be part of the only answer.

Green's dots sit side by side at row 2, columns 3 and 4, and the corner above them has to be filled by someone. Any other color that got into row 1, column 3 would end up stuck at row 1, column 4, with only Green's dot below. So Green loops through the corner, and the solver confirms this is the only answer. The turn backs onto the edge of the board, so no other path is running along its outside to swap with.

So the accurate version of the rule is this. A 2×2 block of one color in a puzzle with one answer is impossible whenever another path runs straight along the outside of the turn, because the two paths could trade cells. When the outside of the turn is the edge of the board, a hole, a dot, or a path that turns away, the trade may not work, and the block can be part of the only answer.

A stricter rule on this site

The puzzle generator on this site goes further than uniqueness requires. It builds every puzzle from paths that never touch themselves. None of the published answers contain a path that runs right next to an earlier part of itself, so none contain a 2×2 block of one color. On this site's puzzles, you can reject any move that makes a path touch itself.

Blue has run along the top edge and turned down at row 2, column 5. Turning left to row 2, column 4 would put Blue directly below its own path at row 1, column 4, so you can reject it on sight. The longer reason is that it cuts Orange off from its partner, as the bottlenecks lesson shows for this same puzzle. This turn backs onto the edge of the board, so the general swap argument wouldn't catch it. Only the site's own rule does, and puzzles from other sources may not follow it.

Using it to rule out moves

In practice, the trick is a fast way to reject options before you test them by thinking ahead. Three patterns come up most:

1. A detour next to a straight run. If a move would send a path on a short detour that another path's straight run could take over, the move is wrong. The same goes for the other path bulging into a straight run beside it.
2. A region with more than one filling. If a move would wall off an area that only one color can enter, count the ways that color could fill it. An empty 3×3 square entered at one corner and left at the opposite corner can be filled row by row or column by column. Two fillings means the move is wrong.
3. A tiebreak between two survivors. When two options both survive a trial, ask whether either one leaves room for a swap. That option is the one to drop.

In a puzzle with one answer, a move ruled out this way would fail anyway. The trick tells you sooner, without making you find the contradiction.

Only for puzzles known to be unique

Everything here depends on the puzzle having exactly one answer. In a puzzle with two answers, the trick can rule out every answer. On the 4×4 board above, it rejects both drawings and leaves you with nothing. Puzzles from books, apps and other sites usually have one answer, but not always, and a board you build yourself in the solver's editor can easily have several. Use the trick freely on this site's puzzles, where uniqueness is proven. Elsewhere, treat it as a strong hint and confirm with ordinary checks before you rely on it.

The swap shape also works in reverse as a test. When you finish a puzzle from somewhere else, look over your answer for a shallow detour beside a straight run. If you find one, the puzzle had at least two answers, and any uniqueness shortcuts you took while solving it weren't safe.

A different kind of global argument, based on coloring the board like a checkerboard, is covered in checkerboard parity.