Why puzzles with one solution feel fair

Puzzles with one solution feel fair because the clues are always enough. When you're stuck between two moves, something on the board decides it, and finding that something is the fun. If a puzzle has two answers, sooner or later you're guessing, and a guess never feels like a win.

You know the moment. You're halfway through, two moves both look possible, and nothing seems to choose between them. On a well made board, that's never a dead end. Look a little longer and something always tips it.

What do puzzles with one solution promise?

A board with a single answer, sometimes called a unique solution, makes you three quiet promises.

• The clues are enough. Everything you need is already on the board, so you never need luck, trivia or a random tap.
• Every line has a reason. The right route is right because every other route breaks a rule somewhere. Sometimes you find that by trying an idea and watching it fail, but you never have to guess and hope.
• A contradiction means a mistake, not a bad board. If your lines stop fitting, the problem is somewhere in what you drew, so you know exactly where to look.

The same promise sits behind most well made logic puzzles for a calm break. Here's what it looks like on a real Linelet board.

Why do two answers feel unfair?

Say a board secretly has two solutions. One of two things happens.

The game might accept only one of them. Then you can follow every rule perfectly and still be told you're wrong, which is about the least fair thing a puzzle can do.

Or the game accepts both. That sounds kind, but it means there was a moment where no amount of thinking could pick between them. You weren't solving any more, you were guessing. Luck helped you finish, and that takes some of the shine off the last line.

Hints suffer too. A good hint points you to the step you missed. On a board with two answers, some of those steps simply don't exist.

How does Linelet make sure each board has one answer?

Linelet is the number line puzzle we make, and a board with two answers counts as broken. Before any board is allowed into the game, it goes through these steps.

1. Build the answer first.
   The generator starts by carving the whole board into lines that fill it exactly. Then it hides the lines and keeps only the matching ends and their numbers. So an answer is guaranteed to exist. The only question left is whether it's the only one.
2. Check the board is well formed.
   A separate checker looks at the shape of the puzzle. Every end has to be on the board and not on a wall, no two ends can share a square, every line length has to be reachable, and all the lengths must add up to the number of open squares.
3. Search for a second answer.
   A solver then tries every legal way to draw every line. It throws out dead ends early, using the same reasoning a good player uses: a line that can't reach its other end in the squares it has left, a line that's an odd number of steps off when it needs an even number, and a pocket of empty squares that nothing can fill. It stops the moment it finds two answers.
4. Only a complete search counts.
   A board is kept only if the search finished completely and found exactly one answer. If the search ever runs out of its allowance before finishing, the board is thrown away. "We didn't find a second answer" isn't the same as "there is no second answer".
5. Never trust the generator's word.
   When the library is built, every board is checked again, separately from the part that made it, before it's accepted.
6. Readable for everyone.
   Two pairs may share a number, but never the same number and the same shape. That way a player who can't tell colours apart can still tell every pair apart. A board that breaks this rule is rejected too.
7. Checked again, every time.
   Every finished board is stored in full, along with the seed it was made from. The app's automated tests re-check all 572 boards in the main packs, and they fail if any one is badly formed, has more than one answer, or wasn't fully searched. The twist and Very hard boards, which take far longer to search, are re-checked by sample.

Daily puzzles, event boards and freshly made boards are built on your phone by the same generator, and they go through the same search before you see them.

One small, true detail from building the library: proving that the biggest ten by ten boards have a single answer is hard work for a computer. In one pack, adding two more walls to each board cut the build time from about 47 minutes to about 8 seconds. Every wall removes routes the solver would otherwise have to rule out one by one.

How are Easy, Medium and Advanced decided?

Nobody picks the labels by hand. Each one comes from what the board actually contains: how many ways each line could go on its own, how many lines have only one possible route, how much the lines compete for the same squares, and how much searching the solver needed.

So a board always earns the same label, on every phone. There's more on that in our guide to how Linelet boards get harder from 5x5 to 13x12.

Using the one answer rule when you play

Knowing there's exactly one answer changes how you can play. A few habits make the most of it.

• About to guess? Look again. There's a reason somewhere. Count the gap between a pair's ends, or look for a corner or pocket that only one line can reach.
• Treat a contradiction as a clue. It tells you the last choice you made was wrong, so undo back to it.
• Use a hint to learn, not just to finish. It points at the line worth thinking about, which makes the next board easier.

Our guide on how to solve number line puzzles covers the deductions that do most of the work. You can try them right away on today's free Linelet puzzle.

Questions people ask

Does every logic puzzle have only one solution?

Well made ones usually do, and good puzzle setters treat it as a basic rule. But not every game or generator checks, so you can run into boards with more than one answer, especially in apps that make puzzles automatically.

Can a computer really prove there's only one answer?

Yes, for boards this size. The solver tries every possible way to draw every line, using shortcuts that rule out dead ends early. Because the search is complete, not a sample, finishing it with one answer is a proof.

If every board has one answer, why does Linelet have hints?

One answer doesn't mean an easy answer. Hints climb gently: first the line worth thinking about, then the first few squares of its route. You get those two free on every board, and the one-time purchase adds a third that shows the whole line.

What if I think I've found a different solution?

If your lines follow every rule, with each line exactly its number of squares, no crossings and every open square filled, then they're the answer. On a board with one solution, there's no other way to be right.