How to start a thermometer puzzle: 5 opening moves

Every thermometer puzzle on this site can be solved without guessing, but the first deduction is often the hardest to see. Experienced solvers don't wait for inspiration. They run the same short checklist every time. Here it is. (New to the rules? Read how to play first.)

Move 1: Find the zeros and the full lines

A clue of 0 means the whole row or column is air. A clue equal to the grid size n means the whole line is mercury. These are free deductions, and they are worth more than they look: a full line fills every thermometer that crosses it back to its bulb, and an empty line empties every crossing thermometer forward to its tip.

Next, check clues of 1 and n − 1. They aren't free, but they leave so little freedom that moves 2 and 3 usually settle them.

Move 2: The overlap rule for long thermometers

When a thermometer lies along a line, the rest of the line has n − L cells, where L is the thermometer's length. Even if every one of those cells were mercury, they could supply at most n − L. So if the clue c is larger than that, this thermometer has to make up the difference:

cells certainly filled = L + c − n  (when this is above 0)

Because mercury rises from the bulb, those certain cells are the first L + c − n cells counted from the bulb.

This is the same idea as the "simple boxes" technique in nonograms. If you know that puzzle, you already know this move. See how the two puzzles compare.

Move 3: Small clues put a ceiling on thermometers

The mirror image of move 2: a thermometer lying along a line with clue c can never fill more than c of its cells in that line. Every cell beyond the c-th from the bulb is therefore air, whatever the rest of the line does.

Small clues on long lines are the best place to use this. A clue of 1 on a row made of long horizontal thermometers removes almost every cell at once.

Move 4: Spread every certain cell along its tube

Each time you place mercury, trace the thermometer back to its bulb and fill every cell on the way. Each time you mark air, trace it forward to the tip and mark every cell on the way. In our game, auto-flood does this for you, but it pays to notice which other lines the spreading touched, because those lines have just changed.

Move 5: Re-scan only the lines you touched

After a change, you don't need to re-read the whole grid. Only the rows and columns that contain newly decided cells can produce new deductions. Keep a mental (or pencil) list of "dirty" lines and work through it:

1. Has the line reached its clue? Then every open cell in it is air.
2. Are its open cells exactly enough to reach the clue? Then they are all mercury.
3. Otherwise, re-apply moves 2 and 3 with the cells now known.

In the game you can click a clue to finish its line automatically whenever case 1 or 2 applies.

Putting it together

On a 6×6 puzzle, these five moves are often enough to solve the whole grid. Our solved 6×6 example uses nothing else. On 10×10 and larger grids they get you started, and the full squeeze and ceiling reasoning takes over once lines are partly filled.

Questions