Play Thermometers Online
Free thermometer puzzles — fill mercury from the bulb using row and column clues. Choose your grid size and difficulty. No account needed.
Created by Brian Hamilton
How to play Thermometers
Fill each thermometer with mercury so that every row and column holds exactly the number of filled squares its clue gives.
- Mercury always fills from the bulb and never has a gap — fill a square and every square below it is filled too.
- The reverse also holds: rule a square out and everything above it is empty.
- A thermometer may be empty, partly filled, or completely full.
- Every puzzle has exactly one solution.
Controls: click or drag to fill squares, and click again to mark them empty. Check highlights mistakes.
The bulb rule cuts both ways
Everyone reads the rule the same way: mercury rises from the bulb, so filling a square means every square below it is filled too. That half is obvious. The other half is where the puzzle is actually solved.
Ruling a square out works downwards from the tip. If a square cannot be filled, then nothing above it can be either — mercury would have to jump the gap. So every deduction you make travels in one direction or the other along the whole thermometer, never just one square.
That is why a thermometer is not really a set of independent squares. A thermometer of length L has only L + 1 possible states — empty, filled to one, filled to two, and so on. A five-square thermometer has six states, not thirty-two.
| Length | 2 | 3 | 4 | 5 |
|---|---|---|---|---|
| Share | 57% | 24% | 13% | 5% |
| Possible states | 3 | 4 | 5 | 6 |
Most thermometers are only two squares long, which means three states: empty, half, full. Once you think in states rather than squares, a 7×7 board stops being 49 unknowns and becomes about 17 small multiple-choice questions.
Start with the zeros
The fastest opening is a row or column whose clue is 0. Every square in that line is empty, and because of the bulb rule each of those empties then propagates up its own thermometer, often clearing squares several rows away.
About 14% of all lines on a generated board have a clue of zero — on a 9×9 that is typically two or three lines you can clear before doing any real thinking.
The mirror image is a line whose clue equals the number of thermometer squares in it: every one of those is filled. That is rarer, but worth checking, and note the count is of thermometer squares, not of squares in the row.
A worked board
The productive move after the zeros is cross-referencing. A thermometer that runs across several rows is constrained by every one of them, so a tight clue anywhere along its length limits how far the mercury can rise.
Where solvers get stuck
Treating squares as independent. Marking one square without following the consequence up or down its thermometer wastes most of the information. Every mark should travel.
Only ever filling. Proving squares empty is at least as useful, and usually easier — roughly seven squares in ten end up empty. A board is mostly emptiness, and finding it is how the filled squares get pinned down.
Forgetting a thermometer can be completely empty. There is no rule that a thermometer holds any mercury at all. Assuming every one has at least one filled square is a common and expensive mistake.
Two things you can check about these puzzles
Every puzzle has exactly one solution. The generator lays out thermometers, fills them, derives the clues, and then runs a solver that counts solutions and stops at two. A board is only kept when the count comes back as exactly one. 72 boards across every size and difficulty were re-checked against a separate counter written from the rules alone; all 72 had a single solution.
The board is now mostly thermometer. Earlier versions covered only about half the grid — 52% at 7×7 and 32% at 9×9 — leaving a lot of blank filler and a quarter of all lines with a clue of zero. The thermometer count has been raised as far as uniqueness allows: coverage is now around 90% at 5×5 and 7×7 and about 75% at 9×9, with generation still taking milliseconds.
Thermometers and its relatives
Thermometers is a modern puzzle rather than a Nikoli classic, and its closest relative is the Nonogram: both give you counts along rows and columns and ask you to reconstruct a picture. The difference is the shape constraint. A Nonogram's clue describes runs that can sit anywhere in the line; a thermometer's mercury must start at a fixed end, which is a much stronger restriction and makes the deductions more local.
If you like reasoning from line totals, Nonograms is the deeper version, and Tents and Trees uses the same count-along-the-edge idea with a placement rule instead.
More puzzles with row and column counts
If you like reconstructing a grid from its totals, try these: