Thermometer puzzle strategies
Ten techniques that take you from beginner to expert, each with a diagram showing the board before and after. They are listed roughly in the order you'll need them.
The fastest way to solve a thermometer puzzle: fill or empty lines with clues of 0 or the full size, spread every certain cell along its thermometer (mercury back to the bulb, air forward to the tip), then compare each line's clue with the minimum and maximum its thermometers can supply. Repeat until solved.
1. Empty and full lines
A clue of 0 means every cell in that line is air. A clue equal to the grid size means every cell is mercury. These lines are always the best place to start, and they often unlock whole thermometers, because a full cell fills its bulb side and an empty cell empties its tip side.
2. Mercury flows down to the bulb
If you know a cell contains mercury, every cell between it and the bulb also contains mercury. On curved thermometers this can fill cells in several other rows and columns at once, which is often what breaks a hard puzzle open.
3. Air flows up to the tip
The reverse also holds. If a cell is air, every cell beyond it toward the tip is air, because mercury can never jump a gap. Marking one empty cell can clear a long thermometer.
4. The squeeze (minimum fill)
Compare what a line needs with what the other thermometers in it can supply. Below, the row needs 4. The single cell on the right can give at most 1, so the long thermometer must supply at least 3. Mercury fills from the bulb, so its first three cells are certain.
5. The ceiling (maximum fill)
The opposite of the squeeze. A thermometer lying along a line can never add more mercury to it than the clue allows. With a clue of 1, each thermometer below can fill at most its bulb, so every cell after the bulb is air.
6. Capacity of crossing thermometers
A thermometer that crosses a line only adds one cell to it, and only if its mercury reaches that far. Count the most each line could hold. If that maximum equals the clue, every crossing thermometer must reach the line.
7. Facing thermometers
When two thermometers point toward each other in the same line, each can only fill from its own end. A high clue forces both bulb ends to fill, and the uncertainty stays in the middle.
8. Finish lines with counting
Keep a running count. When a line already has as many mercury cells as its clue, every open cell is air. When its open cells are exactly enough to reach the clue, they are all mercury. In our game you can click the clue to do either automatically.
9. Bridge two lines
On larger boards, look at two neighbouring rows or columns together. Add their clues and compare the total with the most mercury the thermometers covering both lines could supply. A curved thermometer that passes through both lines often has to fill to a certain point to make the combined total work, even when neither line alone forces it.
The same idea extends to any block of lines. The sum of all row clues always equals the sum of all column clues, and that equal total is a handy sanity check on your progress.
10. Proof by contradiction
When nothing else works, choose a cell with only two possibilities and suppose it is mercury. Follow the consequences: fill to the bulb, update the counts, apply the techniques above. If you reach an impossible line (too many or too few mercury cells), the supposition was wrong, so the cell must be air. Our puzzles never require this, but it can be faster than finding a long direct chain.
A solving checklist
- Scan for 0 and full-size clues.
- Apply flow: mercury back to the bulb, air forward to the tip.
- For each line, ask: what is the least and the most it could still contain?
- Finish any line whose count is already met or whose open cells are exactly enough.
- Repeat steps 2–4 until nothing changes, then try bridging two lines.
- If you're still stuck, use a hint. It shows a cell that follows from the current board.
Common mistakes
- Forgetting where the bulb is. On curved thermometers the bulb may be far from the cell you're looking at. Trace the tube back.
- Counting a crossing thermometer as more than one cell. A vertical thermometer adds at most one cell to any row it crosses.
- Leaving a gap. Turn on Highlight rule breaks in Settings to flag gaps the moment they appear.
Put it into practice
10×10 puzzles need nearly every technique on this page.