How to solve thermometer puzzles: the complete guide

1. The rules in thirty seconds

A thermometer puzzle is a square grid covered by thermometers. You fill some of them with mercury. Mercury always starts at the round bulb and rises toward the flat tip with no gaps. The number beside each row and above each column says how many cells in that line contain mercury. When every line matches its clue, the puzzle is solved.

Full lesson: How to play thermometer puzzles covers the rules, valid and invalid fills, the game controls, and a solved walkthrough of this exact 4×4 puzzle.

2. The key idea: each thermometer is one number

Because mercury can't skip a cell, a thermometer's state is just its fill level, the number of cells filled from the bulb. A 4-cell thermometer can be at level 0, 1, 2, 3 or 4, and nothing else. Every deduction you make is a statement about a fill level: "at least 2" fills two cells from the bulb, and "at most 1" turns everything after the bulb into air. Thinking in levels instead of single cells is the biggest step from beginner to confident solver. The math behind this is in the math of thermometer puzzles.

3. How to start: five opening moves

Every solve begins with the same scan:

1. Zeros and full lines. A clue of 0 empties a line; a clue equal to the grid size fills it.
2. The overlap rule. A thermometer of length L lying along a line with clue c in a grid of size n has at least L + c − n certain cells, counted from the bulb.
3. The ceiling shortcut. The same thermometer can fill at most c cells of that line; the rest are air.
4. Flow. Spread every certain cell along its tube.
5. Re-scan. Check only the lines that just changed.

Full lesson: How to start a thermometer puzzle: 5 opening moves.

4. The core techniques

Once the opening scan runs dry, ten techniques cover nearly everything you will meet: empty and full lines, flow to the bulb, air to the tip, the squeeze, the ceiling, capacity of crossing thermometers, facing thermometers, finishing lines by counting, bridging two lines, and proof by contradiction. The strategy guide shows each one with a before-and-after diagram.

Four of them, the squeeze, the ceiling, capacity and facing thermometers, are really one rule. Pick a line and a thermometer in it, and work out the least and most the rest of the line can supply. The thermometer must supply the difference:

share ≥ clue − (rest max)   ·   share ≤ clue − (rest min)

Full lessons: 10 solving techniques and, for the rule above in depth, the squeeze and the ceiling.

5. Watch complete solves

Seeing the techniques chain together on a real puzzle is the fastest way to learn them. We publish two fully solved puzzles, generated by a solver that uses only one row or column at a time, so every step is one a person can follow:

• Solved example: 6×6: 10 steps, each with a diagram and the exact counts behind it.
• Solved example: curved 8×8: 22 steps, showing deductions spreading around corners.

6. Curved thermometers

Curved thermometers bend around corners, so one tube can put several cells into the same line and carry a deduction across the board. The rules don't change, but the counting does: you track how many of a tube's cells fall in each line at each fill level.

Full lesson: How to solve curved thermometer puzzles.

7. Large boards: 12×12 and 15×15

Big grids use the same techniques with far more bookkeeping. A good workflow is an opening sweep of every line, then finishing one region at a time while keeping a list of lines that changed. When single lines give nothing, combine two neighbouring lines. On the largest boards, single-line reasoning alone usually fills most of the grid but not all of it, so expect to bridge lines or use a careful supposition near the end.

Full lesson: How to solve large thermometer puzzles, with measured statistics on what makes them hard.

8. Avoiding mistakes

Three slips cause most failed solves: leaving a gap in the mercury, filling from the wrong end when the bulb is on the right or bottom, and counting a crossing thermometer as more than one cell in a line. Turning on Highlight rule breaks in the game catches the first two instantly.

Full lesson: 8 common thermometer puzzle mistakes.

9. Vocabulary

Bulb, tip, mercury, air, fill level, flow, squeeze, ceiling, capacity, given cell: every term used in these lessons is defined in the thermometer puzzle glossary.

10. A learning path by level

For background on the puzzle itself, including how it compares with Thermo-Sudoku and nonograms and how our puzzles are made, see the complete guide to thermometer puzzles.

Every lesson in this collection

1. How to playThe three rules, the controls and a solved 4×4 example.Start here
2. Your first movesA five-step opening scan that cracks open any thermometer puzzle.Beginner
3. Solved example: 6×6A complete 6×6 puzzle solved one line at a time, with a diagram for every step.Beginner
4. Solving strategiesTen techniques with before-and-after diagrams.Core
5. The squeeze and the ceilingThe minimum-and-maximum fill reasoning behind most deductions, in depth.Intermediate
6. Solving curved thermometersHow bent tubes change the logic, and how to trace them.Intermediate
7. Solved example: curved 8×8A curved 8×8 puzzle solved step by step, showing flow across lines.Intermediate
8. Solving 12×12 and 15×15A workflow for big boards: regions, bookkeeping and when to bridge lines.Advanced
9. Common mistakesEight errors that trip up solvers, and how to avoid each one.All levels
10. GlossaryDefinitions of bulb, tip, flow, squeeze, ceiling and more.Reference

Questions