Thermo Sudoku: thermometer sudoku rules and tips

In thermo sudoku, grey thermometers snake across the grid. Digits must rise from the round bulb to the far tip, which turns every thermometer into a little ladder of inequalities.

Thermometer sudoku is one of the friendliest variants to learn and one of the deepest to master: long thermometers squeeze their cells into tiny ranges, and the interplay with rows and boxes does the rest.

The rules

• Normal sudoku rules apply: 1 to 9 once in every row, column and 3×3 box.
• Digits strictly increase along each thermometer, starting at the bulb.
• “Strictly” means no two cells on a thermometer are equal. Steps can be any size: 2, 5, 9 is fine.
• Thermometers can bend, including diagonally, and can cross box borders.

How to start

Count the cells

On a thermometer of L cells, the cell in position k from the bulb is at least k (there are k − 1 smaller digits below it) and at most 9 − L + k (there are L − k larger digits above it). So a seven-cell thermometer has its bulb in 1–3 and its tip in 7–9.

Work from both ends

Any cell above the bulb is at least 2, and any cell below the tip is at most 8. Where a thermometer passes through a box, those small facts often decide where the 1s and 9s can go. Once one cell is known, push the bounds along the thermometer in both directions.

Let the box finish the job

A long thermometer inside one box uses most of that box’s digits. If a five-cell thermometer sits wholly in a box, the other four cells share the four digits it leaves out.

Common mistakes

• Thinking digits must rise by exactly one. Any increase is allowed.
• Reading a thermometer backwards. The bulb is always the smallest digit.
• Allowing two equal digits when a bent thermometer crosses into another box. Strictly increasing means no repeats at all.
• Forgetting that the same digit can appear on two different thermometers.

How the ranks scale difficulty

Every puzzle type on Hoot’s Puzzles comes in four ranks, named after the stages of a mathematical idea. Lemma is the warm-up: about a dozen more digits are given than logic strictly needs, and singles plus the variant’s own rule will carry you through. Proposition needs a real idea or two, typically pointing and claiming. Theorem is near-minimal, so almost every clue is load-bearing and you may need pairs. Conjecture gives the fewest digits and sparser rule clues: expect to think hard.

Higher ranks use fewer thermometers: a Lemma typically has around nine, a Conjecture around five, so each one has to work harder. Thermometers here are three to seven cells long.

There are 99,999 puzzles per rank, each with its own number, so you can replay one or share it. Every type also has a 30-puzzle collection that climbs from Lemma to Conjecture. Every puzzle is checked to have exactly one solution that you can reach by logic alone, with no guessing.

Keep exploring

• Arrow SudokuThe circled digit equals the sum along its arrow. Bound circles and arrows.
• German WhispersNeighbours on a green line differ by at least 5. Fives are impossible there.
• Between LinesEvery digit on a line sits strictly between its two circled ends.
• Solving techniquesSingles, pairs, cage arithmetic, line logic and Latin squares, all in one guide.
• All puzzle guidesEvery puzzle type on Hoot’s Puzzles, with rules and tips.