Thermometer Sudoku: Rules & How to Solve
Digits climb along each thermometer from bulb to tip. Learn how length alone pins down ranges of candidates.
The rule
Along each thermometer, digits must strictly increase starting from the rounded bulb and moving toward the flat tip. Standard sudoku rules apply everywhere else.
Thermometer range table
A thermometer's length does most of the work for you. Because digits strictly increase from the bulb, the cell in position k of an L-cell thermo is at least k and at most 9 − (L − k). That single fact narrows candidates before you look at a row or column.
| Thermo length | Bulb at most | Tip at least | Bulb range | Tip range |
|---|---|---|---|---|
| 2 | 8 | 2 | 1–8 | 2–9 |
| 3 | 7 | 3 | 1–7 | 3–9 |
| 4 | 6 | 4 | 1–6 | 4–9 |
| 5 | 5 | 5 | 1–5 | 5–9 |
| 6 | 4 | 6 | 1–4 | 6–9 |
| 7 | 3 | 7 | 1–3 | 7–9 |
| 8 | 2 | 8 | 1–2 | 8–9 |
| 9 | 1 | 9 | 1–1 | 9–9 |
Read the extremes off the ends: a 9-cell thermometer is completely forced — it must be 1, 2, 3, 4, 5, 6, 7, 8, 9 from bulb to tip. A 5-cell thermo already caps its bulb at 5 and floors its tip at 5.
VariantDoku shows the same reference inside the app: open any puzzle, tap the rules sheet, and each constraint on the grid gets its own tips card.
How to solve thermometer sudoku
Length sets the range
On a thermo with five cells, the bulb is at most 5 and the tip is at least 5 — every extra cell squeezes both ends inward.
The bulb is the floor
Because values only rise, the bulb holds the smallest digit on the thermometer. A long thermo forces a low bulb.
Crossings take the intersection
Where two thermometers share a cell, that cell must fit both thermometers' ranges at once — usually the tightest cell on the board, and the place to start.
Count backwards from the tip
Rather than starting at the bulb, ask what the tip can be. On a length-L thermo the tip is at least L, so a six-cell thermo has a tip of 6, 7, 8 or 9 before you look at anything else.
Two thermos that cross
Where thermometers intersect, the shared cell must satisfy both range windows at once. Intersecting a long thermo with a short one usually collapses the shared cell to one or two candidates.
Step by step: techniques from the hint engine
Each example is drawn the way VariantDoku's hints show it: red candidates are eliminated, blue ones are what the step proves, and green cells are solved.
Squeeze the cells around a known digit
A placed digit splits a thermometer into two shorter ones. Above the 7 on this five-cell thermo, two cells must both exceed 7 and keep increasing, and the only way is 8 then 9. Below it, the bulb and the second cell must climb to something under 7: the second cell is 2–6 and the bulb 1–5.
Two thermometers through one cell
Where thermometers cross, the shared cell must fit both ranges at once. It is the second cell of the horizontal thermo (2–6) and the fourth cell of the vertical one (4–8), so only 4, 5 and 6 survive. Every cell further along either thermometer is then pushed up, and every cell before it is pushed down.
A worked example
Take a four-cell thermometer running into a box you have partly solved. From the table, the bulb is at most 6 and the tip at least 4. Suppose the third cell is already known to be 7. Everything after it must be larger, so the tip is 8 or 9; everything before it must be smaller and strictly increasing, so the bulb and second cell form an increasing pair drawn from 1–6.
The bulb and second cell are still open, but a row can close them: if the bulb's row already contains 1 and 2, the bulb is at least 3, so the pair is {3,4}, {3,5}, {3,6}, {4,5}, {4,6} or {5,6} — and the tip still has only 8 or 9 to choose from.
On 4×4 and 6×6 boards
Ranges scale with the board. On an N×N board, cell k of an L-cell thermometer lies between k and N − L + k. A thermometer as long as the board is completely fixed — 1-2-3-4 on 4×4, 1 to 6 on 6×6 — and one cell shorter leaves every cell with exactly two candidates.
| Board | Length | Candidates, bulb to tip |
|---|---|---|
| 4×4 | 2 cells | 1–3 · 2–4 |
| 4×4 | 3 cells | 1–2 · 2–3 · 3–4 |
| 4×4 | 4 cells | 1 · 2 · 3 · 4 |
| 6×6 | 2 cells | 1–5 · 2–6 |
| 6×6 | 3 cells | 1–4 · 2–5 · 3–6 |
| 6×6 | 4 cells | 1–3 · 2–4 · 3–5 · 4–6 |
| 6×6 | 5 cells | 1–2 · 2–3 · 3–4 · 4–5 · 5–6 |
| 6×6 | 6 cells | 1 · 2 · 3 · 4 · 5 · 6 |
Try one from the catalog
Real VariantDoku puzzles with this rule. Each one opens in the app.
Frequently asked questions
Can digits repeat on a thermometer?
No — values strictly increase, so each cell on a thermometer is larger than the previous one.
Where does a thermometer start?
At the bulb, the rounded end. Digits increase from the bulb toward the flat tip.
How long can a thermometer be?
Up to nine cells in a 9×9 grid (a 1-to-9 run), though most are shorter, which is what makes the range logic useful.
What is the highest digit a thermometer bulb can hold?
10 minus the thermometer's length. A three-cell thermo has a bulb of at most 7; a six-cell thermo has a bulb of at most 4.
Can two thermometers share a cell?
Yes. The shared cell must obey both thermometers simultaneously, which is usually the most constrained cell on the grid and a good place to start.
Play Thermometer Sudoku in VariantDoku
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