1 of 3
Tap every digit that could go in the bulb.
Select all that apply
Level 3 · Lines
How to solve thermo sudoku
Digits rise from the bulb to the tip. Longer thermos squeeze the digits hard.
A thermometer is a thick line with a round bulb at one end. Digits along it must strictly increase as you move away from the bulb, so no two cells on a thermo are equal. Normal sudoku rules still apply.
Long thermometers squeeze the digits hard. A 9-cell thermo can only be 1 to 9 in order, and even a 4-cell thermo rules out several digits in every cell.
How setters write it
The same rule, worded the ways you’ll see it on puzzles.
- Digits along a thermometer increase from the bulb end.
- Digits strictly increase from the bulb.
- Slow thermo: digits increase or stay the same.
Worth memorizing
- On a thermo of n cells, the bulb is at most 10 − n and the tip is at least n. A 4-cell bulb is 1 to 6.
- Equal digits aren’t allowed unless the puzzle says “slow” thermo.
- A 9-cell thermo is exactly 1 to 9 in order.
Solving tips
Bound the bulb and tip
On a thermo of n cells, the bulb is at most 10 - n and the tip is at least n. A 4-cell thermo has a bulb of 1 to 6 and a tip of 4 to 9.
Give every cell a range
Count the cells below and above each cell. The third cell of a 5-cell thermo has two cells below it and two above, so it is 3 to 7.
Tight gaps fill in
If a thermo has a 2 and then a 6 with three cells between them, those cells must be 3, 4 and 5 in order.
Shared bulbs go low
When one bulb starts two thermos, it is smaller than every cell on both. A bulb that starts a 5-cell thermo and a 4-cell thermo is at most 5.
A worked example: a 5-cell thermo with a 4 in the middle
A 5-cell thermometer runs left to right along one row, with the bulb on the left. The third cell is a given 4. Elsewhere in the same row there is a 3 and an 8, and the column under the tip already has a 9.
- Start with the ranges from the table on this page. On a 5-cell thermo the cells run 1 to 5, 2 to 6, 3 to 7, 4 to 8 and 5 to 9. The 4 in the middle fits its range of 3 to 7, so nothing is broken yet.
- The 4 splits the thermo in two. The bulb and the second cell must both be below 4, so they come from 1, 2 and 3. The fourth and fifth cells must both be above 4, so they come from 5 to 9.
- The row already has a 3, so the two cells below the 4 must use 1 and 2. They increase from the bulb, so the bulb is 1 and the second cell is 2.
- Now the top half. The row already has an 8, and the tip's column has a 9. So the tip can only be 5, 6 or 7, and the fourth cell must be smaller than the tip.
- The tip can't be 5, because the fourth cell would need a digit above 4 and below 5. That leaves the tip as 6 or 7 and the fourth cell as 5 or 6. Write those as center marks and wait for another clue to break the tie.
Practice
2 of 3
What goes in the middle?
3 of 3
Tap every digit that could go at the tip.
Select all that apply
Reference table
| Cells on thermo | Bulb | Tip | Each cell, bulb to tip |
|---|---|---|---|
| 2 | 1 to 8 | 2 to 9 | 1 to 8, 2 to 9 |
| 3 | 1 to 7 | 3 to 9 | 1 to 7, 2 to 8, 3 to 9 |
| 4 | 1 to 6 | 4 to 9 | 1 to 6, 2 to 7, 3 to 8, 4 to 9 |
| 5 | 1 to 5 | 5 to 9 | 1 to 5, 2 to 6, 3 to 7, 4 to 8, 5 to 9 |
| 6 | 1 to 4 | 6 to 9 | 1 to 4, 2 to 5, 3 to 6, 4 to 7, 5 to 8, 6 to 9 |
| 7 | 1 to 3 | 7 to 9 | 1 to 3, 2 to 4, 3 to 5, 4 to 6, 5 to 7, 6 to 8, 7 to 9 |
| 8 | 1 to 2 | 8 to 9 | 1 to 2, 2 to 3, 3 to 4, 4 to 5, 5 to 6, 6 to 7, 7 to 8, 8 to 9 |
| 9 | 1 | 9 | 1, 2, 3, 4, 5, 6, 7, 8, 9 |
Going deeper
Where two thermos meet
Thermos often share cells. Two thermos can start from one bulb, one thermo can branch into two, or a thermo can run into the side of another. The rule applies along every path, so a shared cell must be bigger than everything below it on any path and smaller than everything above it on any path.
To bound a shared cell, count the longest path of cells below it and the longest path above it. If the longest path below a cell has 3 cells and the longest path above has 4 cells, the cell is at least 4 and at most 9 minus 4, which is 5. So it is 4 or 5, even when both thermos look short on their own.
Finding 1 and 9 in a crowded box
On a normal thermo, a 1 can only sit in a bulb, since every other cell has something smaller below it. A 9 can only sit at a tip, since every other cell has something bigger above it.
Use this when a box is packed with thermo cells. List the bulbs in the box and the cells with no thermo. The box's 1 must be in one of those. Do the same for 9 with the tips. When only one cell qualifies, place the digit. In the same way, a 2 in the middle of a thermo needs a 1 directly below it, so 2 can only go in a bulb or right after a cell that can be 1.
Slow thermos and equal neighbors
A slow thermo lets digits stay the same as well as rise. Two cells that share a row, column or box still can't be equal. Cells that touch side by side always share a row or column, so those neighbors still have to rise.
Equal neighbors are only possible where the line steps diagonally from one box into another. Look for those diagonal box crossings first. Everywhere else, a slow thermo behaves like a normal one, and the ranges in the table still apply to any stretch without such a crossing.
Common mistakes
Assuming each step goes up by 1
A thermo only needs digits to rise. The steps can be any size, so 2, 5, 9 is fine on a 3-cell thermo. Only a 9-cell thermo, or a stretch squeezed between two fixed digits, is forced into steps of 1.
Reading the thermo from the wrong end
The bulb is the round end, and it holds the smallest digit. On a thin line with no clear bulb, check the rules text. Some puzzles draw the bulb small, and a thermo that doubles back can make the ends hard to spot.
Using only the length ranges
The ranges from length are a starting point. The real gains come from the row, column and box. A digit already placed in the thermo's row removes it from every thermo cell in that row, which can shift the whole chain up or down.
In mixed-rule puzzles
Thermos mix well with sum rules. A thermo inside a killer cage gives the cage a minimum and maximum, since its digits must be all different and rising. A 3-cell thermo inside a cage of 7 must be 1, 2, 4. On a between line, a thermo cell has a range from both rules at once, so take the overlap. Where a thermo overlaps a German whispers line, the overlap is at most 2 cells long, because a whispers line switches between low and high while a thermo keeps rising. A white Kropki dot on a thermo means that step rises by exactly 1, and a black dot means the next cell is exactly double the one before it.
What the video shows
- Digits rise from the bulb end of a thermometer.
- 7 then 3 goes down, so this breaks the thermometer.
- 2, 5, 7, 8 climbs all the way. Equal digits aren’t allowed either.
Questions solvers ask
How do thermometers work in sudoku?
Digits increase along the thermometer, starting from the bulb. The bulb holds the smallest digit and the far end holds the largest.
Can digits repeat on a sudoku thermometer?
No. Each cell must be bigger than the one before it. Only a slow thermo allows equal digits, and the puzzle will say so.
What is a slow thermo in sudoku?
A slow thermo lets digits increase or stay the same along the line. Equal digits still can't share a row, column or box.
What goes in the bulb of a sudoku thermometer?
The smallest digit on the line. On a thermo of n cells, the bulb can be no higher than 10 minus n.
Do thermometer digits have to be consecutive?
No. Each digit must be bigger than the one before it, by any amount. 1, 4, 8 is a valid 3-cell thermo.
What happens when two thermometers share a cell?
The shared cell must satisfy both. It is bigger than every cell below it and smaller than every cell above it, on either thermo.
Can a sudoku thermometer cross into another box?
Yes. It keeps rising across the border. Its digits never repeat, since each one is bigger than the last.
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