Solving thermometers
Digits must rise from the bulb, so each cell needs room for the cells on either side. On a long thermometer that alone rules out most digits, before you look at anything else.
Why it works
On the board above:
The thermometer runs from the bulb in r3c2 to the tip in r3c6. That is 5 cells, and each is bigger than the one before.
r3c2 has 4 cells above it, each at least one bigger. So r3c2 is 1 to 5.
The same count from the other end: r3c6 has 4 smaller cells below it, so r3c6 is 5 to 9.
The cells in between follow the pattern: r3c3 is 2 to 6, r3c4 is 3 to 7 and r3c5 is 4 to 8.
More moves
A given squeezes the line
A digit on the thermo caps every cell below it and lifts every cell above it. Low givens near the bulb force the cells before them.
r4c8 is a 3, and two cells sit below it on the thermo: r2c8 and r3c8.
They must rise to something less than 3, so they can only be 1 and then 2.
So r2c8 is 1 and r3c8 is 2.
A high bulb fills the line
When the bulb is high, the cells above it have few digits left. The same works for a low tip.
The bulb in r8c4 is a 6, and three cells rise from it.
Above 6 there are only 7, 8 and 9. The three cells take them in order.
So r8c5 is 7, r8c6 is 8, and the tip in r7c6 is 9.
How to use it
Count the cells
Note how many cells sit below and above the one you are looking at.
Work out the range
The lowest it can be is one more than the cells below it. The highest is 9 minus the cells above it.
Squeeze with givens
A digit on the line caps everything below it and lifts everything above it.
Check the houses
Cells of the thermo that share a row, column or box with a given still can’t repeat it.
Practice
8 boards. Each has one right answer, checked by our solver.
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Try it in a real puzzle
Each of these starts with thermo ranges.
Tips
The quick formula
On a thermo of n cells, cell k from the bulb is between k and 9 − n + k.
Two digits pin the middle
If the bulb and the tip are known and the gap equals the number of cells, the middle is forced.
Look at bent thermos too
A thermo that turns a corner still rises one step at a time. The count is the same.
Thermos meet
When two thermos share a bulb, the bulb is below the smallest cell of either.