Thermometer Sudoku: How to Use Order and Extremes
A Thermometer Sudoku draws thermometers across the grid. Each one starts at a round bulb and runs to a tip, and the digits along it must increase strictly from the bulb onwards.
Strictly is the operative word. Each digit is larger than the one before it, by any amount. A thermometer reading 1, 4, 9 is perfectly ordinary; only 1, 4, 4 and 4, 1 are wrong.
The rules that do not change
Every row, column and 3x3 box still holds each of 1 to 9 exactly once, and the puzzle still has one solution reachable without guessing. The rules and the step-by-step guide cover the base game; naked singles and hidden singles is the technique underneath everything below.
Length alone tells you something
Before you know a single digit on a thermometer, its length has already constrained it. Count the cells below a position and the cells above it.
Each of the three thermometers here is three cells long. The bulb has two cells above it that must both be larger, so the bulb is at most 7. The tip has two below it that must both be smaller, so the tip is at least 3. The middle cell sits between 2 and 8.
That scales, and it scales sharply. A four-cell thermometer has a bulb of at most 6 and a tip of at least 4. A nine-cell thermometer is not a clue at all — it is the answer, 1 through 9 in order, with nothing left to decide.
The lesson is the one people find backwards at first: a longer thermometer is easier, not harder. Length is constraint, and constraint is what you are looking for.
One known digit reaches the whole thermometer
A digit anywhere on a thermometer pushes outward from where it sits. Everything before it must be smaller, everything after it must be larger, and the push does not stop at the neighbouring cell — it runs to the end.
The highlighted 5 sits at a bulb, r3c5. The two marked cells are the rest of that same thermometer: r4c4 first, then r5c5.
Take them in order. r4c4 comes straight after the bulb, so it is larger than 5. Ordinary Sudoku had it down to a 1 or a 6, and a 1 cannot be larger than 5, so r4c4 is the 6.
r5c5 is the one worth slowing down for. It sits two steps past the 5, and each of those steps has to increase, so “larger than 5” is not the real bound — r4c4 has to fit strictly in between, which leaves r5c5 at 7 or more. Its candidates were 1, 2, 6 and 8, and that bar clears out the 6 along with the small pair. r5c5 is the 8.
The 6 is the part to remember. It survives the loose reading of the rule and dies to the exact one, and the distance from the clue is the whole difference. Note also that none of this needed r4c4 solved first: the bound on r5c5 came from the bulb, two cells away.
Why the ends carry the information
The bulb and the tip are the most constrained cells on any thermometer, and the middle is the least. A bulb has every other cell on the thermometer above it; a tip has every other cell below it. A cell in the middle is squeezed from both sides but has fewer cells on each side doing the squeezing.
So when a thermometer puzzle stalls, work the ends. A bulb is where small digits are forced to live, and a tip is where large ones are. If you have a 1 to place in a region, a bulb is the first place to look for it; if you have a 9, look at the tips.
Try it
In Sudoku Fog, Thermometer is one of the variants you unlock rather than start with. Unlock points come from wins on Hard or above and from the daily challenge — one each, once a day — with bonus points at streak milestones.
Four ways this goes wrong
Reading the thermometer as a run of consecutive digits. Strictly increasing means each digit beats the one before it, not that it beats it by exactly one. 2, 5, 6 is a legal thermometer.
Losing track of which end is the bulb. The bulb is the round one, and it is the small end. A thermometer read the wrong way round gives you bounds that are exactly backwards, and every deduction after that is wrong.
Treating a shared cell as two separate problems. Thermometers are allowed to cross, and a cell where two of them meet has to satisfy both at once, taking the tighter bound from each side. That usually makes it the most constrained cell in sight. The three on this page keep to themselves, so nothing above depends on it, but it arrives the moment a board gets busier.
Working the middle. It is the loosest part of any thermometer. The ends are where the digits are pinned.
Where to go next
Kropki is the other variant whose clues describe relationships between cells rather than the contents of one, and its best trick is the opposite of this one: there, the clue that is missing is the clue that matters. King’s Move, Anti-Knight and X-Sudoku covers the variants that add excluded cells instead.
If you want the version where most of the grid is hidden while you solve, fog of war Sudoku is that. Everything published so far is on the Learn index.
Sudoku Fog is on Android — What Sudoku Fog is, Get it on Google Play.