Thermometer Sudoku Rules: How to Play, Strategies and Variations

What is Thermometer Sudoku?

Thermometer Sudoku, usually shortened to Thermo Sudoku, is a classic Sudoku with extra shapes drawn on the grid. Each shape looks like a thermometer: a round bulb at one end and a thick line running from it through a chain of neighboring cells. The rule is simple. Digits along a thermometer must strictly increase as you move away from the bulb, just as the mercury rises in a real one.

All the normal Sudoku rules still apply. Each row, column and 3×3 box must contain the digits 1 to 9 exactly once. The thermometers add ordering information on top, so a puzzle can start with very few givens, sometimes none at all, and still have a single logical solution.

That makes Thermo Sudoku a great first step into variant Sudoku. There is no arithmetic, as in Killer Sudoku; you only need to compare digits, the same skill used in Futoshiki. Yet the thermometers produce a style of deduction that feels quite different from standard Sudoku.

Where Thermo Sudoku comes from

As Sudoku spread around the world in the mid-2000s, variants with extra constraints drawn directly on the grid multiplied. Thermometers proved one of the most elegant: they are easy to explain, easy to see, and they combine with the Sudoku rules in deep ways. Thermo Sudoku is now a familiar sight in variant puzzle books and in puzzle competitions, including championships held under the World Puzzle Federation.

The thermometer idea also turned up in other puzzles. A separate logic puzzle called Thermometers asks you to fill each thermometer with mercury from the bulb upward so that row and column counts are met. It shares the shape and the bulb-first logic, but it is a shading puzzle rather than a number puzzle.

In recent years Thermo Sudoku has found a large new audience through online variant Sudoku, where setters often mix thermometers with other rules to build puzzles with striking, one-of-a-kind solving paths. For a wider view of the family, see the overview of Sudoku variants on Wikipedia.

The rules of Thermometer Sudoku

A Thermo Sudoku uses the three Sudoku rules plus one thermometer rule. Here is the complete set:

1. 1Rows contain 1 to 9Each row must contain every digit from 1 to 9 exactly once.
2. 2Columns contain 1 to 9Each column must contain every digit from 1 to 9 exactly once.
3. 3Boxes contain 1 to 9Each 3×3 box must contain every digit from 1 to 9 exactly once.
4. 4Thermometers rise from the bulbAlong every thermometer, each digit must be strictly greater than the one before it, starting from the bulb. Two cells on the same thermometer can never be equal.
5. 5Branches follow the same ruleSome thermometers split into branches. Every path from the bulb to any tip must increase, so the shared part is smaller than everything beyond it.

Thermometers do not have to be straight. They can turn corners, run diagonally and cross box borders, and the increasing rule applies all the way along.

A worked example

This mini 6×6 Thermo Sudoku uses the digits 1 to 6, with 2×3 boxes. There are only four givens, yet the thermometers are enough to force a unique solution. Look at the long five-cell thermometer in the bottom-left box. Because each step must climb by at least one, its bulb can only be 1 or 2 and its tip only 5 or 6. Every cell along it has a narrow range, and those ranges interact with the rows, columns and boxes to settle the grid.

Follow any thermometer in the solution from its bulb and you will see the digits climb: 1, 2, 3, 4, 5 along the longest one, and 3, 4, 5 on the short one at the top right.

How to play: a step-by-step method

Thermo puzzles reward a methodical start. This routine works on most grids:

1. Measure every thermometer. Count the cells on each thermometer. On a 9×9 grid, a thermometer of length L means the cell in position k can only hold a digit from k to 9 minus L plus k. A three-cell thermo has a bulb of 1 to 7 and a tip of 3 to 9.
2. Pencil in the ranges. Write the possible range next to each thermometer cell. Long thermometers often start with only two or three candidates per cell, which is a huge head start.
3. Look for forced digits. A thermometer of nine cells must read 1 to 9 in order. One of eight cells leaves only two options per cell. Where a range collapses to a single digit, place it.
4. Combine ranges with Sudoku logic. If a thermo bulb can only be 1 or 2, and 1 is already in its row, the bulb is 2, and the next cell must be at least 3.
5. Push limits along the thermometer. Every placed or restricted digit tightens its neighbors. A cell of 6 means everything before it is at most 5, 4, 3 and so on, and everything after it is at least 7, 8, 9.
6. Finish as a normal Sudoku. Once the thermometers are largely resolved, the rest of the grid usually falls to singles and pairs.

Strategies strong solvers use

Thermometers create their own family of techniques. These are the ones that come up again and again:

Keep your ranges written down and update them after every placement. In Thermo Sudoku the ranges are the puzzle, and most breakthroughs come from noticing that a range has quietly shrunk.

Popular variations

Thermometers mix well with other constraints. Common versions include:

• Slow thermometers. Digits must never decrease along the thermometer, but equal neighbors are allowed where the Sudoku rules permit. The rules will say when a thermometer is slow.
• Mini Thermo Sudoku. Smaller 4×4 or 6×6 grids like the example above, ideal for learning the range-counting technique.
• Thermo with no givens. Some puzzles start with an empty grid and rely entirely on the thermometers for information.
• Mixed variant puzzles. Thermometers are often combined with Killer cages, diagonal rules or inequality signs, each adding another layer of logic.
• Irregular Thermo. The 3×3 boxes are replaced by irregular nine-cell regions, so the thermometers interact with unfamiliar shapes.

Common mistakes

The classic error is reading a thermometer backwards. Always start from the round bulb: it holds the smallest digit, and values rise toward the far end. If a thermometer seems impossible, double-check which end the bulb is on.

Another is forgetting that the rule is strict. Neighboring cells on a standard thermometer can never be equal, even when they sit in different rows, columns and boxes. Only slow thermometers allow equal digits, and puzzles always say so.

Solvers also tend to under-use the ranges. It is easy to note that a bulb is small without writing down exactly how small. Write the full range for every thermometer cell at the start; it takes a minute and saves many more.

Finally, do not forget ordinary Sudoku. Thermometers usually cover only part of the grid, and hidden singles in the uncovered cells often provide the breakthrough you need.

How Thermo Sudoku compares

Compared with standard Sudoku, Thermo Sudoku asks you to think about order as well as placement. That makes the opening very different: rather than hunting for singles among the givens, you begin by working out ranges along each thermometer.

Compared with Killer Sudoku, there are no sums to calculate. Many solvers find thermometers a gentler introduction to variant Sudoku for exactly that reason. If you enjoy the comparison logic, Futoshiki is the natural next puzzle: it uses greater-than signs between neighboring cells on a Latin square grid.

Difficulty varies widely. A puzzle with a few long thermometers can be surprisingly gentle, because long thermos narrow their cells so much. Short thermometers scattered across an almost empty grid tend to be harder, since each carries less information on its own.

Frequently asked questions