How to Solve Thermo Sudoku

A thermometer gives one rule: the digits increase from the bulb to the tip. That single rule removes many candidates before you place a digit, and it keeps removing candidates every time you place one. This page shows how to read a thermometer, which thermometers to use first, and how to combine them with the normal sudoku rules. The rules of the variant are in What is thermo sudoku?.

1. Every thermometer cell has a minimum and a maximum

Count the cells of a thermometer. The bulb is the first cell. A cell that is the k-th cell from the bulb must be at least k, because k−1 smaller digits come before it. A cell that has m cells after it must be at most 9−m, because m larger digits come after it.

So in a thermometer of six cells, the bulb is 1 to 4, the second cell 2 to 5, the third 3 to 6, the fourth 4 to 7, the fifth 5 to 8 and the tip 6 to 9. Write these ranges as pencil marks before anything else. A thermometer of nine cells is fully solved by this alone: 1 to 9 in order.

Long thermometers are the strongest clues. Start with the longest one, then the second longest.

2. Placed digits tighten the ranges

Every digit on a thermometer changes the ranges of the cells on both sides of it. If the fourth cell of a thermometer is 5, the first three cells are at most 2, 3 and 4, and the cells after it are at least 6, 7, 8 and so on. Update the whole thermometer each time one of its cells is filled.

The same applies to digits that are not on the thermometer. If the bulb's row already has a 1 and a 2 elsewhere, the bulb is at least 3, and every later cell moves up by the same amount. Check the row, column and box of each thermometer cell for digits that push its range.

3. Bulbs and tips

The bulb is the smallest digit on the thermometer and the tip is the largest. A bulb on a long thermometer is a small digit, often 1 or 2. A tip on a long thermometer is a large digit, often 8 or 9. When a box contains the tips of two long thermometers, the 9 of that box is in one of them, and the 8 is probably in the other.

Look at where the small digits of a row can go. If the row contains a bulb with range 1 to 2 and no other cell can be 1, the bulb is 1. Thermometers make this kind of hidden single common, because their cells have narrow ranges.

4. Two thermometers in one region

When two thermometers pass through the same box, or run along the same row, their ranges compete. A box can hold only one 9, so two tips in one box cannot both be 9. One of them is at most 8, and that lowers every cell before it on that thermometer.

Thermometers that cross a row twice are useful too. If a thermometer has two cells in one row, the second cell along the thermometer is larger than the first, which fixes their order no matter where the other digits are.

5. Thermometers through a bend

Thermometers on this site may bend and may run diagonally. A diagonal step goes from one box into a neighbouring box without sharing a row or column, so those two cells are only linked by the thermometer. Do not treat them as if they were in the same row. A bend that turns back on itself puts several cells of the thermometer into the same box, and the box rule then applies: those cells hold different digits, which the thermometer already says, but they also use up the box's digits in a known order.

6. A solving order that works

  1. Write the range of every thermometer cell, longest thermometers first.
  2. Check each thermometer cell's row, column and box for digits that narrow the range further.
  3. Place the cells whose range is a single digit. Update the ranges along those thermometers.
  4. Look for hidden singles: a digit that fits only one cell of a row, column or box, now that the thermometer cells have narrow ranges.
  5. Continue with normal sudoku logic, returning to the thermometers after every placed digit.

A fully and correctly filled thermometer fades on the screen grid. That tells you the thermometer is done and you can concentrate on the others.

7. Mistakes to avoid

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