Thermo Sudoku: Rules and Beginner Strategy

Understand the increasing thermometer rule

Thermo Sudoku keeps the standard requirement that every row, column, and 3 by 3 box contains the digits 1 through 9 once. The extra rule applies to each thermometer: digits must increase strictly from the round bulb toward the narrow tip. Strictly means every next cell is larger, not merely equal or larger. A three-cell thermometer could contain 2, 5, 8, but not 2, 2, 8 or 6, 4, 9. Digits do not need to increase by exactly one unless the puzzle states an additional rule. Thermometers may cross rows, columns, and boxes, so their order restriction interacts with ordinary Sudoku peers. Shapedoku itself uses classic Sudoku rules and unordered shapes, so thermometers are not part of its normal boards. Still, the habit of checking rows, columns, and boxes carefully transfers directly between the two forms.

Calculate position bounds before listing candidates

Every thermometer position has a natural minimum and maximum. On a five-cell thermometer, the bulb cannot exceed 5, because four larger digits must still fit after it. The tip cannot be below 5, because four smaller digits must fit before it. More generally, the first cell of an N-cell thermometer can be at most 10 minus N, and the final cell can be at least N. Interior cells inherit bounds from both directions. On a five-cell line, the third position must be at least 3 and at most 7. These basic limits often remove many candidates before any placed digit is considered. Then tighten the bounds with clues. If the second cell is 4, every later cell must be above 4 and the bulb must be below 4. If the fourth cell is 7, earlier cells must fit below 7 while the tip must exceed it. Write compact candidate ranges rather than every digit at first.

Combine order with classic Sudoku restrictions

A thermometer does not override rows, columns, or boxes. If two cells on the same thermometer also share a box, they must be different for two independent reasons. Use placed digits in crossing units to narrow each range, then propagate the new limits along the entire line. Suppose a middle thermometer cell can only be 5 or 6. If its row already contains 6, that cell becomes 5, which immediately lowers the maximum values before it and raises the minimum values after it. Long thermometers deserve early attention because their positional bounds are tighter. Short two-cell thermometers are weaker, but they can still resolve a pair when one cell is limited to high digits and the other to low digits. Keep the order visible after every placement. A legal digit for one cell may still be impossible because it leaves no increasing sequence for the remaining cells.

1. Mark the bulb-to-tip direction on every thermometer.
2. Write the basic minimum and maximum for each position.
3. Apply row, column, and box eliminations within those bounds.
4. Propagate every new placement forward and backward along the thermometer.
5. Check whether each remaining candidate leaves enough smaller and larger digits.
6. Return to ordinary singles and subsets after the thermometer ranges tighten.