Thermo Sudoku: prove the range before placing the digit

A thermometer is an ordered relationship

A grey line can turn several empty cells into a useful chain before any digit is placed. Grandmaster Puzzles’ Thermo rule requires numbers to rise strictly from the round bulb toward the flat end. [1] A sequence such as 1,3,7,9 obeys that order. It does not need to increase by exactly one.

The bulb fixes the direction even if the path bends or runs upward on the page. A repeated value is impossible along one thermometer, and a falling step is impossible too. Read the actual path through its cells; geometric closeness between two unconnected thermometers gives no extra rule.

Rows, columns and boxes still matter. A path sequence can be increasing yet collide with a fixed digit elsewhere in a row. Nikoli’s classic constraints supply those ordinary checks. [2] A valid candidate must survive both kinds of restriction.

Derive the limits instead of memorising them

Count the bulb as the first cell. On a six-cell thermometer, the first digit needs five larger digits after it. Its largest possibility is 4, because 4,5,6,7,8,9 just fits. The last digit needs five smaller predecessors, so it cannot be below 6.

More generally, position k on a length-L thermometer has lower bound k and upper bound 9−L+k. The lower bound leaves k−1 smaller values before it; the upper bound leaves L−k larger values afterward. These are deductions from strict increase and the digit range, independently calculated here.

For four cells, the bulb can be as high as 6 and the tip as low as 4. A two-cell thermo can have a bulb of 8 followed by 9. “Bulbs are small” is only a rough impression; the exact length determines the limit.

Use a known value as an internal boundary

A proved value splits the path into a smaller side and a larger side. If the third cell of a five-cell thermometer is 7, the fourth and fifth must be 8 and 9. There are exactly two available digits above 7, and their order is fixed. The first two cells are still unresolved.

Count distance when tightening a more remote position. A cell two steps before a known 7 cannot exceed 5, since an intervening larger value must fit before 7. A cell two steps after a known 4 cannot be below 6. These restrictions can propagate even when the intermediate cell is not solved.

The source example combining Killer and Thermo explicitly applies both constraints. [3] A cage sum on such a puzzle does not cancel the ordering rule. Every printed rule still participates.

Intervals are only the first filter

Suppose three successive thermometer cells have current candidates A={1,4}, B={2,5}, C={3,6}. Both 1,2,3 and 4,5,6 are increasing, and mixed paths may also work. Now a house restriction removes 6 from C, leaving C=3. B must fall below 3, so B=2, and A must fall below 2, so A=1.

This backward sweep is more precise than saying that a low tip makes the bulb low. It checks whether each candidate has support in the next cell. You can also sweep forward: a candidate below every surviving predecessor has no legal path into it and can be removed.

Do not reverse a sound comparison because a column contains high digits. A tip whose column already has 8 and 9 loses 8 and 9; it does not automatically lose 6. The old high-end values may indirectly restrict earlier cells, but each removal still needs its own inequality.

Read unusual shapes literally

If a puzzle presents branching thermometers, follow each bulb-to-tip path described by its rules. Two branch ends need not be ordered relative to one another merely because they share a bulb. Their ordinary house relationships may forbid equality, but that is a separate reason.

Likewise, a gap in the numbering is legal: 2 followed by 5 is an increase. Borrowing the Consecutive rule here would remove valid candidates. For a new shape or a combined variant, reread the author’s rule text before importing a shortcut from another puzzle.

Test exact bounds, not temperature metaphors

The current ZUDOKU Thermo destination is a Plus preview. The interactive check here works independently of it. For a complete puzzle, use the cited constructor example rather than treating these schematic paths as a uniquely solvable board.