Solved example: a 6×6 thermometer puzzle, step by step

This walkthrough was produced by our solver, set to use only the reasoning a person would use: one line at a time, plus the rule that mercury fills from the bulb. It picks the simplest available line at each step, the same order an experienced solver would choose. If any term is unfamiliar, the glossary and the lesson on first moves explain them.

The puzzle

This is puzzle #1000 (6×6, straight thermometers). It has 12 thermometers and no pre-filled cells. Cells are named by row and column: R2C5 is row 2, column 5, counting from the top-left. Want to solve it yourself first? Open it in the game and come back if you get stuck.

Step-by-step solution

Each step uses one row or column (highlighted) plus the flow rule. Newly decided cells are outlined in blue. The whole solve takes 10 steps: 5 use a simple count, 0 use the squeeze, 0 use the ceiling, and in 1 the deduction spreads through tubes into other lines.

1. Step 1: Row 2

   Row 2 has a clue of 6, the full width of the grid, so every cell in it is mercury. Following the tubes, 2 cells back toward the bulb fill with mercury outside row 2.

   8 cells decided (outlined)

2. Step 2: Row 4

   Row 4 needs 4 mercury cells. Apart from the thermometer with its bulb at R4C1, the rest of row 4 can supply exactly 0 mercury cells, so this thermometer must put exactly 4 of its cells into the line. That fixes the highlighted cells: the mercury ones are needed to reach the clue and the air ones would overshoot it.

   6 cells decided (outlined)

3. Step 3: Row 3

   Row 3 needs 4 mercury cells. Apart from the thermometer with its bulb at R3C2, the rest of row 3 can supply exactly 1 mercury cell, so this thermometer must put exactly 3 of its cells into the line. That fixes the highlighted cells: the mercury ones are needed to reach the clue and the air ones would overshoot it.

   5 cells decided (outlined)

4. Step 4: Column 4

   Column 4 already contains its 3 mercury cells, so its remaining open cells must be air.

   3 cells decided (outlined)

5. Step 5: Row 1

   Row 1 needs 5 mercury cells: 1 is already filled and exactly 4 open cells are left, so all of them are mercury.

   4 cells decided (outlined)

6. Step 6: Column 1

   Column 1 needs 5 mercury cells. Apart from the thermometer with its bulb at R5C1, the rest of column 1 can supply exactly 4 mercury cells, so this thermometer must put exactly 1 of its cells into the line. That fixes the highlighted cells: the mercury ones are needed to reach the clue and the air ones would overshoot it.

   2 cells decided (outlined)

7. Step 7: Column 2

   Column 2 already contains its 4 mercury cells, so its remaining open cells must be air.

   2 cells decided (outlined)

8. Step 8: Column 3

   Column 3 needs 5 mercury cells. Apart from the thermometer with its bulb at R5C3, the rest of column 3 can supply exactly 4 mercury cells, so this thermometer must put exactly 1 of its cells into the line. That fixes the highlighted cells: the mercury ones are needed to reach the clue and the air ones would overshoot it.

   2 cells decided (outlined)

9. Step 9: Row 5

   Row 5 needs 3 mercury cells. Apart from the thermometer with its bulb at R5C6, the rest of row 5 can supply exactly 2 mercury cells, so this thermometer must put exactly 1 of its cells into the line. That fixes the highlighted cells: the mercury ones are needed to reach the clue and the air ones would overshoot it.

   2 cells decided (outlined)

10. Step 10: Row 6

    Row 6 needs 2 mercury cells and has exactly 2 open cells, so every one of them is mercury.

    2 cells decided (outlined)

The finished grid

What to notice

• Start with the extremes. The first steps use the clues that leave the least freedom.
• One line opens the next. Almost every step uses a cell decided in an earlier step.
• Use the "apart from" check. In the squeeze and ceiling steps, ask what the rest of the line can supply, and the thermometer's share follows. This is explained fully in the squeeze and the ceiling.

Ready for something harder? The curved 8×8 example shows how bent thermometers carry deductions across the board.

Questions