Solved example: a curved 8×8 thermometer puzzle

Before reading, it helps to know how curved thermometers change the logic. The walkthrough below was generated by our solver using only single-line reasoning plus flow, and every step has the exact counts behind it.

The puzzle

This is puzzle #2000 (8×8, curved thermometers). It has 16 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 22 steps: 12 use a simple count, 7 use the squeeze, 2 use the ceiling, and in 8 the deduction spreads through tubes into other lines.

1. Step 1: Row 5

   Row 5 has a clue of 0, so none of its cells can hold mercury. Following the tubes, 2 cells toward the tip become air outside row 5.

   10 cells decided (outlined)

2. Step 2: Row 1

   Row 1 needs 7 mercury cells. Apart from the thermometer with its bulb at R1C4, the rest of row 1 can supply between 0 and 6 mercury cells, so this thermometer must put at least 1 of its cells into the line. Apart from the thermometer with its bulb at R1C1, the rest of row 1 can supply between 0 and 5 mercury cells, so this thermometer must put at least 2 of its cells into the line. Since mercury rises from the bulb, that fills the highlighted cells (the squeeze). Following the tubes, 2 cells back toward the bulb fill with mercury outside row 1.

   7 cells decided (outlined)

3. Step 3: Row 2

   Row 2 needs 3 mercury cells. Apart from the thermometer with its bulb at R1C4 (which crosses row 2), the rest of row 2 can supply between 2 and 5 mercury cells, so this thermometer must put at most 1 of its cells into the line. Apart from the thermometer with its bulb at R1C1 (which crosses row 2), the rest of row 2 can supply between 0 and 4 mercury cells, so this thermometer must put at most 3 of its cells into the line. Anything beyond that would overshoot the clue, so the highlighted cells are air (the ceiling).

   3 cells decided (outlined)

4. Step 4: Column 1

   Column 1 needs 6 mercury cells. Apart from the thermometer with its bulb at R4C1, the rest of column 1 can supply between 2 and 5 mercury cells, so this thermometer must put at least 1 of its cells into the line. Apart from the thermometer with its bulb at R6C1, the rest of column 1 can supply between 2 and 4 mercury cells, so this thermometer must put at least 2 of its cells into the line. Since mercury rises from the bulb, that fills the highlighted cells (the squeeze).

   3 cells decided (outlined)

5. Step 5: Row 3

   Row 3 needs 5 mercury cells. Apart from the thermometer with its bulb at R2C8 (which crosses row 3), the rest of row 3 can supply between 0 and 4 mercury cells, so this thermometer must put at least 1 of its cells into the line. Since mercury rises from the bulb, that fills the highlighted cells (the squeeze). Following the tubes, 1 cell back toward the bulb fills with mercury outside row 3.

   2 cells decided (outlined)

6. Step 6: Row 2

   Row 2 already contains its 3 mercury cells, so its remaining open cells must be air.

   2 cells decided (outlined)

7. Step 7: Column 7

   Column 7 needs 4 mercury cells. Apart from the thermometer with its bulb at R8C6 (which crosses column 7), the rest of column 7 can supply between 1 and 3 mercury cells, so this thermometer must put at least 1 of its cells into the line. Since mercury rises from the bulb, that fills the highlighted cells (the squeeze). Following the tubes, 1 cell back toward the bulb fills with mercury outside column 7.

   2 cells decided (outlined)

8. Step 8: Column 6

   Column 6 already contains its 1 mercury cell, so its remaining open cells must be air. Following the tubes, 2 cells toward the tip become air outside column 6.

   7 cells decided (outlined)

9. Step 9: Row 6

   Row 6 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)

10. Step 10: Row 1

    Row 1 needs 7 mercury cells: 5 are already filled and exactly 2 open cells are left, so all of them are mercury.

    2 cells decided (outlined)

11. Step 11: Row 3

    Row 3 needs 5 mercury cells. Apart from the thermometer with its bulb at R3C4, the rest of row 3 can supply between 1 and 4 mercury cells, so this thermometer must put at least 1 of its cells into the line. Apart from the thermometer with its bulb at R4C1 (which crosses row 3), the rest of row 3 can supply between 1 and 4 mercury cells, so this thermometer must put at least 1 of its cells into the line. Since mercury rises from the bulb, that fills the highlighted cells (the squeeze).

    2 cells decided (outlined)

12. Step 12: Row 4

    Row 4 needs 4 mercury cells. Apart from the thermometer with its bulb at R4C4, the rest of row 4 can supply between 1 and 3 mercury cells, so this thermometer must put at least 1 of its cells into the line. Since mercury rises from the bulb, that fills the highlighted cells (the squeeze).

    1 cell decided (outlined)

13. Step 13: Column 4

    Column 4 needs 5 mercury cells. Apart from the thermometer with its bulb at R7C4, the rest of column 4 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. Following the tubes, 2 cells toward the tip become air outside column 4.

    4 cells decided (outlined)

14. Step 14: Column 1

    Column 1 already contains its 6 mercury cells, so its remaining open cells must be air.

    1 cell decided (outlined)

15. Step 15: Row 8

    Row 8 needs 4 mercury cells. Apart from the thermometer with its bulb at R8C3, the rest of row 8 can supply between 2 and 3 mercury cells, so this thermometer must put at least 1 of its cells into the line. Since mercury rises from the bulb, that fills the highlighted cells (the squeeze).

    1 cell decided (outlined)

16. Step 16: Column 5

    Column 5 needs 2 mercury cells: 1 is already filled and exactly 1 open cell is left, so all of them are mercury.

    1 cell decided (outlined)

17. Step 17: Row 4

    Row 4 needs 4 mercury cells. Apart from the thermometer with its bulb at R4C4, the rest of row 4 can supply between 2 and 3 mercury cells, so this thermometer must put between 1 and 2 of its cells into the line. Anything beyond that would overshoot the clue, so the highlighted cells are air (the ceiling).

    1 cell decided (outlined)

18. Step 18: Column 2

    Column 2 needs 6 mercury cells: 3 are already filled and exactly 3 open cells are left, so all of them are mercury. Following the tubes, 1 cell back toward the bulb fills with mercury outside column 2.

    4 cells decided (outlined)

19. Step 19: Row 7

    Row 7 already contains its 4 mercury cells, so its remaining open cells must be air. Following the tubes, 1 cell toward the tip becomes air outside row 7.

    3 cells decided (outlined)

20. Step 20: Column 3

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

    2 cells decided (outlined)

21. Step 21: Row 3

    Row 3 needs 5 mercury cells: 4 are already filled and exactly 1 open cell is left, so all of them are mercury.

    1 cell decided (outlined)

22. Step 22: Row 4

    Row 4 needs 4 mercury cells: 3 are already filled and exactly 1 open cell is left, so all of them are mercury.

    1 cell decided (outlined)

The finished grid

What to notice

• Flow does a lot of the work. Look at how often a step decides cells outside its own line.
• Bulbs can be far away. When a step fills a cell in the middle of a tube, the mercury runs back around corners to the bulb.
• Lines are counted, not tubes. A curved thermometer can put two or three cells into the same row, so count its cells line by line.

Next, try the advice for large 12×12 and 15×15 puzzles, or test yourself on a curved puzzle.

Questions