QED Logic quod erat demonstrandum
Logic Ascent

How to solve this 9×9 logic puzzle

9×9 · sharp · 20 steps

A
B
C
D
E
F
G
H
I
1
2
3
4
5
6
7
8
9
1
2
3
4🐱
5
🐱
🐱
6
🐱
🐱
7🐱
8
🐱
9
🐱
🐱

This 9×9 board rates as sharp (difficulty score 1023). Solving it from scratch takes 31 logical steps, using 7 techniques: Single cell, Row/column exclusion, Adjacency clear, Region confinement, Line confinement, Shape squeeze, Forced chain. Every step below is forced: nothing here is a guess.

  1. First, place a marker at I1. Region has exactly one open cell left. Placing here clears the rest of row 1, column I, and its region. Placing here also clears its 1 touching neighbour.
  2. Next, every open cell left in this region sits in row 2, so its marker has to land there; that clears every other open cell in the row.
  3. Then, every open cell left in this region sits in column A, so its marker has to land there; that clears every other open cell in the column.
  4. After that, every open cell left in this region sits in column B, so its marker has to land there; that clears every other open cell in the column.
  5. Now, place a marker at C8. Region has exactly one open cell left. Placing here clears the rest of row 8, column C, and its region. Placing here also clears its 3 touching neighbours.
  6. From there, every open cell in row 9 belongs to the same region, so that region's marker has to be somewhere in this row; that clears every other open cell in the region.
  7. Following that, every open cell in column D belongs to the same region, so that region's marker has to be somewhere in this column; that clears every other open cell in the region.
  8. First, every remaining candidate in this region touches E3 and F3, so whichever candidate ends up marked will clear them, and they can be ruled out now regardless of which one wins.
  9. Next, every open cell left in this region sits in column G, so its marker has to land there; that clears every other open cell in the column.
  10. Then, every remaining candidate in this region touches H3 and F4 and H4, so whichever candidate ends up marked will clear them, and they can be ruled out now regardless of which one wins.
  11. After that, every remaining candidate in this region touches B6, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  12. Now, if F2 were the marker, it would force G4 (the last open cell in region 3), then E5 (the last open cell in region 6), then B3 (the last open cell in region 1), and column D would end up with no open cell for its marker, which can't happen. So it can't go there.
  13. From there, place a marker at E2. Region has exactly one open cell left. Placing here clears the rest of row 2, column E, and its region. Placing here also clears its 1 touching neighbour.
  14. Following that, if G3 were the marker, it would force B5 (the last open cell in region 1), then A7 (the last open cell in region 7), and region 6 would end up with no open cell for its marker, which can't happen. So it can't go there.
  15. First, place a marker at G4. Region has exactly one open cell left. Placing here clears the rest of row 4, column G, and its region. Placing here also clears its 2 touching neighbours.
  16. Next, place a marker at F7. Region has exactly one open cell left. Placing here clears the rest of row 7, column F, and its region.
  17. Then, place a marker at A6. Region has exactly one open cell left. Placing here clears the rest of row 6, column A, and its region. Placing here also clears its 1 touching neighbour.
  18. After that, place a marker at B3. Region has exactly one open cell left.
  19. Now, place a marker at D5. Region has exactly one open cell left.
  20. Finally, place a marker at H9. Region has exactly one open cell left.

The key move here was Forced chain. Once you can spot that, this board falls quickly. ∎