QED Logic quod erat demonstrandum

An Expert 9×9 Puzzle Solved with Forced Chain in 24 Steps

9×9 · expert · 24 steps

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This 9×9 board rates as expert (difficulty score 1274). Solving it from scratch takes 24 logical steps, using 7 techniques: Single cell, Row/column exclusion, Adjacency clear, Region confinement, Paired regions, Shape squeeze, Forced chain. Every step below is forced: nothing here is a guess.

  1. First, every open cell left in this region sits in column E, so its marker has to land there; that clears every other open cell in the column.
  2. Next, together, two regions have open cells only in rows 7 and 8; with one marker each, those two markers have to fill exactly those two rows, so every other region's open cells there can be cleared.
  3. Then, every open cell left in this region sits in row 6, so its marker has to land there; that clears every other open cell in the row.
  4. After that, every open cell left in this region sits in row 9, so its marker has to land there; that clears every other open cell in the row.
  5. Now, place a marker at A5. Region has exactly one open cell left. Placing here clears the rest of row 5, column A, and its region. Placing here also clears its 1 touching neighbour.
  6. From there, every remaining candidate in this region touches H3 and H4, so whichever candidate ends up marked will clear them, and they can be ruled out now regardless of which one wins.
  7. Following that, every remaining candidate in this region touches D7 and D8, so whichever candidate ends up marked will clear them, and they can be ruled out now regardless of which one wins.
  8. First, place a marker at E8. Row 8 has exactly one open cell left. Placing here clears the rest of row 8, column E, and its region. Placing here also clears its 2 touching neighbours.
  9. Next, if B1 were the marker, it would force C7 (the last open cell in region 7), then F2 cleared (every open cell in column D belongs to the same region, so that region's marker has to be somewhere in this column), then F3, G3 and F4 cleared (every open cell in row 2 belongs to the same region, so that region's marker has to be somewhere in this row), and column F would end up with no open cell for its marker, which can't happen. So it can't go there.
  10. Then, that one can't be the marker: at C1 it would force B7 (the last open cell in region 7), then F2 cleared (every open cell in column D belongs to the same region, so that region's marker has to be somewhere in this column), then F3, G3 and F4 cleared (every open cell in row 2 belongs to the same region, so that region's marker has to be somewhere in this row), and column F would be left with no open cell for its marker.
  11. After that, marking G1 would immediately leave column F with no open cell for its marker, so it can't be the marker there.
  12. Now, if H1 were the marker, it would force F2 (the last open cell in column F), 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, that one can't be the marker: at I1 it would force G4 (the last open cell in region 4), then H6 (the last open cell in region 6), then C9 (the last open cell in region 9), and the chain continues, and row 3 would be left with no open cell for its marker.
  14. Following that, marking B2 would force C7 (the last open cell in region 7), then F1 (the last open cell in row 1), and then column D would be left with no open cell for its marker. So B2 can't be the marker there.
  15. First, if C2 were the marker, it would force B7 (the last open cell in region 7), then F1 (the last open cell in row 1), and column D would end up with no open cell for its marker, which can't happen. So it can't go there.
  16. Next, if F2 were the marker, region 3 would end up with no open cell for its marker, which can't happen. So it can't go there.
  17. Then, marking D1 would force C4 (the last open cell in region 2), then I3 (the last open cell in region 4), and then column F would be left with no open cell for its marker. So D1 can't be the marker there.
  18. After that, place a marker at F1. Row 1 has exactly one open cell left. Placing here clears the rest of row 1, column F, and its region. Placing here also clears its 1 touching neighbour.
  19. Now, place a marker at D2. Column 4 has exactly one open cell left. Placing here clears the rest of row 2, column D, and its region.
  20. From there, place a marker at G3. Region has exactly one open cell left. Placing here clears the rest of row 3, column G, and its region.
  21. Following that, place a marker at I4. Region has exactly one open cell left. Placing here clears the rest of row 4, column I, and its region.
  22. First, place a marker at H6. Region has exactly one open cell left. Placing here clears the rest of row 6, column H, and its region.
  23. Next, place a marker at C9. Region has exactly one open cell left. Placing here clears the rest of row 9, column C, and its region.
  24. Finally, place a marker at B7. Region has exactly one open cell left.

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