QED Logic quod erat demonstrandum

An Expert 9×9 Puzzle Solved with Forced Chain in 28 Steps

9×9 · expert · 28 steps

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This 9×9 board rates as expert (difficulty score 1386). Solving it from scratch takes 28 logical steps, using 8 techniques: Single cell, Row/column exclusion, Adjacency clear, Region confinement, Line confinement, Paired regions, Shape squeeze, Forced chain. Every step below is forced: nothing here is a guess.

  1. First, every open cell in column A 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.
  2. Next, together, two regions have open cells only in rows 1 and 2; 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 3, so its marker has to land there; that clears every other open cell in the row.
  4. After that, every remaining candidate in this region touches H2 and H4 and I4, so whichever candidate ends up marked will clear them, and they can be ruled out now regardless of which one wins.
  5. Now, every remaining candidate in this region touches G5, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  6. From there, that one can't be the marker: place it at F1, and row 2 would be left with no open cell for its marker.
  7. Following that, marking G1 would immediately leave row 2 with no open cell for its marker, so it can't be the marker there.
  8. First, if H1 were the marker, it would force F2 (the last open cell in region 2), then I3 (the last open cell in region 4), then G4 (the last open cell in region 6), and region 8 would end up with no open cell for its marker, which can't happen. So it can't go there.
  9. Next, that one can't be the marker: at I1 it would force F2 (the last open cell in region 2), then H3 (the last open cell in region 4), and region 6 would be left with no open cell for its marker.
  10. Then, place a marker at G2. Region has exactly one open cell left. Placing here clears the rest of row 2, column G, and its region. Placing here also clears its 1 touching neighbour.
  11. After that, place a marker at E1. Region has exactly one open cell left. Placing here clears the rest of row 1, column E, and its region.
  12. Now, place a marker at I3. Region has exactly one open cell left. Placing here clears the rest of row 3, column I, and its region.
  13. From there, every open cell left in this region sits in column F, so its marker has to land there; that clears every other open cell in the column.
  14. Following that, every open cell left in this region sits in column H, so its marker has to land there; that clears every other open cell in the column.
  15. First, marking A4 would force F5 (the last open cell in region 6), then H9 (the last open cell in region 8), then D8 (the last open cell in region 9), and the chain continues, and then row 6 would be left with no open cell for its marker. So A4 can't be the marker there.
  16. Next, if B4 were the marker, it would force F5 (the last open cell in region 6), then H9 (the last open cell in region 8), then D8 (the last open cell in region 9), and column C would end up with no open cell for its marker, which can't happen. So it can't go there.
  17. Then, that one can't be the marker: at C4 it would force F5 (the last open cell in region 6), then H9 (the last open cell in region 8), then B6 (the last open cell in region 5), and row 7 would be left with no open cell for its marker.
  18. After that, marking D4 would force F5 (the last open cell in region 6), then H9 (the last open cell in region 8), and then region 9 would be left with no open cell for its marker. So D4 can't be the marker there.
  19. Now, place a marker at F4. Row 4 has exactly one open cell left. Placing here clears the rest of row 4, column F, and its region.
  20. From there, every remaining candidate in this region touches C6, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  21. Following that, that one can't be the marker: place it at A5, and region 5 would be left with no open cell for its marker.
  22. First, marking B5 would force H9 (the last open cell in region 8), then D8 (the last open cell in region 9), and then row 6 would be left with no open cell for its marker. So B5 can't be the marker there.
  23. Next, if C5 were the marker, it would force H9 (the last open cell in region 8), then D8 (the last open cell in region 9), then B7 (the last open cell in region 7), and row 6 would end up with no open cell for its marker, which can't happen. So it can't go there.
  24. Then, place a marker at B6. Region has exactly one open cell left. Placing here clears the rest of row 6, column B, and its region. Placing here also clears its 2 touching neighbours.
  25. After that, place a marker at D7. Row 7 has exactly one open cell left. Placing here clears the rest of row 7, column D, and its region.
  26. Now, 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.
  27. From there, place a marker at H5. Region has exactly one open cell left.
  28. Finally, place a marker at A8. Region has exactly one open cell left.

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