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 1360). 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, marking A1 would force G2, H2 and I2 cleared (every open cell left in this region sits in row 2, so its marker has to land there), then I3 (the last open cell in region 3), then H8 and H9 cleared (every open cell left in this region sits in column H, so its marker has to land there), and then region 8 would be left with no open cell for its marker. So A1 can't be the marker there.
  2. Next, marking D1 would force A8, E8, F8, and 4 more cleared (together, two regions have open cells only in rows 8 and 9), then I7 (the last open cell in region 8), then A2 cleared (every open cell left in this region sits in row 2, so its marker has to land there), and the chain continues, and then region 4 would be left with no open cell for its marker. So D1 can't be the marker there.
  3. Then, if E1 were the marker, it would force F9, G2, G3, and 2 more cleared (together, two regions have open cells only in columns F and G), then D9 (the last open cell in region 9), then B8 (the last open cell in region 6), and the chain continues, and region 5 would end up with no open cell for its marker, which can't happen. So it can't go there.
  4. After that, that one can't be the marker: at F1 it would force H3, H4, H5, and 5 more cleared (together, two regions have open cells only in columns H and I), then G3 (the last open cell in region 5), then I2 (the last open cell in region 3), and the chain continues, and region 7 would be left with no open cell for its marker.
  5. Now, marking G1 would force A2 and C2 cleared (every open cell left in this region sits in row 2, so its marker has to land there), then A9, B9, C9, and 2 more cleared (every open cell left in this region sits in row 9, so its marker has to land there), then B3, C3, D3, and 10 more cleared (every open cell in column A belongs to the same region, so that region's marker has to be somewhere in this column), and the chain continues, and then column D would be left with no open cell for its marker. So G1 can't be the marker there.
  6. From there, if H1 were the marker, it would force A2 and C2 cleared (every open cell left in this region sits in row 2, so its marker has to land there), then I4, I5 and I6 cleared (every open cell left in this region sits in column I, so its marker has to land there), then G3 (the last open cell in region 5), and the chain continues, and region 6 would end up with no open cell for its marker, which can't happen. So it can't go there.
  7. Following that, that one can't be the marker: at I1 it would force A2 and C2 cleared (every open cell left in this region sits in row 2, so its marker has to land there), then H3, H4, H5, and 2 more cleared (every open cell left in this region sits in column H, so its marker has to land there), then G3 (the last open cell in region 5), and the chain continues, and region 6 would be left with no open cell for its marker.
  8. First, every open cell in row 1 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.
  9. Next, every remaining candidate in this region touches C2, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  10. Then, every remaining candidate in this region touches H3, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  11. After that, that one can't be the marker: at A2 it would force C1 (the last open cell in region 2), then I3 (the last open cell in region 3), then H8 and H9 cleared (every open cell left in this region sits in column H, so its marker has to land there), and region 8 would be left with no open cell for its marker.
  12. Now, every open cell in row 2 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.
  13. From there, marking H2 would force I7, I8 and I9 cleared (every open cell left in this region sits in column I, so its marker has to land there), and then region 8 would be left with no open cell for its marker. So H2 can't be the marker there.
  14. Following that, if I2 were the marker, it would force H4, H5, H6, and 1 more cleared (every open cell left in this region sits in column H, so its marker has to land there), then G3 (the last open cell in region 5), then E7, E8 and E9 cleared (every open cell left in this region sits in column E, so its marker has to land there), and the chain continues, 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 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.
  16. Next, 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.
  17. Then, every open cell in row 3 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.
  18. After that, place a marker at A3. Column 1 has exactly one open cell left. Placing here clears the rest of row 3, column A, and its region.
  19. Now, together, two regions have open cells only in columns E and F; with one marker each, those two markers have to fill exactly those two columns, so every other region's open cells there can be cleared.
  20. From there, place a marker at D9. Region has exactly one open cell left. Placing here clears the rest of row 9, column D, and its region. Placing here also clears its 2 touching neighbours.
  21. Following that, place a marker at B8. Region has exactly one open cell left. Placing here clears the rest of row 8, column B, and its region.
  22. First, place a marker at C1. Region has exactly one open cell left.
  23. Next, place a marker at I7. Region has exactly one open cell left. Placing here clears the rest of row 7, column I, and its region. Placing here also clears its 1 touching neighbour.
  24. Then, 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.
  25. After that, every remaining candidate in this region touches F5, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  26. Now, place a marker at F6. Region has exactly one open cell left. Placing here clears the rest of row 6, column F, and its region. Placing here also clears its 1 touching neighbour.
  27. From there, place a marker at E4. Region has exactly one open cell left. Placing here clears the rest of row 4, column E, and its region.
  28. Finally, place a marker at H5. Region has exactly one open cell left.

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