QED Logic quod erat demonstrandum

An Expert 9×9 Puzzle Solved with Forced Chain in 38 Steps

9×9 · expert · 38 steps

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This 9×9 board rates as expert (difficulty score 1716). Solving it from scratch takes 38 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 remaining candidate in this region touches H2, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  2. Next, marking A1 would force F2 and I2 cleared (every open cell left in this region sits in row 2, so its marker has to land there), then D3, E3, F3, and 1 more cleared (every open cell left in this region sits in row 3, so its marker has to land there), and then region 4 would be left with no open cell for its marker. So A1 can't be the marker there.
  3. Then, if B1 were the marker, it would force F2 and I2 cleared (every open cell left in this region sits in row 2, so its marker has to land there), then D3, E3, F3, and 1 more cleared (every open cell left in this region sits in row 3, so its marker has to land there), and region 4 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 C1 it would force F2 and I2 cleared (every open cell left in this region sits in row 2, so its marker has to land there), then D3, E3, F3, and 1 more cleared (every open cell left in this region sits in row 3, so its marker has to land there), and region 4 would be left with no open cell for its marker.
  5. Now, if E1 were the marker, it would force A3, B3, C3, and 3 more cleared (every open cell left in this region sits in row 3, so its marker has to land there), then I2 (the last open cell in region 3), then B4, D4, A5, and 4 more cleared (together, two regions have open cells only in rows 4 and 5), and the chain continues, and region 8 would end up with no open cell for its marker, which can't happen. So it can't go there.
  6. From there, that one can't be the marker: at F1 it would force A3, B3, C3, and 3 more cleared (every open cell left in this region sits in row 3, so its marker has to land there), then I2 (the last open cell in region 3), then B4, D4, E4, and 5 more cleared (together, two regions have open cells only in rows 4 and 5), and the chain continues, and region 8 would be left with no open cell for its marker.
  7. Following that, marking G1 would force A3, B3, C3, and 3 more cleared (every open cell left in this region sits in row 3, so its marker has to land there), then I2 (the last open cell in region 3), then B4, D4, E4, and 5 more cleared (together, two regions have open cells only in rows 4 and 5), and the chain continues, and then region 8 would be left with no open cell for its marker. So G1 can't be the marker there.
  8. First, if H1 were the marker, it would force A2, B2, C2, and 1 more cleared (every open cell left in this region sits in row 2, so its marker has to land there), then A3, B3, C3, and 1 more cleared (every open cell left in this region sits in row 3, so its marker has to land there), then A4, B4, C4, and 2 more cleared (every open cell left in this region sits in row 4, so its marker has to land there), and the chain continues, 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 A2 it would force D1 (the last open cell in region 2), then E3, F3 and G3 cleared (every open cell left in this region sits in row 3, so its marker has to land there), and region 4 would be left with no open cell for its marker.
  10. Then, marking B2 would force D1 (the last open cell in region 2), then E3, F3 and G3 cleared (every open cell left in this region sits in row 3, so its marker has to land there), and then region 4 would be left with no open cell for its marker. So B2 can't be the marker there.
  11. After that, if C2 were the marker, region 2 would end up with no open cell for its marker, which can't happen. So it can't go there.
  12. Now, every remaining candidate in this region touches B4, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  13. From there, that one can't be the marker: at D2 it would force I1 (the last open cell in row 1), then A3 and B3 cleared (every open cell left in this region sits in row 3, so its marker has to land there), then E4, A5, B5, and 3 more cleared (together, two regions have open cells only in rows 4 and 5), and the chain continues, and region 8 would be left with no open cell for its marker.
  14. Following that, that one can't be the marker: at I1 it would force F2 cleared (every open cell left in this region sits in row 2, so its marker has to land there), then A3, B3, C3, and 1 more cleared (every open cell left in this region sits in row 3, so its marker has to land there), then D4, E4, A5, and 5 more cleared (together, two regions have open cells only in rows 4 and 5), and the chain continues, and region 8 would be left with no open cell for its marker.
  15. First, place a marker at D1. Row 1 has exactly one open cell left. Placing here clears the rest of row 1, column D, and its region.
  16. Next, together, two regions have open cells only in rows 2 and 3; 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.
  17. Then, together, two regions have open cells only in rows 4 and 5; 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.
  18. After that, if F2 were the marker, it would force H4 and I4 cleared (every remaining candidate in this region touches H4 and I4), then G8, G9, H3, and 3 more cleared (together, two regions have open cells only in columns G and H), then I3 (the last open cell in region 3), and region 8 would end up with no open cell for its marker, which can't happen. So it can't go there.
  19. Now, place a marker at I2. Row 2 has exactly one open cell left. Placing here clears the rest of row 2, column I, and its region.
  20. From there, every remaining candidate in this region touches F4, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  21. Following that, marking F3 would force H6, H7, H8, and 1 more cleared (every open cell left in this region sits in column H, so its marker has to land there), then G8 (the last open cell in region 8), and then region 6 would be left with no open cell for its marker. So F3 can't be the marker there.
  22. First, if G3 were the marker, it would force H5 (the last open cell in region 7), then F6 (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.
  23. Next, place a marker at E3. Region has exactly one open cell left. Placing here clears the rest of row 3, column E, and its region.
  24. Then, every remaining candidate in this region touches B7, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  25. After that, that one can't be the marker: at C4 it would force H5 (the last open cell in region 7), then A7 and A8 cleared (every open cell in row 9 belongs to the same region, so that region's marker has to be somewhere in this row), then F9 and G9 cleared (together, two regions have open cells only in columns F and G), and the chain continues, and column F would be left with no open cell for its marker.
  26. Now, marking G4 would force C5 (the last open cell in region 1), then A6 cleared (every open cell left in this region sits in row 6, so its marker has to land there), then B8 (the last open cell in region 5), and then column A would be left with no open cell for its marker. So G4 can't be the marker there.
  27. From there, 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.
  28. Following that, 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.
  29. First, together, two regions have open cells only in columns F and G; 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.
  30. Next, every remaining candidate in this region touches F7, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  31. Then, every open cell left in this region sits in row 8, so its marker has to land there; that clears every other open cell in the row.
  32. After that, every remaining candidate in this region touches G7, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  33. Now, place a marker at C7. Row 7 has exactly one open cell left. Placing here clears the rest of row 7, column C, and its region.
  34. From there, place a marker at A4. Region has exactly one open cell left. Placing here clears the rest of row 4, column A, and its region.
  35. Following that, place a marker at H5. Region has exactly one open cell left. Placing here also clears its 1 touching neighbour.
  36. First, 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.
  37. Next, place a marker at G8. Region has exactly one open cell left.
  38. Finally, place a marker at B9. Region has exactly one open cell left.

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