QED Logic quod erat demonstrandum

An Expert 9×9 Puzzle Solved with Forced Chain in 29 Steps

9×9 · expert · 29 steps

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A
B
C
D
E
F
G
H
I
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9
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9×
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This 9×9 board rates as expert (difficulty score 1349). Solving it from scratch takes 29 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 left in this region sits in column A, 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 columns B and C; 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.
  3. Then, every open cell left in this region sits in column D, so its marker has to land there; that clears every other open cell in the column.
  4. After that, every remaining candidate in this region touches C7 and E7 and E8, 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 F6, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  6. From there, if B1 were the marker, it would force G2 (the last open cell in region 2), then C6 (the last open cell in region 6), then D8 (the last open cell in region 8), and the chain continues, and region 4 would end up with no open cell for its marker, which can't happen. So it can't go there.
  7. Following that, 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.
  8. First, if E1 were the marker, it would force F7 (the last open cell in region 3), then D8 (the last open cell in region 8), then A9 (the last open cell in region 7), and region 9 would end up with no open cell for its marker, which can't happen. So it can't go there.
  9. Next, marking B2 would force C6 (the last open cell in region 6), then D8 (the last open cell in region 8), then A9 cleared (every open cell left in this region sits in row 9, 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 B2 can't be the marker there.
  10. Then, that one can't be the marker: at F1 it would force E3, E4 and E9 cleared (every open cell left in this region sits in column E, so its marker has to land there), then G3, H3, I3, and 7 more cleared (every open cell in row 2 belongs to the same region, so that region's marker has to be somewhere in this row), then B4 (the last open cell in row 4), and row 3 would be left with no open cell for its marker.
  11. After that, 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.
  12. Now, marking E2 would force F7 (the last open cell in region 3), then D8 (the last open cell in region 8), then A9 (the last open cell in region 7), and then region 9 would be left with no open cell for its marker. So E2 can't be the marker there.
  13. From there, 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.
  14. Following that, if H1 were the marker, it would force F2 (the last open cell in region 5), then E4 and E9 cleared (every open cell left in this region sits in column E, so its marker has to land there), then B4 (the last open cell in row 4), and row 3 would end up with no open cell for its marker, which can't happen. So it can't go there.
  15. First, that one can't be the marker: at I1 it would force F2 (the last open cell in region 5), then E4 and E9 cleared (every open cell left in this region sits in column E, so its marker has to land there), then B4 (the last open cell in row 4), and row 3 would be left with no open cell for its marker.
  16. Next, place a marker at G1. Region has exactly one open cell left. Placing here clears the rest of row 1, column G, and its region. Placing here also clears its 1 touching neighbour.
  17. Then, place a marker at I2. Region has exactly one open cell left. Placing here clears the rest of row 2, column I, and its region.
  18. After that, that one can't be the marker: at B3 it would force C6 (the last open cell in region 6), then D8 (the last open cell in region 8), then A9 cleared (every open cell left in this region sits in row 9, so its marker has to land there), and the chain continues, and row 4 would be left with no open cell for its marker.
  19. Now, that one can't be the marker: at E3 it would force B4 (the last open cell in region 1), and row 5 would be left with no open cell for its marker.
  20. From there, place a marker at C3. Row 3 has exactly one open cell left. Placing here clears the rest of row 3, column C, and its region.
  21. Following that, every open cell in row 4 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.
  22. First, every open cell in column H 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.
  23. Next, every remaining candidate in this region touches E5, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  24. Then, place a marker at B5. Row 5 has exactly one open cell left. Placing here clears the rest of row 5, column B, and its region.
  25. After that, place a marker at E6. Row 6 has exactly one open cell left. Placing here clears the rest of row 6, column E, and its region. Placing here also clears its 1 touching neighbour.
  26. Now, place a marker at F4. Region has exactly one open cell left.
  27. From there, place a marker at D8. Region has exactly one open cell left. Placing here clears the rest of row 8, column D, and its region.
  28. Following that, place a marker at H9. Region has exactly one open cell left. Placing here clears the rest of row 9, column H, and its region.
  29. Finally, place a marker at A7. Region has exactly one open cell left.

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