QED Logic quod erat demonstrandum

A Severe 9×9 Puzzle Solved with Forced Chain in 30 Steps

9×9 · severe · 30 steps

Play this board yourself →
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I
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This 9×9 board rates as severe (difficulty score 1317). Solving it from scratch takes 30 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, 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.
  3. Then, every remaining candidate in this region touches B5, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  4. After that, every remaining candidate in this region touches B8 and C8, so whichever candidate ends up marked will clear them, and they can be ruled out now regardless of which one wins.
  5. Now, if B1 were the marker, it would force C9 (the last open cell in region 8), then D2, D3, D4, and 1 more cleared (every open cell left in this region sits in column D, so its marker has to land there), then E5, E6, F6, and 1 more cleared (together, two regions have open cells only in columns E and F), 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.
  6. From there, that one can't be the marker: at C1 it would force B9 (the last open cell in region 8), then D3, D4, D5, and 1 more cleared (every open cell left in this region sits in column D, so its marker has to land there), then E5, E6, F6, and 1 more cleared (together, two regions have open cells only in columns E and F), and the chain continues, and region 6 would be left with no open cell for its marker.
  7. Following that, marking D1 would force F6, F7, E5, and 1 more cleared (together, two regions have open cells only in columns F and E), then C3, C5 and B6 cleared (together, two regions have open cells only in columns C and B), then G6 (the last open cell in region 4), and the chain continues, and then region 7 would be left with no open cell for its marker. So D1 can't be the marker there.
  8. First, if E1 were the marker, it would force F6, F7 and F8 cleared (every open cell left in this region sits in column F, so its marker has to land there), then D8 (the last open cell in region 9), then B9 (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.
  9. 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.
  10. 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.
  11. After that, every remaining candidate in this region touches E6 and E7, so whichever candidate ends up marked will clear them, and they can be ruled out now regardless of which one wins.
  12. Now, 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.
  13. From there, every open cell in column I 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.
  14. Following that, every remaining candidate in this region touches G6, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  15. First, 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.
  16. Next, every open cell left in this region sits in row 1, so its marker has to land there; that clears every other open cell in the row.
  17. Then, 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.
  18. After that, if H1 were the marker, it would force B2 (the last open cell in row 2), then C9 (the last open cell in region 8), then I3 (the last open cell in row 3), and the chain continues, and row 6 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 G1. Region has exactly one open cell left. Placing here clears the rest of row 1, column G, and its region.
  20. From there, every remaining candidate in this region touches I6, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  21. Following that, if I2 were the marker, it would force B3 (the last open cell in row 3), and row 4 would end up with no open cell for its marker, which can't happen. So it can't go there.
  22. First, place a marker at B2. Row 2 has exactly one open cell left. Placing here clears the rest of row 2, column B, and its region.
  23. Next, place a marker at C9. Region has exactly one open cell left.
  24. Then, place a marker at I3. Row 3 has exactly one open cell left. Placing here clears the rest of row 3, column I, and its region.
  25. After that, place a marker at A4. Row 4 has exactly one open cell left. Placing here clears the rest of row 4, column A, and its region.
  26. Now, marking H5 would force F6 (the last open cell in region 4), then D7 (the last open cell in region 7), and then row 8 would be left with no open cell for its marker. So H5 can't be the marker there.
  27. From there, place a marker at E5. Row 5 has exactly one open cell left. Placing here clears the rest of row 5, column E, and its region. Placing here also clears its 1 touching neighbour.
  28. Following that, place a marker at D7. Region has exactly one open cell left. Placing here clears the rest of row 7, column D, and its region.
  29. First, place a marker at H6. Region has exactly one open cell left.
  30. Finally, place a marker at F8. Region has exactly one open cell left.

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