QED Logic quod erat demonstrandum

An Expert 9×9 Puzzle Solved with Forced Chain in 24 Steps

9×9 · expert · 24 steps

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A
B
C
D
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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 1146). Solving it from scratch takes 24 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 I, 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, place a marker at A8. Region has exactly one open cell left. Placing here clears the rest of row 8, column A, and its region. Placing here also clears its 1 touching neighbour.
  4. After that, every open cell left in this region sits in column B, so its marker has to land there; that clears every other open cell in the column.
  5. Now, together, two regions have open cells only in columns H 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.
  6. 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.
  7. Following that, every remaining candidate in this region touches D2, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  8. 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.
  9. Next, every remaining candidate in this region touches C3 and C4, so whichever candidate ends up marked will clear them, and they can be ruled out now regardless of which one wins.
  10. Then, that one can't be the marker: at C1 it would force E5, D7 and E7 cleared (every open cell in row 6 belongs to the same region, so that region's marker has to be somewhere in this row), and row 7 would be left with no open cell for its marker.
  11. After that, every open cell in column C 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.
  12. Now, if E1 were the marker, it would force C2, B3 and F3 cleared (together, two regions have open cells only in rows 2 and 3), then B4 (the last open cell in region 1), then F5 (the last open cell in region 6), and the chain continues, and column D would end up with no open cell for its marker, which can't happen. So it can't go there.
  13. From there, if H1 were the marker, it would force E2 (the last open cell in region 2), then I3 (the last open cell in region 4), then B4 (the last open cell in region 1), and the chain continues, and column D would end up with no open cell for its marker, which can't happen. So it can't go there.
  14. Following that, that one can't be the marker: at I1 it would force E2 (the last open cell in region 2), then H3 (the last open cell in region 3), then B4 (the last open cell in region 1), and the chain continues, and column D 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. Placing here also clears its 1 touching neighbour.
  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, place a marker at B4. Region has exactly one open cell left. Placing here clears the rest of row 4, column B, and its region. Placing here also clears its 1 touching neighbour.
  18. After that, place a marker at F5. Region has exactly one open cell left. Placing here clears the rest of row 5, column F, and its region. Placing here also clears its 1 touching neighbour.
  19. Now, place a marker at E7. Region has exactly one open cell left. Placing here clears the rest of row 7, column E, and its region.
  20. From there, place a marker at C6. Region has exactly one open cell left.
  21. Following that, 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.
  22. 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.
  23. Next, place a marker at I3. Region has exactly one open cell left.
  24. Finally, place a marker at H9. Region has exactly one open cell left.

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