QED Logic quod erat demonstrandum

An Expert 9×9 Puzzle Solved with Forced Chain in 23 Steps

9×9 · expert · 23 steps

Play this board yourself →
A
B
C
D
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G
H
I
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9×
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This 9×9 board rates as expert (difficulty score 1131). Solving it from scratch takes 23 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 row 1, so its marker has to land there; that clears every other open cell in the row.
  2. Next, every open cell in column A 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.
  3. Then, 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.
  4. After that, 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.
  5. Now, if H1 were the marker, it would force I6, I7, I8, and 1 more cleared (every open cell left in this region sits in column I, so its marker has to land there), then G3, G4, G5, and 2 more cleared (every open cell left in this region sits in column G, so its marker has to land there), then F4, F5, F7, and 2 more cleared (every open cell left in this region sits in column F, so its marker has to land there), and the chain continues, and column C 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 I1 it would force H4, H6, H7, and 2 more cleared (every open cell left in this region sits in column H, so its marker has to land there), then G2, G3, G4, and 3 more cleared (every open cell left in this region sits in column G, so its marker has to land there), then F4, F5, F7, and 2 more cleared (every open cell left in this region sits in column F, so its marker has to land there), and the chain continues, and column C would be left with no open cell for its marker.
  7. Following that, 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.
  8. First, together, two regions have open cells only in columns I and H; 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.
  9. Next, place a marker at F3. Region has exactly one open cell left. Placing here clears the rest of row 3, column F, and its region. Placing here also clears its 2 touching neighbours.
  10. Then, every open cell left in this region sits in row 2, so its marker has to land there; that clears every other open cell in the row.
  11. After that, every open cell left in this region sits in column E, so its marker has to land there; that clears every other open cell in the column.
  12. Now, 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.
  13. From there, every remaining candidate in this region touches D6, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  14. Following that, every remaining candidate in this region touches C8 and D8, so whichever candidate ends up marked will clear them, and they can be ruled out now regardless of which one wins.
  15. First, that one can't be the marker: at D2 it would force C9 (the last open cell in region 9), and column B would be left with no open cell for its marker.
  16. Next, place a marker at B2. Region has exactly one open cell left. Placing here clears the rest of row 2, column B, and its region.
  17. Then, that one can't be the marker: at I4 it would force H7 (the last open cell in region 8), then D5 (the last open cell in region 6), and column E would be left with no open cell for its marker.
  18. 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.
  19. Now, place a marker at I8. Row 8 has exactly one open cell left. Placing here clears the rest of row 8, column I, and its region.
  20. From there, place a marker at H5. Region has exactly one open cell left. Placing here clears the rest of row 5, column H, and its region.
  21. Following 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.
  22. First, place a marker at C7. Region has exactly one open cell left. Placing here clears the rest of row 7, column C, and its region.
  23. Finally, place a marker at D9. Region has exactly one open cell left.

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