QED Logic quod erat demonstrandum

An Expert 9×9 Puzzle Solved with Forced Chain in 20 Steps

9×9 · expert · 20 steps

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This 9×9 board rates as expert (difficulty score 1143). Solving it from scratch takes 20 logical steps, using 7 techniques: Single cell, Row/column exclusion, Adjacency clear, Region 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 I7, I8, I9, and 5 more cleared (together, two regions have open cells only in columns I and H), 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 F2, F3, F4, and 3 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 then region 2 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 I7, I8, I9, and 5 more cleared (together, two regions have open cells only in columns I and H), 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 F2, F3, F4, and 3 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 region 2 would end up with no open cell for its marker, which can't happen. So it can't go there.
  4. After that, marking D1 would force I7, I8, I9, and 4 more cleared (together, two regions have open cells only in columns I and H), then G5, G6 and G9 cleared (every open cell left in this region sits in column G, so its marker has to land there), then F5, F6 and F7 cleared (every open cell left in this region sits in column F, so its marker has to land there), and the chain continues, and then column G would be left with no open cell for its marker. So D1 can't be the marker there.
  5. Now, if E1 were the marker, it would force I7, I8, I9, and 4 more cleared (together, two regions have open cells only in columns I and H), then G5, G6 and G9 cleared (every open cell left in this region sits in column G, so its marker has to land there), then F7, F8 and F9 cleared (every open cell left in this region sits in column F, so its marker has to land there), and region 9 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 I7, I8, I9, and 4 more cleared (together, two regions have open cells only in columns I and H), then G9 (the last open cell in region 9), and region 8 would be left with no open cell for its marker.
  7. Following that, marking G1 would force I8, I9, H4, and 1 more cleared (together, two regions have open cells only in columns I and H), then F3, F4, F5, and 2 more cleared (every open cell left in this region sits in column F, so its marker has to land there), then E5 (the last open cell in region 6), and then region 2 would be left with no open cell for its marker. So G1 can't be the marker there.
  8. First, together, two regions have open cells only in columns H and I; 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, every open cell left in this region sits in column G, so its marker has to land there; that clears every other open cell in the column.
  10. Then, 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.
  11. After that, place a marker at E5. Region has exactly one open cell left. Placing here clears the rest of row 5, column E, and its region. Placing here also clears its 2 touching neighbours.
  12. Now, place a marker at C1. Region has exactly one open cell left. Placing here clears the rest of row 1, column C, and its region. Placing here also clears its 2 touching neighbours.
  13. From there, place a marker at H3. Column 8 has exactly one open cell left. Placing here clears the rest of row 3, column H, and its region. Placing here also clears its 1 touching neighbour.
  14. Following that, place a marker at I6. Region has exactly one open cell left. Placing here clears the rest of row 6, column I, and its region.
  15. First, place a marker at A2. Row 2 has exactly one open cell left. Placing here clears the rest of row 2, column A, and its region.
  16. Next, place a marker at B4. Row 4 has exactly one open cell left. Placing here clears the rest of row 4, column B, and its region.
  17. Then, every remaining candidate in this region touches F8, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  18. After that, place a marker at F9. Region has exactly one open cell left. Placing here clears the rest of row 9, column F, and its region. Placing here also clears its 1 touching neighbour.
  19. Now, place a marker at G7. Region has exactly one open cell left. Placing here clears the rest of row 7, column G, and its region.
  20. Finally, place a marker at D8. Region has exactly one open cell left.

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