QED Logic quod erat demonstrandum

An Expert 9×9 Puzzle Solved with Forced Chain in 28 Steps

9×9 · expert · 28 steps

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A
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C
D
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H
I
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This 9×9 board rates as expert (difficulty score 1376). Solving it from scratch takes 28 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 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.
  2. Next, marking A1 would force E2, F2, G2, and 1 more cleared (every open cell left in this region sits in row 2, so its marker has to land there), then F3 (the last open cell in region 4), then D4 (the last open cell in region 3), and then region 6 would be left with no open cell for its marker. So A1 can't be the marker there.
  3. Then, that one can't be the marker: at C1 it would force B3 (the last open cell in region 6), then E2 and I2 cleared (every open cell left in this region sits in row 2, so its marker has to land there), then D4 (the last open cell in region 3), and the chain continues, and row 5 would be left with no open cell for its marker.
  4. After that, marking D1 would immediately leave region 2 with no open cell for its marker, so it can't be the marker there.
  5. Now, if E1 were the marker, it would force C2 (the last open cell in region 2), 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 F1 it would force A2, B2 and I2 cleared (every open cell left in this region sits in row 2, so its marker has to land there), then H4, H6, E7, and 2 more cleared (every open cell in column G belongs to the same region, so that region's marker has to be somewhere in this column), and column H would be left with no open cell for its marker.
  7. Following that, that one can't be the marker: at I1 it would force A2, B2, E2, and 2 more cleared (every open cell left in this region sits in row 2, so its marker has to land there), then F3 (the last open cell in region 4), then D4 (the last open cell in region 3), and region 6 would be left with no open cell for its marker.
  8. First, that one can't be the marker: place it at A2, and region 2 would be left with no open cell for its marker.
  9. Next, marking B2 would immediately leave region 2 with no open cell for its marker, so it can't be the marker there.
  10. Then, every remaining candidate in this region touches B4, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  11. After that, if C2 were the marker, region 6 would end up with no open cell for its marker, which can't happen. So it can't go there.
  12. Now, marking G1 would force D2 (the last open cell in region 2), and then region 3 would be left with no open cell for its marker. So G1 can't be the marker there.
  13. From there, if H1 were the marker, it would force D2 (the last open cell in region 2), and region 3 would end up with no open cell for its marker, which can't happen. So it can't go there.
  14. Following that, place a marker at B1. Row 1 has exactly one open cell left. Placing here clears the rest of row 1, column B, and its region.
  15. First, every open cell left in this region sits in column C, so its marker has to land there; that clears every other open cell in the column.
  16. Next, 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.
  17. Then, 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.
  18. After that, 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.
  19. Now, place a marker at G2. Column 7 has exactly one open cell left. Placing here clears the rest of row 2, column G, and its region.
  20. From there, together, two regions have open cells only in rows 3 and 4; 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.
  21. Following that, place a marker at H6. Region has exactly one open cell left. Placing here clears the rest of row 6, column H, and its region. Placing here also clears its 2 touching neighbours.
  22. First, place a marker at I8. Region has exactly one open cell left. Placing here clears the rest of row 8, column I, and its region.
  23. Next, place a marker at F9. Column 6 has exactly one open cell left. Placing here clears the rest of row 9, column F, and its region.
  24. Then, every remaining candidate in this region touches D3 and D4, so whichever candidate ends up marked will clear them, and they can be ruled out now regardless of which one wins.
  25. After that, place a marker at E3. Region has exactly one open cell left. Placing here clears the rest of row 3, column E, and its region.
  26. Now, place a marker at C4. Region has exactly one open cell left. Placing here also clears its 1 touching neighbour.
  27. From there, 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.
  28. Finally, place a marker at A5. Region has exactly one open cell left.

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