QED Logic quod erat demonstrandum

An Expert 10×10 Puzzle Solved with Forced Chain in 27 Steps

10×10 · expert · 27 steps

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A
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8
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9
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This 10×10 board rates as expert (difficulty score 1145). Solving it from scratch takes 27 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 10, so its marker has to land there; that clears every other open cell in the row.
  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 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.
  4. After that, together, two regions have open cells only in columns A and B; 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.
  5. Now, every open cell left in this region sits in row 7, so its marker has to land there; that clears every other open cell in the row.
  6. From there, every open cell left in this region sits in row 6, so its marker has to land there; that clears every other open cell in the row.
  7. Following that, every remaining candidate in this region touches I2, 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 I4, 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 D5 and E5 and D7, so whichever candidate ends up marked will clear them, and they can be ruled out now regardless of which one wins.
  10. Then, 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. Placing here also clears its 2 touching neighbours.
  11. After that, place a marker at E6. Region 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.
  12. Now, 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.
  13. From there, place a marker at B10. Region has exactly one open cell left.
  14. Following that, together, two regions have open cells only in rows 1 and 2; 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.
  15. First, marking D1 would force I3 cleared (every remaining candidate in this region touches I3), then I5 (the last open cell in column I), then F4 (the last open cell in row 4), and then column G would be left with no open cell for its marker. So D1 can't be the marker there.
  16. Next, 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.
  17. Then, if H1 were the marker, it would force F9 (the last open cell in region 9), then D2 (the last open cell in region 1), then G4 cleared (every open cell left in this region sits in column G, so its marker has to land there), and the chain continues, and column I would end up with no open cell for its marker, which can't happen. So it can't go there.
  18. After that, if F2 were the marker, it would force H9 (the last open cell in region 9), then G5 cleared (every open cell in column D belongs to the same region, so that region's marker has to be somewhere in this column), then G4 (the last open cell in column G), and row 5 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 D2. Region has exactly one open cell left. Placing here clears the rest of row 2, column D, and its region.
  20. From there, every remaining candidate in this region touches G4, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  21. Following that, every open cell in column G 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.
  22. First, 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.
  23. Next, every remaining candidate in this region touches H4, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  24. Then, place a marker at J4. Row 4 has exactly one open cell left. Placing here clears the rest of row 4, column J, and its region. Placing here also clears its 1 touching neighbour.
  25. After that, place a marker at I1. Region has exactly one open cell left.
  26. Now, 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.
  27. Finally, place a marker at G3. Region has exactly one open cell left.

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