QED Logic quod erat demonstrandum

An Expert 10×10 Puzzle Solved with Forced Chain in 29 Steps

10×10 · expert · 29 steps

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A
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This 10×10 board rates as expert (difficulty score 1330). Solving it from scratch takes 29 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 open cell left in this region sits in column J, 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 column I, so its marker has to land there; that clears every other open cell in the column.
  3. Then, every remaining candidate in this region touches G8, 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 E7, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  5. Now, every remaining candidate in this region touches B9, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  6. From there, marking A1 would force D10, E10, F10, and 2 more cleared (every open cell left in this region sits in row 10, so its marker has to land there), then H9 cleared (every open cell left in this region sits in row 9, so its marker has to land there), then F2, B3, C3, and 3 more cleared (together, two regions have open cells only in rows 2 and 3), and the chain continues, and then region 8 would be left with no open cell for its marker. So A1 can't be the marker there.
  7. Following that, if B1 were the marker, it would force F2, A3, C3, and 3 more cleared (together, two regions have open cells only in rows 2 and 3), then G4, G5, G6, and 6 more cleared (together, two regions have open cells only in columns G and H), then F4 (the last open cell in region 4), and the chain continues, and region 8 would end up with no open cell for its marker, which can't happen. So it can't go there.
  8. First, that one can't be the marker: at C1 it would force F2, A3, B3, and 3 more cleared (together, two regions have open cells only in rows 2 and 3), then G4, G5, G6, and 6 more cleared (together, two regions have open cells only in columns G and H), then F4 (the last open cell in region 4), and the chain continues, and region 8 would be left with no open cell for its marker.
  9. Next, marking D1 would force F2, A3, B3, and 4 more cleared (together, two regions have open cells only in rows 2 and 3), then G4, G5, G6, and 6 more cleared (together, two regions have open cells only in columns G and H), then F4 (the last open cell in region 4), and the chain continues, and then region 8 would be left with no open cell for its marker. So D1 can't be the marker there.
  10. Then, if E1 were the marker, it would force F3, F4, F6, and 2 more cleared (every open cell left in this region sits in column F, so its marker has to land there), then A3, B3, C3, and 2 more cleared (together, two regions have open cells only in rows 2 and 3), then G7, G9, G10, and 4 more cleared (together, two regions have open cells only in columns G and H), and region 7 would end up with no open cell for its marker, which can't happen. So it can't go there.
  11. After that, marking G1 would force H3, H4, H5, and 2 more cleared (every open cell left in this region sits in column H, so its marker has to land there), then F5, F6, F7, and 3 more cleared (every open cell left in this region sits in column F, so its marker has to land there), then E8 (the last open cell in region 8), and the chain continues, and then column B would be left with no open cell for its marker. So G1 can't be the marker there.
  12. Now, if H1 were the marker, it would force G7 (the last open cell in region 7), then E8 (the last open cell in region 8), then F5 and F10 cleared (every open cell left in this region sits in column F, so its marker has to land there), and the chain continues, and row 9 would end up with no open cell for its marker, which can't happen. So it can't go there.
  13. From there, marking J1 would force G4, G5, G6, and 7 more cleared (together, two regions have open cells only in columns G and H), then F5, F6, F7, and 3 more cleared (every open cell left in this region sits in column F, so its marker has to land there), then E8 (the last open cell in region 8), and the chain continues, and then column B would be left with no open cell for its marker. So J1 can't be the marker there.
  14. Following that, place a marker at F1. Row 1 has exactly one open cell left. Placing here clears the rest of row 1, column F, and its region. Placing here also clears its 1 touching neighbour.
  15. First, place a marker at E8. Region has exactly one open cell left. Placing here clears the rest of row 8, column E, and its region. Placing here also clears its 2 touching neighbours.
  16. Next, 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.
  17. Then, together, two regions have open cells only in rows 9 and 10; 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.
  18. After that, 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.
  19. Now, every remaining candidate in this region touches I3, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  20. From there, place a marker at I4. Region has exactly one open cell left. Placing here clears the rest of row 4, column I, and its region. Placing here also clears its 3 touching neighbours.
  21. Following that, place a marker at J2. Region has exactly one open cell left. Placing here clears the rest of row 2, column J, and its region.
  22. First, together, two regions have open cells only in columns D and C; 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.
  23. Next, every remaining candidate in this region touches G6 and H6, so whichever candidate ends up marked will clear them, and they can be ruled out now regardless of which one wins.
  24. Then, place a marker at G5. Region has exactly one open cell left. Placing here clears the rest of row 5, column G, and its region.
  25. After that, place a marker at D3. Region has exactly one open cell left. Placing here clears the rest of row 3, column D, and its region.
  26. Now, place a marker at H7. Region has exactly one open cell left.
  27. From there, place a marker at C9. Region has exactly one open cell left. Placing here clears the rest of row 9, column C, and its region. Placing here also clears its 1 touching neighbour.
  28. Following that, place a marker at A10. Region has exactly one open cell left. Placing here clears the rest of row 10, column A, and its region.
  29. Finally, place a marker at B6. Region has exactly one open cell left.

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