QED Logic quod erat demonstrandum

An Expert 8×8 Puzzle Solved with Forced Chain in 21 Steps

8×8 · expert · 21 steps

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A
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C
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F
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H
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This 8×8 board rates as expert (difficulty score 1192). Solving it from scratch takes 21 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 row 1 belongs to the same region, so that region's marker has to be somewhere in this row; that clears every other open cell in the region.
  2. Next, every remaining candidate in this region touches H7, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  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 rows 6 and 7; 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.
  5. Now, place a marker at A5. Region has exactly one open cell left. Placing here clears the rest of row 5, column A, and its region. Placing here also clears its 1 touching neighbour.
  6. From there, every remaining candidate in this region touches G7, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  7. Following that, 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.
  8. First, if B1 were the marker, it would force D7 (the last open cell in region 6), then C3 (the last open cell in region 2), and region 4 would end up with no open cell for its marker, which can't happen. So it can't go there.
  9. Next, that one can't be the marker: at C1 it would force B3 (the last open cell in region 2), then D4 (the last open cell in region 4), and row 7 would be left with no open cell for its marker.
  10. Then, marking D1 would force B7 (the last open cell in region 6), then C3 (the last open cell in region 2), and then region 4 would be left with no open cell for its marker. So D1 can't be the marker there.
  11. After that, that one can't be the marker: at F1 it would force B3 and C3 cleared (every open cell in row 2 belongs to the same region, so that region's marker has to be somewhere in this row), then B7 (the last open cell in column B), then G3, G4 and H4 cleared (together, two regions have open cells only in columns G and H), and the chain continues, and row 3 would be left with no open cell for its marker.
  12. Now, marking G1 would force H6 (the last open cell in region 7), then F8 (the last open cell in region 8), then E3 cleared (every open cell left in this region sits in column E, so its marker has to land there), and the chain continues, and then column C 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 G6 (the last open cell in region 7), then F8 (the last open cell in region 8), then E3 cleared (every open cell left in this region sits in column E, 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.
  14. Following that, place a marker at E1. Region has exactly one open cell left. Placing here clears the rest of row 1, column E, and its region. Placing here also clears its 2 touching neighbours.
  15. First, that one can't be the marker: at G2 it would force H6 (the last open cell in region 7), then F8 (the last open cell in region 8), then D3 cleared (every open cell left in this region sits in row 3, so its marker has to land there), and the chain continues, and column C would be left with no open cell for its marker.
  16. Next, place a marker at C2. Row 2 has exactly one open cell left. Placing here clears the rest of row 2, column C, and its region. Placing here also clears its 1 touching neighbour.
  17. Then, place a marker at B7. Column 2 has exactly one open cell left. Placing here clears the rest of row 7, column B, and its region.
  18. After that, place a marker at D4. Column 4 has exactly one open cell left. Placing here clears the rest of row 4, column D, and its region.
  19. Now, place a marker at G3. Region has exactly one open cell left. Placing here clears the rest of row 3, column G, and its region.
  20. From there, 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.
  21. Finally, place a marker at F8. Region has exactly one open cell left.

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