QED Logic quod erat demonstrandum

An Expert 8×8 Puzzle Solved with Forced Chain in 26 Steps

8×8 · expert · 26 steps

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A
B
C
D
E
F
G
H
1
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8
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This 8×8 board rates as expert (difficulty score 1246). Solving it from scratch takes 26 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 column A, so its marker has to land there; that clears every other open cell in the column.
  2. Next, together, two regions have open cells only in columns G and H; 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.
  3. 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.
  4. After that, every remaining candidate in this region touches E5 and G5, so whichever candidate ends up marked will clear them, and they can be ruled out now regardless of which one wins.
  5. Now, every remaining candidate in this region touches B7 and B8, so whichever candidate ends up marked will clear them, and they can be ruled out now regardless of which one wins.
  6. From there, if B1 were the marker, it would force G2 (the last open cell in region 2), then E6 and F6 cleared (every open cell left in this region sits in row 6, so its marker has to land there), then C4 cleared (every open cell in row 3 belongs to the same region, so that region's marker has to be somewhere in this row), and the chain continues, and row 4 would end up with no open cell for its marker, which can't happen. So it can't go there.
  7. Following that, every remaining candidate in this region touches D2, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  8. First, that one can't be the marker: at C1 it would force G2 (the last open cell in region 2), then D4, D5, D7, and 1 more cleared (every open cell in column E belongs to the same region, so that region's marker has to be somewhere in this column), then E7 and E8 cleared (together, two regions have open cells only in rows 7 and 8), and the chain continues, and row 3 would be left with no open cell for its marker.
  9. Next, every remaining candidate in this region touches E2, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  10. Then, marking D1 would force G2 (the last open cell in region 2), then C5, C7 and C8 cleared (every open cell in column E belongs to the same region, so that region's marker has to be somewhere in this column), then E7 and E8 cleared (together, two regions have open cells only in rows 7 and 8), and the chain continues, and then row 3 would be left with no open cell for its marker. So D1 can't be the marker there.
  11. After that, every open cell left in this region sits in column E, so its marker has to land there; that clears every other open cell in the column.
  12. Now, marking G1 would force E3 (the last open cell in region 1), then F6 cleared (every open cell left in this region sits in row 6, so its marker has to land there), then F5 (the last open cell in region 4), and the chain continues, and then row 4 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 E3 (the last open cell in region 1), then F6 and G6 cleared (every open cell left in this region sits in row 6, so its marker has to land there), then F5 (the last open cell in region 4), and the chain continues, and row 4 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 G2. Region has exactly one open cell left. Placing here clears the rest of row 2, column G, and its region.
  15. First, place a marker at E1. Row 1 has exactly one open cell left. Placing here clears the rest of row 1, column E, and its region.
  16. Next, together, two regions have open cells only in rows 7 and 8; 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.
  17. Then, every remaining candidate in this region touches C4, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  18. After that, that one can't be the marker: place it at B3, and region 3 would be left with no open cell for its marker.
  19. Now, 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.
  20. From there, every open cell in row 3 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.
  21. Following that, place a marker at B6. Column 2 has exactly one open cell left. Placing here clears the rest of row 6, column B, and its region. Placing here also clears its 2 touching neighbours.
  22. First, 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.
  23. Next, place a marker at H7. Region has exactly one open cell left.
  24. Then, place a marker at C3. Column 3 has exactly one open cell left. Placing here clears the rest of row 3, column C, and its region. Placing here also clears its 1 touching neighbour.
  25. After that, place a marker at D5. Region has exactly one open cell left. Placing here clears the rest of row 5, column D, and its region.
  26. Finally, place a marker at F4. Region has exactly one open cell left.

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