QED Logic quod erat demonstrandum

An Expert 8×8 Puzzle Solved with Forced Chain in 20 Steps

8×8 · expert · 20 steps

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A
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This 8×8 board rates as expert (difficulty score 1186). Solving it from scratch takes 20 logical steps, using 6 techniques: Single cell, Row/column exclusion, Adjacency clear, Region confinement, 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 4, so its marker has to land there; that clears every other open cell in the row.
  2. Next, every remaining candidate in this region touches B3 and B5, so whichever candidate ends up marked will clear them, and they can be ruled out now regardless of which one wins.
  3. Then, every remaining candidate in this region touches B7, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  4. After that, marking A1 would force B6 and C5 cleared (together, two regions have open cells only in columns B and C), then D2, D3 and D8 cleared (every open cell left in this region sits in column D, so its marker has to land there), then E2 and 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 row 6 would be left with no open cell for its marker. So A1 can't be the marker there.
  5. Now, if B1 were the marker, it would force A5, A6 and C5 cleared (together, two regions have open cells only in columns A and C), then D2, D3 and D8 cleared (every open cell left in this region sits in column D, so its marker has to land there), then E2 and 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 row 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 C1 it would force A5, A6 and B6 cleared (together, two regions have open cells only in columns A and B), then D3 and D8 cleared (every open cell left in this region sits in column D, so its marker has to land there), then E2 and 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 D would be left with no open cell for its marker.
  7. Following that, if E1 were the marker, region 2 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 F1 it would force A8, B8, C8, and 2 more cleared (every open cell left in this region sits in row 8, so its marker has to land there), then D3 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 A5, C5, A6, and 1 more cleared (every open cell in column D belongs to the same region, so that region's marker has to be somewhere in this column), and the chain continues, and column D would be left with no open cell for its marker.
  9. Next, marking G1 would force H8 (the last open cell in column H), and then column F would be left with no open cell for its marker. So G1 can't be the marker there.
  10. Then, if H1 were the marker, it would force G8 (the last open cell in column G), then F2 (the last open cell in column F), 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 row 3 would end up with no open cell for its marker, which can't happen. So it can't go there.
  11. After that, place a marker at D1. Row 1 has exactly one open cell left. Placing here clears the rest of row 1, column D, and its region. Placing here also clears its 1 touching neighbour.
  12. Now, 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.
  13. From there, place a marker at F2. Region has exactly one open cell left. Placing here clears the rest of row 2, column F, and its region. Placing here also clears its 1 touching neighbour.
  14. Following that, place a marker at H3. Row 3 has exactly one open cell left. Placing here clears the rest of row 3, column H, and its region.
  15. First, place a marker at G8. Region has exactly one open cell left. Placing here clears the rest of row 8, column G, and its region.
  16. Next, every remaining candidate in this region touches B6, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  17. Then, place a marker at B4. Column 2 has exactly one open cell left. Placing here clears the rest of row 4, column B, and its region. Placing here also clears its 2 touching neighbours.
  18. After that, place a marker at A6. Region has exactly one open cell left. Placing here clears the rest of row 6, column A, and its region.
  19. Now, 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.
  20. Finally, place a marker at E5. Region has exactly one open cell left.

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