QED Logic quod erat demonstrandum

An Expert 8×8 Puzzle Solved with Forced Chain in 18 Steps

8×8 · expert · 18 steps

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A
B
C
D
E
F
G
H
1
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8
1×
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7×
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This 8×8 board rates as expert (difficulty score 1157). Solving it from scratch takes 18 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 remaining candidate in this region touches B4, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  2. Next, 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.
  3. Then, marking A1 would force B8 (the last open cell in region 7), then C3 (the last open cell in region 5), then D5 (the last open cell in region 6), and the chain continues, and then region 3 would be left with no open cell for its marker. So A1 can't be the marker there.
  4. After that, if B1 were the marker, it would force E2 and F2 cleared (every open cell left in this region sits in row 2, so its marker has to land there), then A3, A4 and A5 cleared (every open cell left in this region sits in column A, so its marker has to land there), then C3 (the last open cell in region 5), and the chain continues, and region 2 would end up with no open cell for its marker, which can't happen. So it can't go there.
  5. Now, that one can't be the marker: at C1 it would force E2 and F2 cleared (every open cell left in this region sits in row 2, so its marker has to land there), then G3, H3, G4, and 11 more cleared (every open cell in column F belongs to the same region, so that region's marker has to be somewhere in this column), then G2 (the last open cell in column G), and column H would be left with no open cell for its marker.
  6. From there, marking D1 would force A2, B2 and F2 cleared (every open cell left in this region sits in row 2, so its marker has to land there), and then region 1 would be left with no open cell for its marker. So D1 can't be the marker there.
  7. Following that, if E1 were the marker, it would force A2, B2 and C2 cleared (every open cell left in this region sits in row 2, so its marker has to land there), then D3 (the last open cell in region 1), then A6, A7 and A8 cleared (every open cell left in this region sits in column A, so its marker has to land there), 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, marking G1 would force H3, H4, H5, and 5 more cleared (every open cell in column F belongs to the same region, so that region's marker has to be somewhere in this column), and then column H would be left with no open cell for its marker. So G1 can't be the marker there.
  9. Next, if H1 were the marker, it would force G3, G4, G5, and 5 more cleared (every open cell in column F belongs to the same region, so that region's marker has to be somewhere in this column), and column G would end up with no open cell for its marker, which can't happen. So it can't go there.
  10. Then, 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.
  11. After that, place a marker at H2. Region has exactly one open cell left. Placing here clears the rest of row 2, column H, and its region. Placing here also clears its 1 touching neighbour.
  12. Now, 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. Placing here also clears its 1 touching neighbour.
  13. From there, 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.
  14. Following that, place a marker at B8. Region has exactly one open cell left. Placing here clears the rest of row 8, column B, and its region. Placing here also clears its 1 touching neighbour.
  15. First, place a marker at C5. Region has exactly one open cell left. Placing here clears the rest of row 5, column C, and its region.
  16. Next, place a marker at A4. Region has exactly one open cell left. Placing here clears the rest of row 4, column A, and its region.
  17. Then, place a marker at E7. Region has exactly one open cell left. Placing here clears the rest of row 7, column E, and its region.
  18. Finally, place a marker at G6. Region has exactly one open cell left.

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