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
B
C
D
E
F
G
H
1
2
3
4
5
6
7
8
1×
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2×
3×
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4×
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5×
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6×
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7
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8×
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This 8×8 board rates as expert (difficulty score 1184). Solving it from scratch takes 21 logical steps, using 7 techniques: Single cell, Row/column exclusion, Adjacency clear, Region confinement, Line 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 8, 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 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 and C8 cleared (together, two regions have open cells only in columns B and C), and then row 8 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 D7, E7, F7, and 2 more cleared (every open cell left in this region sits in row 7, so its marker has to land there), then F2, G2 and G3 cleared (every open cell in column H belongs to the same region, so that region's marker has to be somewhere in this column), then E5, E6 and F6 cleared (every open cell in column G 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 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 E4 and E7 cleared (every open cell in column D belongs to the same region, so that region's marker has to be somewhere in this column), then A8 and B8 cleared (together, two regions have open cells only in columns A and B), and row 8 would be left with no open cell for its marker.
  6. From there, if E1 were the marker, it would force D3 cleared (every open cell left in this region sits in column D, so its marker has to land there), then G4, G5, G6, and 2 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 H2, H3, H4, and 2 more cleared (every open cell in column G belongs to the same region, so that region's marker has to be somewhere in this column), and column H would end up with no open cell for its marker, which can't happen. So it can't go there.
  7. Following that, that one can't be the marker: at F1 it would force H7 (the last open cell in column H), and column G would be left with no open cell for its marker.
  8. First, marking G1 would force H7 (the last open cell in column H), then B6 (the last open cell in region 7), then A2 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 then row 2 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 E5, E6, F6, and 1 more cleared (every open cell in column G belongs to the same region, so that region's marker has to be somewhere in this column), then D2, E2 and E3 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 C3, D3 and C4 cleared (every open cell in row 2 belongs to the same region, so that region's marker has to be somewhere in this row), and the chain continues, and column E 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 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.
  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, 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.
  13. From there, every open cell in row 2 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.
  14. Following that, every remaining candidate in this region touches E4 and G4, so whichever candidate ends up marked will clear them, and they can be ruled out now regardless of which one wins.
  15. First, 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.
  16. Next, place a marker at B6. Region 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.
  17. 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. Placing here also clears its 1 touching neighbour.
  18. After that, place a marker at H2. Region has exactly one open cell left.
  19. Now, place a marker at F3. Region has exactly one open cell left. Placing here clears the rest of row 3, column F, and its region.
  20. From there, 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.
  21. Finally, place a marker at C8. Region has exactly one open cell left.

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