QED Logic quod erat demonstrandum

An Expert 7×7 Puzzle Solved with Forced Chain in 22 Steps

7×7 · expert · 22 steps

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A
B
C
D
E
F
G
1
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5
6
7
1×
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5×
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6
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7×
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This 7×7 board rates as expert (difficulty score 1279). Solving it from scratch takes 22 logical steps, using 7 techniques: Single cell, Row/column exclusion, Adjacency clear, Line confinement, Paired regions, Shape squeeze, Forced chain. Every step below is forced: nothing here is a guess.

  1. First, together, two regions have open cells only in columns A and B; 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.
  2. Next, every remaining candidate in this region touches F6, 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 B7 (the last open cell in region 6), then D2, D3 and D4 cleared (every open cell left in this region sits in column D, so its marker has to land there), then E2, E3 and E4 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 5 would be left with no open cell for its marker. So A1 can't be the marker there.
  4. After that, that one can't be the marker: at C1 it would force B7 (the last open cell in column B), then D3 and D4 cleared (every open cell left in this region sits in column D, so its marker has to land there), then E3 and E4 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 5 would be left with no open cell for its marker.
  5. Now, if E1 were the marker, it would force F7 (the last open cell in region 7), then B2 (the last open cell in column B), then D3 (the last open cell in row 3), and row 4 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 F1 it would force E4 cleared (every open cell left in this region sits in column E, so its marker has to land there), then D3 and D4 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 C4 (the last open cell in row 4), and the chain continues, and row 2 would be left with no open cell for its marker.
  7. Following that, that one can't be the marker: at A2 it would force B7 (the last open cell in region 6), then D1, D3 and D4 cleared (every open cell left in this region sits in column D, so its marker has to land there), then G1 (the last open cell in row 1), and the chain continues, and row 5 would be left with no open cell for its marker.
  8. First, if C2 were the marker, it would force A3 (the last open cell in region 1), then B7 (the last open cell in region 6), then G1 (the last open cell in row 1), and the chain continues, and row 5 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 D2 it would force A3 (the last open cell in region 1), then B7 (the last open cell in region 6), and column C would be left with no open cell for its marker.
  10. Then, marking E2 would force A3 (the last open cell in region 1), then B7 (the last open cell in region 6), and then column C would be left with no open cell for its marker. So E2 can't be the marker there.
  11. After that, together, two regions have open cells only in columns D and C; 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.
  12. Now, marking G1 would force B2 (the last open cell in row 2), and then row 3 would be left with no open cell for its marker. So G1 can't be the marker there.
  13. From there, 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.
  14. Following that, 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.
  15. First, place a marker at B2. Column 2 has exactly one open cell left. Placing here clears the rest of row 2, column B, and its region.
  16. Next, 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.
  17. Then, every open cell in row 4 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.
  18. After that, place a marker at G3. Column 7 has exactly one open cell left. Placing here clears the rest of row 3, column G, and its region. Placing here also clears its 1 touching neighbour.
  19. Now, place a marker at E4. Region has exactly one open cell left. Placing here clears the rest of row 4, column E, and its region.
  20. From there, place a marker at F7. Region has exactly one open cell left. Placing here clears the rest of row 7, column F, and its region.
  21. Following that, place a marker at C6. Region has exactly one open cell left. Placing here clears the rest of row 6, column C, and its region.
  22. Finally, place a marker at A5. Region has exactly one open cell left.

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