QED Logic quod erat demonstrandum

An Expert 7×7 Puzzle Solved with Forced Chain in 27 Steps

7×7 · expert · 27 steps

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A
B
C
D
E
F
G
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7
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2×
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4
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7×
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This 7×7 board rates as expert (difficulty score 1343). Solving it from scratch takes 27 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 remaining candidate in this region touches C5, 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 B6, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  3. Then, every remaining candidate in this region touches D5, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  4. After that, 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.
  5. Now, every remaining candidate in this region touches C3 and D3, so whichever candidate ends up marked will clear them, and they can be ruled out now regardless of which one wins.
  6. From there, that one can't be the marker: at C1 it would force D4 (the last open cell in region 4), then E6 (the last open cell in region 5), then G7 (the last open cell in region 7), and region 2 would be left with no open cell for its marker.
  7. Following that, marking D1 would force C4 (the last open cell in region 4), then A5, A6 and A7 cleared (every open cell left in this region sits in column A, so its marker has to land there), then B7 (the last open cell in region 6), and the chain continues, and then row 5 would be left with no open cell for its marker. So D1 can't be the marker there.
  8. First, every remaining candidate in this region touches F2, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  9. Next, if E1 were the marker, it would force A6, F6 and G6 cleared (every open cell left in this region sits in row 6, so its marker has to land there), then A7, B7 and C7 cleared (every open cell left in this region sits in row 7, so its marker has to land there), then F5 and G5 cleared (every open cell left in this region sits in row 5, 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.
  10. Then, that one can't be the marker: at A2 it would force G3 cleared (every open cell in row 1 belongs to the same region, so that region's marker has to be somewhere in this row), then F5, G5 and G6 cleared (every open cell in row 3 belongs to the same region, so that region's marker has to be somewhere in this row), then C7 cleared (every open cell in column B belongs to the same region, so that region's marker has to be somewhere in this column), and the chain continues, and row 5 would be left with no open cell for its marker.
  11. After that, marking B2 would immediately leave region 1 with no open cell for its marker, so it can't be the marker there.
  12. Now, that one can't be the marker: at F1 it would force A7, B7 and C7 cleared (every open cell left in this region sits in row 7, so its marker has to land there), then A3 cleared (every open cell in row 2 belongs to the same region, so that region's marker has to be somewhere in this row), then G5 and G6 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 be left with no open cell for its marker.
  13. From there, marking G1 would force A3 cleared (every open cell in row 2 belongs to the same region, so that region's marker has to be somewhere in this row), then F5 cleared (every open cell in row 3 belongs to the same region, so that region's marker has to be somewhere in this row), then B5, B7 and C7 cleared (every open cell in column A belongs to the same region, so that region's marker has to be somewhere in this column), 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.
  14. Following that, every open cell in row 1 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.
  15. First, 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.
  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 remaining candidate in this region touches F3, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  18. After that, every open cell in column F belongs to the same region, so that region's marker has to be somewhere in this column; that clears every other open cell in the region.
  19. Now, place a marker at G2. Column 7 has exactly one open cell left. Placing here clears the rest of row 2, column G, and its region.
  20. From there, every remaining candidate in this region touches E6, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  21. Following that, that one can't be the marker: at B3 it would force A1 (the last open cell in region 1), then D4 (the last open cell in region 4), and row 5 would be left with no open cell for its marker.
  22. First, place a marker at E3. Region has exactly one open cell left. Placing here clears the rest of row 3, column E, and its region. Placing here also clears its 1 touching neighbour.
  23. Next, place a marker at C4. Region has exactly one open cell left. Placing here clears the rest of row 4, column C, and its region. Placing here also clears its 1 touching neighbour.
  24. Then, place a marker at D6. Region has exactly one open cell left. Placing here clears the rest of row 6, column D, and its region.
  25. After that, 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.
  26. Now, place a marker at A5. Region has exactly one open cell left. Placing here clears the rest of row 5, column A, and its region.
  27. Finally, place a marker at B1. Region has exactly one open cell left.

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