QED Logic quod erat demonstrandum

An Expert 7×7 Puzzle Solved with Forced Chain in 20 Steps

7×7 · expert · 20 steps

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This 7×7 board rates as expert (difficulty score 1166). Solving it from scratch takes 20 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 F2, 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 D5, 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 E3, 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), then E2 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 C2, D2, D3, and 2 more cleared (every open cell in column E belongs to the same region, so that region's marker has to be somewhere in this column), and the chain continues, and then region 7 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 E3, 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), then E2 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, D3, C4, and 1 more cleared (every open cell in column E belongs to the same region, so that region's marker has to be somewhere in this column), and the chain continues, and region 7 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 D6 (the last open cell in region 6), then E4 (the last open cell in region 2), then G2 (the last open cell in row 2), and the chain continues, and region 7 would be left with no open cell for its marker.
  6. From there, every open cell left in this region sits in row 2, so its marker has to land there; that clears every other open cell in the row.
  7. Following that, every remaining candidate in this region touches A3 and B3, so whichever candidate ends up marked will clear them, and they can be ruled out now regardless of which one wins.
  8. First, marking D1 would force G3 (the last open cell in region 3), then C7 cleared (every open cell left in this region sits in column C, so its marker has to land there), then A5 and B5 cleared (every open cell in row 4 belongs to the same region, so that region's marker has to be somewhere in this row), and the chain continues, and then column F would be left with no open cell for its marker. So D1 can't be the marker there.
  9. Next, 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.
  10. Then, if E1 were the marker, it would force 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), and column G would end up with no open cell for its marker, which can't happen. So it can't go there.
  11. After that, marking G1 would force E3, E6, D7, and 1 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 D3, C4 and D4 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 D6 (the last open cell in column D), and the chain continues, and then region 7 would be left with no open cell for its marker. So G1 can't be the marker there.
  12. Now, place a marker at F1. Region has exactly one open cell left. Placing here clears the rest of row 1, column F, and its region.
  13. From there, every open cell in column G 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.
  14. Following that, every open cell in column E 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.
  15. First, place a marker at C3. Row 3 has exactly one open cell left. Placing here clears the rest of row 3, column C, and its region. Placing here also clears its 1 touching neighbour.
  16. Next, place a marker at A2. Region has exactly one open cell left. Placing here clears the rest of row 2, column A, and its region.
  17. 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. Placing here also clears its 1 touching neighbour.
  18. After that, 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.
  19. Now, place a marker at B7. Region has exactly one open cell left. Placing here clears the rest of row 7, column B, and its region.
  20. Finally, place a marker at G5. Region has exactly one open cell left.

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