QED Logic quod erat demonstrandum

An Expert 7×7 Puzzle Solved with Forced Chain in 25 Steps

7×7 · expert · 25 steps

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A
B
C
D
E
F
G
1
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5
6
7
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3×
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4
5×
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6×
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7×
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This 7×7 board rates as expert (difficulty score 1302). Solving it from scratch takes 25 logical steps, using 8 techniques: Single cell, Row/column exclusion, Adjacency clear, Region confinement, Line confinement, Paired regions, Shape squeeze, Forced chain. Every step below is forced: nothing here is a guess.

  1. First, every remaining candidate in this region touches B5, 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 C4, 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 B6 (the last open cell in region 4), 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 A3 and A7 cleared (every open cell left in this region sits in column A, so its marker has to land there), then A5, E5, F5, and 1 more cleared (every open cell left in this region sits in row 5, so its marker has to land there), then D4, D6, D7, and 2 more cleared (together, two regions have open cells only in columns D and C), 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, marking D1 would force B6 cleared (every remaining candidate in this region touches B6), then A2, A3 and A7 cleared (every open cell left in this region sits in column A, so its marker has to land there), then B7, C6 and C7 cleared (together, two regions have open cells only in columns B and C), and then region 7 would be left with no open cell for its marker. So D1 can't be the marker there.
  6. From there, that one can't be the marker: at F1 it would force A3, B3, C3, and 1 more 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 G3 (the last open cell in row 3), then E7 (the last open cell in column E), and the chain continues, and row 5 would be left with no open cell for its marker.
  7. Following that, every remaining candidate in this region touches D2, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  8. First, marking G1 would force F7 (the last open cell in column F), then A6 and B6 cleared (every open cell left in this region sits in row 6, so its marker has to land there), then A2 and A3 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 6 would be left with no open cell for its marker. So G1 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, that one can't be the marker: at A2 it would force B6 (the last open cell in region 4), and region 7 would be left with no open cell for its marker.
  11. After that, that one can't be the marker: at C1 it would force F3 and G3 cleared (every open cell left in this region sits in row 3, so its marker has to land there), then D4, E4, F4, and 4 more 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 E7 (the last open cell in column E), and region 7 would be left with no open cell for its marker.
  12. Now, place a marker at E1. Region has exactly one open cell left. Placing here clears the rest of row 1, column E, and its region. Placing here also clears its 1 touching neighbour.
  13. From there, that one can't be the marker: at G2 it would force F7 (the last open cell in column F), then A6 and B6 cleared (every open cell left in this region sits in row 6, so its marker has to land there), then A3 cleared (every open cell left in this region sits in column A, so its marker has to land there), and the chain continues, and column B would be left with no open cell for its marker.
  14. Following that, place a marker at B2. Row 2 has exactly one open cell left. Placing here clears the rest of row 2, column B, and its region.
  15. First, 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.
  16. Next, every open cell left in this region sits in row 5, so its marker has to land there; that clears every other open cell in the row.
  17. Then, 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.
  18. After that, together, two regions have open cells only in columns C and D; 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.
  19. Now, every remaining candidate in this region touches G4, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  20. From there, place a marker at A4. Row 4 has exactly one open cell left. Placing here clears the rest of row 4, column A, and its region.
  21. Following that, every remaining candidate in this region touches C6 and D6, so whichever candidate ends up marked will clear them, and they can be ruled out now regardless of which one wins.
  22. First, place a marker at C7. Region has exactly one open cell left. Placing here clears the rest of row 7, column C, and its region.
  23. Next, place a marker at D5. Region has exactly one open cell left.
  24. Then, place a marker at G6. Region has exactly one open cell left. Placing here clears the rest of row 6, column G, and its region.
  25. Finally, place a marker at F3. Region has exactly one open cell left.

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