QED Logic quod erat demonstrandum

An Expert 7×7 Puzzle Solved with Forced Chain in 24 Steps

7×7 · expert · 24 steps

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A
B
C
D
E
F
G
1
2
3
4
5
6
7
1×
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2×
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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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This 7×7 board rates as expert (difficulty score 1321). Solving it from scratch takes 24 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 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.
  2. Next, every remaining candidate in this region touches D3, 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 B6, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  4. After that, marking A1 would force D7, E7, F7, and 1 more cleared (every open cell left in this region sits in row 7, so its marker has to land there), then F4 (the last open cell in column F), then C3 (the last open cell in row 3), and then row 5 would be left with no open cell for its marker. So A1 can't be the marker there.
  5. Now, that one can't be the marker: at C1 it would force E4, E5, E6, and 1 more cleared (every open cell left in this region sits in column E, so its marker has to land there), then F4 (the last open cell in region 5), then E2 (the last open cell in region 3), and the chain continues, and row 5 would be left with no open cell for its marker.
  6. From there, marking D1 would force E3 (the last open cell in region 3), and then region 5 would be left with no open cell for its marker. So D1 can't be the marker there.
  7. Following that, that one can't be the marker: at A2 it would force E3 (the last open cell in region 3), and region 5 would be left with no open cell for its marker.
  8. First, marking B2 would force E3 (the last open cell in region 3), and then region 5 would be left with no open cell for its marker. So B2 can't be the marker there.
  9. Next, if C2 were the marker, it would force E3 (the last open cell in region 3), and region 5 would end up with no open cell for its marker, which can't happen. So it can't go there.
  10. Then, every open cell left in this region sits in column B, so its marker has to land there; that clears every other open cell in the column.
  11. After that, that one can't be the marker: at G2 it would force E3 (the last open cell in region 3), and region 5 would be left with no open cell for its marker.
  12. Now, 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.
  13. From there, if A3 were the marker, it would force B1 (the last open cell in region 1), then C7 (the last open cell in region 7), and region 6 would end up with no open cell for its marker, which can't happen. So it can't go there.
  14. Following that, marking C3 would force B1 (the last open cell in region 1), then E2 (the last open cell in region 3), then F4 (the last open cell in region 5), and then row 5 would be left with no open cell for its marker. So C3 can't be the marker there.
  15. First, marking E2 would force F4 (the last open cell in region 5), then B3 (the last open cell in row 3), then G1 (the last open cell in row 1), and the chain continues, and then column D would be left with no open cell for its marker. So E2 can't be the marker there.
  16. Next, place a marker at D2. Region has exactly one open cell left. Placing here clears the rest of row 2, column D, and its region.
  17. Then, together, two regions have open cells only in columns A 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.
  18. After that, every open cell left in this region sits in row 7, so its marker has to land there; that clears every other open cell in the row.
  19. Now, place a marker at A6. Region has exactly one open cell left. Placing here clears the rest of row 6, column A, and its region.
  20. From there, 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.
  21. Following that, place a marker at B1. Region has exactly one open cell left. Placing here clears the rest of row 1, column B, and its region.
  22. First, place a marker at E5. Region has exactly one open cell left. Placing here clears the rest of row 5, column E, and its region.
  23. Next, place a marker at G3. Region has exactly one open cell left.
  24. Finally, place a marker at F7. Region has exactly one open cell left.

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