QED Logic quod erat demonstrandum

A Hard 7×7 Puzzle Solved with Forced Chain in 18 Steps

7×7 · hard · 18 steps

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A
B
C
D
E
F
G
1
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7
1×
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2×
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 hard (difficulty score 1114). Solving it from scratch takes 18 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 F2, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  2. Next, marking A1 would force G2 (the last open cell in region 3), then E3 (the last open cell in row 3), then C4 (the last open cell in row 4), and the chain continues, and then row 6 would be left with no open cell for its marker. So A1 can't be the marker there.
  3. Then, if B1 were the marker, it would force G2 (the last open cell in region 3), then F4 and F7 cleared (every open cell left in this region sits in column F, so its marker has to land there), then E6 and E7 cleared (every open cell left in this region sits in column E, so its marker has to land there), and the chain continues, and region 6 would end up with no open cell for its marker, which can't happen. So it can't go there.
  4. After that, that one can't be the marker: at C1 it would force G2 (the last open cell in region 3), then F4 and F7 cleared (every open cell left in this region sits in column F, so its marker has to land there), then E6 and E7 cleared (every open cell left in this region sits in column E, so its marker has to land there), and the chain continues, and region 6 would be left with no open cell for its marker.
  5. Now, marking D1 would force G2 (the last open cell in region 3), then F7 cleared (every open cell left in this region sits in column F, so its marker has to land there), then E7 (the last open cell in region 7), and the chain continues, and then column A would be left with no open cell for its marker. So D1 can't be the marker there.
  6. From there, 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.
  7. Following that, if E1 were the marker, it would force F4 (the last open cell in region 2), and region 5 would end up with no open cell for its marker, which can't happen. So it can't go there.
  8. First, together, two regions have open cells only in columns F and G; 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.
  9. Next, 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.
  10. Then, together, two regions have open cells only in columns D and E; 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.
  11. After 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. Placing here also clears its 2 touching neighbours.
  12. Now, place a marker at E7. Region has exactly one open cell left. Placing here clears the rest of row 7, column E, and its region.
  13. From there, 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.
  14. Following that, every remaining candidate in this region touches A3 and A4, so whichever candidate ends up marked will clear them, and they can be ruled out now regardless of which one wins.
  15. First, 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. Placing here also clears its 1 touching neighbour.
  16. Next, place a marker at B3. Region has exactly one open cell left. Placing here clears the rest of row 3, column B, and its region.
  17. Then, place a marker at G4. Region has exactly one open cell left. Placing here clears the rest of row 4, column G, and its region.
  18. Finally, place a marker at F1. Region has exactly one open cell left.

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