QED Logic quod erat demonstrandum

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

7×7 · hard · 18 steps

Play this board yourself →
A
B
C
D
E
F
G
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7
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5×
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This 7×7 board rates as hard (difficulty score 1045). 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, together, two regions have open cells only in rows 1 and 2; with one marker each, those two markers have to fill exactly those two rows, so every other region's open cells there can be cleared.
  2. Next, every open cell left in this region sits in row 3, so its marker has to land there; that clears every other open cell in the row.
  3. Then, every remaining candidate in this region touches F6, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  4. After that, if E1 were the marker, it would force G2 (the last open cell in region 3), then F4 cleared (every open cell left in this region sits in column F, so its marker has to land there), then C5, C6, C7, and 2 more cleared (together, two regions have open cells only in columns C and D), and the chain continues, and row 4 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: place it at F1, and row 2 would be left with no open cell for its marker.
  6. From there, every open cell left in this region sits in column G, so its marker has to land there; that clears every other open cell in the column.
  7. Following that, marking G1 would force E2 (the last open cell in region 2), then C3 (the last open cell in region 1), then F4 cleared (every open cell left in this region sits in column F, so its marker has to land there), 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.
  8. First, place a marker at G2. Region has exactly one open cell left. Placing here clears the rest of row 2, column G, and its region. Placing here also clears its 1 touching neighbour.
  9. Next, place a marker at D1. Region has exactly one open cell left. Placing here clears the rest of row 1, column D, and its region.
  10. Then, 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.
  11. After that, every open cell in column A 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.
  12. Now, every open cell in column B 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.
  13. From there, every remaining candidate in this region touches A5 and A6, so whichever candidate ends up marked will clear them, and they can be ruled out now regardless of which one wins.
  14. Following that, place a marker at A7. Region has exactly one open cell left. Placing here clears the rest of row 7, column A, and its region. Placing here also clears its 1 touching neighbour.
  15. First, place a marker at B5. Region has exactly one open cell left. Placing here clears the rest of row 5, column B, and its region. Placing here also clears its 1 touching neighbour.
  16. Next, place a marker at E6. Region has exactly one open cell left. Placing here clears the rest of row 6, column E, and its region.
  17. Then, place a marker at C3. Region has exactly one open cell left.
  18. Finally, place a marker at F4. Region has exactly one open cell left.

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