QED Logic quod erat demonstrandum

A Hard 7×7 Puzzle Solved with Forced Chain in 15 Steps

7×7 · hard · 15 steps

Play this board yourself →
A
B
C
D
E
F
G
1
2
3
4
5
6
7
1×
×
2×
×
3×
×
×
×
×
×
×
×
4×
×
×
5×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
6
7×
×
×
×
×
×
×
×
×
×
×
×

This 7×7 board rates as hard (difficulty score 1049). Solving it from scratch takes 15 logical steps, using 7 techniques: Single cell, Row/column exclusion, Adjacency clear, 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 6 and 7; 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 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.
  3. Then, place a marker at F5. Column 6 has exactly one open cell left. Placing here clears the rest of row 5, column F, and its region. Placing here also clears its 1 touching neighbour.
  4. After that, every remaining candidate in this region touches B3, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  5. Now, 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.
  6. From there, marking A1 would force B4 (the last open cell in region 4), then C7 (the last open cell in region 6), and then row 6 would be left with no open cell for its marker. So A1 can't be the marker there.
  7. Following that, if B1 were the marker, it would force E7 (the last open cell in column E), then A6 (the last open cell in region 6), and region 4 would end up with no open cell for its marker, which can't happen. So it can't go there.
  8. First, marking D1 would force E7 (the last open cell in column E), and then column C would be left with no open cell for its marker. So D1 can't be the marker there.
  9. Next, if E1 were the marker, it would force G2 cleared (every open cell left in this region sits in row 2, so its marker has to land there), then G3 (the last open cell in region 3), then C6 cleared (every open cell in column D belongs to the same region, so that region's marker has to be somewhere in this column), and the chain continues, and row 6 would end up with no open cell for its marker, which can't happen. So it can't go there.
  10. Then, place a marker at E7. Column 5 has exactly one open cell left. Placing here clears the rest of row 7, column E, and its region.
  11. After that, 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.
  12. Now, place a marker at B4. Region has exactly one open cell left. Placing here clears the rest of row 4, column B, and its region. Placing here also clears its 1 touching neighbour.
  13. From there, place a marker at C1. Region has exactly one open cell left. Placing here clears the rest of row 1, column C, and its region. Placing here also clears its 1 touching neighbour.
  14. Following that, place a marker at D3. Region has exactly one open cell left. Placing here clears the rest of row 3, column D, and its region.
  15. Finally, place a marker at G2. Region has exactly one open cell left.

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