QED Logic quod erat demonstrandum

A Hard 7×7 Puzzle Solved with Forced Chain in 16 Steps

7×7 · hard · 16 steps

Play this board yourself →
A
B
C
D
E
F
G
1
2
3
4
5
6
7
1×
2×
×
×
3×
×
×
×
×
×
×
4×
×
×
×
×
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5×
×
6×
×
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7
×
×
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This 7×7 board rates as hard (difficulty score 978). Solving it from scratch takes 16 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 left in this region sits in row 1, so its marker has to land there; that clears every other open cell in the row.
  2. Next, 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.
  3. Then, 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.
  4. After that, together, two regions have open cells only in rows 2 and 3; 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.
  5. Now, together, two regions have open cells only in columns B 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.
  6. From there, every remaining candidate in this region touches D2 and E2, so whichever candidate ends up marked will clear them, and they can be ruled out now regardless of which one wins.
  7. Following that, every remaining candidate in this region touches B6 and B7, so whichever candidate ends up marked will clear them, and they can be ruled out now regardless of which one wins.
  8. First, if E1 were the marker, it would force D3 (the last open cell in region 3), then B2 (the last open cell in region 1), then C7 (the last open cell in region 5), 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.
  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, place a marker at F2. Region has exactly one open cell left. Placing here clears the rest of row 2, column F, and its region.
  11. After that, every remaining candidate in this region touches B4, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  12. Now, place a marker at G4. Row 4 has exactly one open cell left. Placing here clears the rest of row 4, column G, and its region.
  13. From there, place a marker at A6. Row 6 has exactly one open cell left. Placing here clears the rest of row 6, column A, and its region. Placing here also clears its 1 touching neighbour.
  14. Following that, 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.
  15. First, place a marker at B3. Region has exactly one open cell left.
  16. Finally, place a marker at E5. Region has exactly one open cell left.

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