QED Logic quod erat demonstrandum

A Hard 7×7 Puzzle Solved with Forced Chain in 17 Steps

7×7 · hard · 17 steps

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A
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This 7×7 board rates as hard (difficulty score 1010). Solving it from scratch takes 17 logical steps, using 7 techniques: Single cell, Row/column exclusion, Adjacency clear, Region confinement, Line confinement, 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 row 2, so its marker has to land there; that clears every other open cell in the row.
  3. Then, 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.
  4. After that, 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.
  5. Now, every open cell in column F 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.
  6. From there, every remaining candidate in this region touches C1 and C3 and D3, 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 G4 and G5, so whichever candidate ends up marked will clear them, and they can be ruled out now regardless of which one wins.
  8. First, if B1 were the marker, it would force D2 (the last open cell in region 2), and row 3 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 A1. Region has exactly one open cell left. Placing here clears the rest of row 1, column A, and its region.
  10. Then, every remaining candidate in this region touches B5 and C5, so whichever candidate ends up marked will clear them, and they can be ruled out now regardless of which one wins.
  11. After that, that one can't be the marker: at D2 it would force B3 (the last open cell in region 4), then C6 (the last open cell in region 3), then E7 (the last open cell in region 7), and column G would be left with no open cell for its marker.
  12. Now, place a marker at C2. Region has exactly one open cell left. Placing here clears the rest of row 2, column C, and its region. Placing here also clears its 1 touching neighbour.
  13. From there, place a marker at E3. Region has exactly one open cell left. Placing here clears the rest of row 3, column E, and its region. Placing here also clears its 1 touching neighbour.
  14. Following that, place a marker at F5. Region 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.
  15. First, place a marker at G7. Region has exactly one open cell left. Placing here clears the rest of row 7, column G, and its region.
  16. Next, place a marker at D6. Region has exactly one open cell left. Placing here clears the rest of row 6, column D, and its region.
  17. Finally, place a marker at B4. Region has exactly one open cell left.

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