QED Logic quod erat demonstrandum

An Expert 7×7 Puzzle Solved with Forced Chain in 20 Steps

7×7 · expert · 20 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 expert (difficulty score 1141). Solving it from scratch takes 20 logical steps, using 6 techniques: Single cell, Row/column exclusion, Adjacency clear, Region confinement, Shape squeeze, Forced chain. Every step below is forced: nothing here is a guess.

  1. First, every remaining candidate in this region touches A2, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  2. Next, every remaining candidate in this region touches F5, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  3. Then, marking A1 would force B3 (the last open cell in region 3), and then region 5 would be left with no open cell for its marker. So A1 can't be the marker there.
  4. After that, if B1 were the marker, it would force A3 (the last open cell in region 3), then D4 (the last open cell in region 4), and region 5 would end up with no open cell for its marker, which can't happen. So it can't go there.
  5. Now, every remaining candidate in this region touches D1, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  6. From there, if E1 were the marker, it would force C2 (the last open cell in region 1), then A3 (the last open cell in region 3), and region 5 would end up with no open cell for its marker, which can't happen. So it can't go there.
  7. Following that, that one can't be the marker: at F1 it would force B2 cleared (every open cell left in this region sits in row 2, so its marker has to land there), then C3, D3 and E3 cleared (every open cell left in this region sits in row 3, so its marker has to land there), then D4 (the last open cell in region 4), and the chain continues, and region 5 would be left with no open cell for its marker.
  8. First, marking G1 would force B2 cleared (every open cell left in this region sits in row 2, so its marker has to land there), then C3, D3, E3, and 1 more cleared (every open cell left in this region sits in row 3, so its marker has to land there), then D4 (the last open cell in region 4), and the chain continues, and then region 5 would be left with no open cell for its marker. So G1 can't be the marker there.
  9. Next, place a marker at C1. Row 1 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.
  10. 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.
  11. After that, place a marker at D4. Region has exactly one open cell left. Placing here clears the rest of row 4, column D, and its region. Placing here also clears its 1 touching neighbour.
  12. Now, 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.
  13. From there, 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.
  14. Following that, every open cell left in this region sits in row 6, so its marker has to land there; that clears every other open cell in the row.
  15. First, 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.
  16. Next, 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.
  17. Then, every remaining candidate in this region touches E7 and F7, so whichever candidate ends up marked will clear them, and they can be ruled out now regardless of which one wins.
  18. After that, 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. Placing here also clears its 1 touching neighbour.
  19. Now, 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.
  20. Finally, place a marker at F2. Region has exactly one open cell left.

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