QED Logic quod erat demonstrandum

A Sharp 8×8 Puzzle Solved with Forced Chain in 20 Steps

8×8 · sharp · 20 steps

Play this board yourself →
A
B
C
D
E
F
G
H
1
2
3
4
5
6
7
8
1×
×
×
2×
×
3×
4×
×
5×
×
×
×
×
×
×
×
×
×
×
×
×
6×
×
×
7×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
8×
×
×
×
×
×
×
×
×
×
×
×
×

This 8×8 board rates as sharp (difficulty score 1095). Solving it from scratch takes 20 logical steps, using 7 techniques: Single cell, Row/column exclusion, Adjacency clear, Region 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 remaining candidate in this region touches F2 and G2, so whichever candidate ends up marked will clear them, and they can be ruled out now regardless of which one wins.
  3. Then, that one can't be the marker: at F1 it would force D2 (the last open cell in region 2), and region 5 would be left with no open cell for its marker.
  4. After that, place a marker at G1. Region has exactly one open cell left. Placing here clears the rest of row 1, column G, and its region. Placing here also clears its 1 touching neighbour.
  5. Now, place a marker at H3. Region has exactly one open cell left. Placing here clears the rest of row 3, column H, and its region.
  6. From there, 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.
  7. Following that, that one can't be the marker: at A2 it would force F6, F7 and F8 cleared (every open cell left in this region sits in column F, so its marker has to land there), then D6, D7 and D8 cleared (every open cell left in this region sits in column D, so its marker has to land there), then E7 and E8 cleared (every open cell left in this region sits in column E, so its marker has to land there), and the chain continues, and row 7 would be left with no open cell for its marker.
  8. First, every remaining candidate in this region touches B4 and B5, so whichever candidate ends up marked will clear them, and they can be ruled out now regardless of which one wins.
  9. Next, marking B2 would force F6, F7 and F8 cleared (every open cell left in this region sits in column F, so its marker has to land there), then E7 and E8 cleared (every open cell left in this region sits in column E, so its marker has to land there), then C8 and D8 cleared (every open cell in row 7 belongs to the same region, so that region's marker has to be somewhere in this row), and then row 8 would be left with no open cell for its marker. So B2 can't be the marker there.
  10. Then, if C2 were the marker, it would force F6, F7 and F8 cleared (every open cell left in this region sits in column F, so its marker has to land there), then E7 and E8 cleared (every open cell left in this region sits in column E, so its marker has to land there), then D8 cleared (every open cell in row 7 belongs to the same region, so that region's marker has to be somewhere in this row), 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.
  11. After that, every open cell left in this region sits in column D, so its marker has to land there; that clears every other open cell in the column.
  12. Now, place a marker at E2. Row 2 has exactly one open cell left. Placing here clears the rest of row 2, column E, and its region.
  13. From there, every open cell left in this region sits in column F, so its marker has to land there; that clears every other open cell in the column.
  14. Following that, together, two regions have open cells only in rows 4 and 5; 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.
  15. First, every remaining candidate in this region touches B7 and C7, so whichever candidate ends up marked will clear them, and they can be ruled out now regardless of which one wins.
  16. Next, place a marker at C8. Region has exactly one open cell left. Placing here clears the rest of row 8, column C, and its region.
  17. Then, place a marker at B6. Region has exactly one open cell left. Placing here clears the rest of row 6, column B, and its region. Placing here also clears its 1 touching neighbour.
  18. After that, place a marker at A4. Region has exactly one open cell left. Placing here clears the rest of row 4, column A, and its region.
  19. Now, place a marker at D5. Region has exactly one open cell left.
  20. Finally, place a marker at F7. Region has exactly one open cell left.

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