QED Logic quod erat demonstrandum

An Expert 8×8 Puzzle Solved with Forced Chain in 27 Steps

8×8 · expert · 27 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 expert (difficulty score 1265). Solving it from scratch takes 27 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 open cell left in this region sits in column B, so its marker has to land there; that clears every other open cell in the column.
  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 left in this region sits in column C, so its marker has to land there; that clears every other open cell in the column.
  4. 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.
  5. Now, every remaining candidate in this region touches E2, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  6. From there, every remaining candidate in this region touches A5 and C5, 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, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  8. First, every remaining candidate in this region touches A4, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  9. Next, every remaining candidate in this region touches F8, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  10. Then, if E1 were the marker, it would force D3 (the last open cell in region 1), then A6 (the last open cell in region 4), then B4 (the last open cell in region 5), and the chain continues, and region 6 would end up with no open cell for its marker, which can't happen. So it can't go there.
  11. After that, that one can't be the marker: at F1 it would force D3 (the last open cell in region 1), then A6 (the last open cell in region 4), then B4 (the last open cell in region 5), and the chain continues, and row 2 would be left with no open cell for its marker.
  12. Now, marking G1 would immediately leave row 2 with no open cell for its marker, so it can't be the marker there.
  13. From there, 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.
  14. Following that, every remaining candidate in this region touches F3 and G3, so whichever candidate ends up marked will clear them, and they can be ruled out now regardless of which one wins.
  15. First, if H1 were the marker, it would force D3 (the last open cell in region 1), then F2 (the last open cell in region 2), then A6 (the last open cell in region 4), and row 5 would end up with no open cell for its marker, which can't happen. So it can't go there.
  16. Next, place a marker at D1. Row 1 has exactly one open cell left. Placing here clears the rest of row 1, column D, and its region.
  17. Then, that one can't be the marker: at G2 it would force E6, E7 and E8 cleared (every open cell left in this region sits in column E, so its marker has to land there), then F7 (the last open cell in region 6), then C6 (the last open cell in region 8), and the chain continues, and column B would be left with no open cell for its marker.
  18. After that, 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. Placing here also clears its 1 touching neighbour.
  19. Now, every open cell left in this region sits in column E, so its marker has to land there; that clears every other open cell in the column.
  20. From there, every remaining candidate in this region touches H4, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  21. Following that, that one can't be the marker: at H3 it would force A6 (the last open cell in region 4), and row 5 would be left with no open cell for its marker.
  22. First, place a marker at A3. Row 3 has exactly one open cell left. Placing here clears the rest of row 3, column A, and its region. Placing here also clears its 1 touching neighbour.
  23. Next, place a marker at B5. Region has exactly one open cell left. Placing here clears the rest of row 5, column B, and its region. Placing here also clears its 1 touching neighbour.
  24. Then, place a marker at G4. Region has exactly one open cell left. Placing here clears the rest of row 4, column G, and its region.
  25. After 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.
  26. Now, place a marker at E8. Region has exactly one open cell left. Placing here clears the rest of row 8, column E, and its region.
  27. Finally, place a marker at H6. Region has exactly one open cell left.

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