QED Logic quod erat demonstrandum

A Hard 9×9 Puzzle Solved with Forced Chain in 18 Steps

9×9 · hard · 18 steps

Play this board yourself →
A
B
C
D
E
F
G
H
I
1
2
3
4
5
6
7
8
9
1×
2×
×
×
×
×
×
3
×
×
×
×
4×
5×
×
×
6×
×
×
×
×
×
×
×
7×
8×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
9×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×

This 9×9 board rates as hard (difficulty score 955). Solving it from scratch takes 18 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 column I, 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 H, so its marker has to land there; that clears every other open cell in the column.
  3. Then, together, two regions have open cells only in rows 1 and 2; 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.
  4. After that, place a marker at I3. Region has exactly one open cell left. Placing here clears the rest of row 3, column I, and its region. Placing here also clears its 1 touching neighbour.
  5. Now, place a marker at H1. Region has exactly one open cell left. Placing here clears the rest of row 1, column H, and its region.
  6. From there, every open cell left in this region sits in row 4, so its marker has to land there; that clears every other open cell in the row.
  7. Following that, place a marker at G5. Region has exactly one open cell left. Placing here clears the rest of row 5, column G, and its region. Placing here also clears its 1 touching neighbour.
  8. First, every open cell left in this region sits in row 9, so its marker has to land there; that clears every other open cell in the row.
  9. Next, every open cell in column A 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.
  10. Then, place a marker at B4. Column 2 has exactly one open cell left. Placing here clears the rest of row 4, column B, and its region.
  11. After that, place a marker at C2. Column 3 has exactly one open cell left. Placing here clears the rest of row 2, column C, and its region.
  12. Now, every open cell in column D 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.
  13. From there, 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.
  14. Following that, if A6 were the marker, it would force F7 (the last open cell in region 4), then D8 (the last open cell in region 9), and row 9 would end up with no open cell for its marker, which can't happen. So it can't go there.
  15. First, place a marker at E6. Row 6 has exactly one open cell left. Placing here clears the rest of row 6, column E, and its region. Placing here also clears its 1 touching neighbour.
  16. Next, place a marker at D8. Region has exactly one open cell left. Placing here clears the rest of row 8, column D, and its region.
  17. Then, place a marker at A7. Region has exactly one open cell left.
  18. Finally, place a marker at F9. Region has exactly one open cell left.

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