QED Logic quod erat demonstrandum

A Hard 9×9 Puzzle Solved with Forced Chain in 16 Steps

9×9 · hard · 16 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 959). Solving it from scratch takes 16 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 column D, so its marker has to land there; that clears every other open cell in the column.
  2. Next, every remaining candidate in this region touches C2 and E2 and C3 and E3, so whichever candidate ends up marked will clear them, and they can be ruled out now regardless of which one wins.
  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 D3. Region has exactly one open cell left. Placing here clears the rest of row 3, column D, and its region. Placing here also clears its 2 touching neighbours.
  5. Now, place a marker at I4. Region has exactly one open cell left. Placing here clears the rest of row 4, column I, and its region. Placing here also clears its 1 touching neighbour.
  6. From there, 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. Placing here also clears its 1 touching neighbour.
  7. Following that, place a marker at B7. Region has exactly one open cell left. Placing here clears the rest of row 7, column B, and its region. Placing here also clears its 2 touching neighbours.
  8. First, place a marker at C9. Region has exactly one open cell left. Placing here clears the rest of row 9, column C, and its region.
  9. Next, every open cell left in this region sits in row 8, so its marker has to land there; that clears every other open cell in the row.
  10. Then, every remaining candidate in this region touches F1, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  11. After that, every remaining candidate in this region touches G2, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  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, place a marker at G6. Column 7 has exactly one open cell left. Placing here clears the rest of row 6, column G, and its region.
  14. Following that, place a marker at E1. Column 5 has exactly one open cell left. Placing here clears the rest of row 1, column E, and its region.
  15. First, place a marker at H2. Region has exactly one open cell left. Placing here clears the rest of row 2, column H, and its region.
  16. Finally, place a marker at F8. Region has exactly one open cell left.

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