QED Logic quod erat demonstrandum

An Expert 9×9 Puzzle Solved with Forced Chain in 19 Steps

9×9 · expert · 19 steps

Play this board yourself →
A
B
C
D
E
F
G
H
I
1
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8
9
1
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4×
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7
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8
9×
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This 9×9 board rates as expert (difficulty score 1132). Solving it from scratch takes 19 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 B, so its marker has to land there; that clears every other open cell in the column.
  2. Next, together, two regions have open cells only in rows 8 and 9; 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.
  3. Then, every remaining candidate in this region touches D6, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  4. After that, every remaining candidate in this region touches C8 and C9, so whichever candidate ends up marked will clear them, and they can be ruled out now regardless of which one wins.
  5. Now, place a marker at D9. Region has exactly one open cell left. Placing here clears the rest of row 9, column D, and its region.
  6. From there, place a marker at B8. Region has exactly one open cell left. Placing here also clears its 2 touching neighbours.
  7. Following that, that one can't be the marker: at C1 it would force E7 (the last open cell in region 6), then A2 (the last open cell in region 1), and region 4 would be left with no open cell for its marker.
  8. First, if E1 were the marker, it would force H2 and I2 cleared (every open cell left in this region sits in row 2, so its marker has to land there), then C2 and C4 cleared (every open cell left in this region sits in column C, so its marker has to land there), then A2 (the last open cell in region 1), and region 4 would end up with no open cell for its marker, which can't happen. So it can't go there.
  9. Next, that one can't be the marker: at F1 it would force H2 and I2 cleared (every open cell left in this region sits in row 2, so its marker has to land there), then C5 and C6 cleared (together, two regions have open cells only in columns A and C), then E7 (the last open cell in region 6), and the chain continues, and region 5 would be left with no open cell for its marker.
  10. Then, marking G1 would force I7 (the last open cell in column I), and then column H would be left with no open cell for its marker. So G1 can't be the marker there.
  11. After that, if H1 were the marker, it would force I7 (the last open cell in column I), then F2 cleared (every open cell left in this region sits in row 2, so its marker has to land there), then A3 cleared (every open cell left in this region sits in row 3, so its marker has to land there), and the chain continues, and row 3 would end up with no open cell for its marker, which can't happen. So it can't go there.
  12. Now, that one can't be the marker: at I1 it would force F2 and G2 cleared (every open cell left in this region sits in row 2, so its marker has to land there), then A3 cleared (every open cell left in this region sits in row 3, so its marker has to land there), then F7 and G7 cleared (every open cell in column H belongs to the same region, so that region's marker has to be somewhere in this column), and the chain continues, and column G would be left with no open cell for its marker.
  13. From there, place a marker at A1. Row 1 has exactly one open cell left. Placing here clears the rest of row 1, column A, and its region.
  14. Following that, place a marker at C4. Region has exactly one open cell left. Placing here clears the rest of row 4, column C, and its region.
  15. First, place a marker at E7. Region has exactly one open cell left. Placing here clears the rest of row 7, column E, and its region. Placing here also clears its 1 touching neighbour.
  16. Next, place a marker at H6. Region has exactly one open cell left. Placing here clears the rest of row 6, column H, and its region. Placing here also clears its 2 touching neighbours.
  17. Then, place a marker at F5. Region has exactly one open cell left. Placing here clears the rest of row 5, column F, and its region.
  18. After that, place a marker at G2. Region has exactly one open cell left. Placing here clears the rest of row 2, column G, and its region.
  19. Finally, place a marker at I3. Region has exactly one open cell left.

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