An Expert 9×9 Puzzle Solved with Forced Chain in 44 Steps

9×9 · expert · 44 steps

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This 9×9 board rates as expert (difficulty score 1947). Solving it from scratch takes 44 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.

First, marking A1 would force I2 (the last open cell in region 3), then D3 (the last open cell in region 2), then H4 (the last open cell in region 4), and the chain continues, and then region 5 would be left with no open cell for its marker. So A1 can't be the marker there.
Next, if B1 were the marker, it would force I2 (the last open cell in region 3), then D3 (the last open cell in region 2), then H4 (the last open cell in region 4), and the chain continues, and row 5 would end up with no open cell for its marker, which can't happen. So it can't go there.
Then, if H1 were the marker, it would force A2, A3, B3, and 4 more cleared (together, two regions have open cells only in rows 2 and 3), then A6, A7, A8, and 1 more cleared (every open cell left in this region sits in column A, so its marker has to land there), then B2, B4, B7, and 2 more cleared (every open cell left in this region sits in column B, so its marker has to land there), and the chain continues, and row 7 would end up with no open cell for its marker, which can't happen. So it can't go there.
After that, that one can't be the marker: at A2 it would force C1 and D1 cleared (every open cell left in this region sits in row 1, so its marker has to land there), then D3 (the last open cell in region 2), then H4 (the last open cell in region 4), and the chain continues, and region 5 would be left with no open cell for its marker.
Now, every remaining candidate in this region touches B4, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
From there, marking B2 would force A6, A7, A8, and 1 more cleared (every open cell left in this region sits in column A, so its marker has to land there), and then region 7 would be left with no open cell for its marker. So B2 can't be the marker there.
Following that, if C2 were the marker, it would force A6, A7, A8, and 1 more cleared (every open cell left in this region sits in column A, so its marker has to land there), then B7, B8 and B9 cleared (every open cell left in this region sits in column B, so its marker has to land there), then I6, G7, I7, and 4 more cleared (every open cell in row 9 belongs to the same region, so that region's marker has to be somewhere in this row), and the chain continues, and row 9 would end up with no open cell for its marker, which can't happen. So it can't go there.
First, if E1 were the marker, it would force D3 (the last open cell in region 2), then F6 (the last open cell in region 6), then A7, A8 and A9 cleared (every open cell left in this region sits in column A, so its marker has to land there), and the chain continues, and column A would end up with no open cell for its marker, which can't happen. So it can't go there.
Next, marking E2 would force C1 (the last open cell in region 2), and then region 3 would be left with no open cell for its marker. So E2 can't be the marker there.
Then, every remaining candidate in this region touches G3, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
After that, if F2 were the marker, it would force I1 (the last open cell in region 3), then D3 (the last open cell in region 2), then E6 (the last open cell in region 6), and the chain continues, and column A would end up with no open cell for its marker, which can't happen. So it can't go there.
Now, that one can't be the marker: at G2 it would force I1 (the last open cell in region 3), then D3 (the last open cell in region 2), then A6, A7, A8, and 1 more cleared (every open cell left in this region sits in column A, so its marker has to land there), and the chain continues, and column A would be left with no open cell for its marker.
From there, marking H2 would force F1 (the last open cell in region 3), then D3 (the last open cell in region 2), then E6 (the last open cell in region 6), and the chain continues, and then column A would be left with no open cell for its marker. So H2 can't be the marker there.
Following that, every remaining candidate in this region touches G4, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
First, if A3 were the marker, it would force H4 (the last open cell in region 4), then B7, B8 and B9 cleared (every open cell left in this region sits in column B, so its marker has to land there), then I6, G7, I7, and 5 more cleared (every open cell in row 9 belongs to the same region, so that region's marker has to be somewhere in this row), and the chain continues, and row 7 would end up with no open cell for its marker, which can't happen. So it can't go there.
Next, marking C3 would force D1 (the last open cell in region 2), then I2 (the last open cell in region 3), then H4 (the last open cell in region 4), and the chain continues, and then row 5 would be left with no open cell for its marker. So C3 can't be the marker there.
Then, together, two regions have open cells only in columns B and A; with one marker each, those two markers have to fill exactly those two columns, so every other region's open cells there can be cleared.
After that, if D3 were the marker, it would force H4 (the last open cell in region 4), then A5 (the last open cell in region 1), and column B would end up with no open cell for its marker, which can't happen. So it can't go there.
Now, that one can't be the marker: at E3 it would force H4 (the last open cell in region 4), then A5 (the last open cell in region 1), and column B would be left with no open cell for its marker.
From there, marking F3 would force A6, A7, A8, and 1 more cleared (every open cell left in this region sits in column A, so its marker has to land there), then I6, G7, I7, and 5 more cleared (every open cell in row 9 belongs to the same region, so that region's marker has to be somewhere in this row), then H8 (the last open cell in row 8), and the chain continues, and then column I would be left with no open cell for its marker. So F3 can't be the marker there.
Following that, 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.
First, every remaining candidate in this region touches I3 and I4, so whichever candidate ends up marked will clear them, and they can be ruled out now regardless of which one wins.
Next, marking D1 would force I2 (the last open cell in region 3), then H4 (the last open cell in region 4), then B3 (the last open cell in row 3), and the chain continues, and then row 5 would be left with no open cell for its marker. So D1 can't be the marker there.
Then, that one can't be the marker: at H3 it would force D2 (the last open cell in row 2), then A6, A7, A8, and 1 more cleared (every open cell left in this region sits in column A, so its marker has to land there), then I6, G7, I7, and 4 more cleared (every open cell in row 9 belongs to the same region, so that region's marker has to be somewhere in this row), and the chain continues, and row 7 would be left with no open cell for its marker.
After that, place a marker at H4. Region has exactly one open cell left. Placing here clears the rest of row 4, column H, and its region. Placing here also clears its 2 touching neighbours.
Now, place a marker at B3. Row 3 has exactly one open cell left. Placing here clears the rest of row 3, column B, and its region.
From there, together, two regions have open cells only in rows 5 and 6; 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.
Following that, every remaining candidate in this region touches E5, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
First, every remaining candidate in this region touches F6, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
Next, that one can't be the marker: at F1 it would force D2 (the last open cell in region 2), and row 5 would be left with no open cell for its marker.
Then, marking G1 would force D2 (the last open cell in region 2), then E6 (the last open cell in region 6), and then row 5 would be left with no open cell for its marker. So G1 can't be the marker there.
After that, 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.
Now, that one can't be the marker: at I1 it would force D2 (the last open cell in region 2), then E6 (the last open cell in region 6), and row 5 would be left with no open cell for its marker.
From there, place a marker at I2. Region has exactly one open cell left. Placing here clears the rest of row 2, column I, and its region.
Following that, place a marker at C1. Region has exactly one open cell left. Placing here clears the rest of row 1, column C, and its region.
First, every remaining candidate in this region touches E8, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
Next, if F5 were the marker, it would force D6 (the last open cell in region 6), and region 8 would end up with no open cell for its marker, which can't happen. So it can't go there.
Then, place a marker at G6. Region has exactly one open cell left. Placing here clears the rest of row 6, column G, and its region. Placing here also clears its 1 touching neighbour.
After that, place a marker at D5. Region has exactly one open cell left. Placing here clears the rest of row 5, column D, and its region.
Now, 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.
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.
Following that, 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.
First, place a marker at A8. Region has exactly one open cell left.
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. ∎