An Expert 11×11 Puzzle Solved with Forced Chain in 48 Steps

11×11 · expert · 48 steps

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This 11×11 board rates as expert (difficulty score 2241). Solving it from scratch takes 48 logical steps, using 7 techniques: Single cell, Row/column exclusion, Adjacency clear, Region confinement, Line confinement, Shape squeeze, Forced chain. Every step below is forced: nothing here is a guess.

First, every open cell left in this region sits in row 11, so its marker has to land there; that clears every other open cell in the row.
Next, every remaining candidate in this region touches B9, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
Then, every remaining candidate in this region touches B10, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
After that, marking A1 would force B8 (the last open cell in region 10), then C11 (the last open cell in region 11), then D9 (the last open cell in region 8), and then region 2 would be left with no open cell for its marker. So A1 can't be the marker there.
Now, if B1 were the marker, it would force A3, A4, A5, and 3 more cleared (every open cell left in this region sits in column A, so its marker has to land there), then C11 (the last open cell in region 11), then D9 (the last open cell in region 8), and region 1 would end up with no open cell for its marker, which can't happen. So it can't go there.
From there, that one can't be the marker: at C1 it would force A2, A3, A4, and 6 more cleared (together, two regions have open cells only in columns A and B), then D4, D8, D9, and 1 more cleared (every open cell left in this region sits in column D, so its marker has to land there), and region 8 would be left with no open cell for its marker.
Following that, that one can't be the marker: at A2 it would force B8 (the last open cell in region 10), then C11 (the last open cell in region 11), then D9 (the last open cell in region 8), and the chain continues, and region 7 would be left with no open cell for its marker.
First, marking B2 would force A4, A5, A6, and 2 more cleared (every open cell left in this region sits in column A, so its marker has to land there), then C11 (the last open cell in region 11), then D9 (the last open cell in region 8), and then region 1 would be left with no open cell for its marker. So B2 can't be the marker there.
Next, if C2 were the marker, it would force A3, A4, A5, and 5 more cleared (together, two regions have open cells only in columns A and B), then D4, D8, D9, and 1 more cleared (every open cell left in this region sits in column D, so its marker has to land there), and region 8 would end up with no open cell for its marker, which can't happen. So it can't go there.
Then, marking H2 would force D1 and E1 cleared (every open cell left in this region sits in row 1, so its marker has to land there), then F5, F6, D8, and 1 more cleared (every open cell in column E belongs to the same region, so that region's marker has to be somewhere in this column), then G7, G8, G9, and 1 more cleared (every open cell in column F belongs to the same region, so that region's marker has to be somewhere in this column), and the chain continues, and then column K would be left with no open cell for its marker. So H2 can't be the marker there.
After that, if A3 were the marker, it would force B8 (the last open cell in region 10), then C11 (the last open cell in region 11), then D9 (the last open cell in region 8), and the chain continues, and region 7 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 B3 it would force A5, A6, A7, and 1 more cleared (every open cell left in this region sits in column A, so its marker has to land there), then C11 (the last open cell in region 11), then D9 (the last open cell in region 8), and region 1 would be left with no open cell for its marker.
From there, marking C3 would force A4, A5, A6, and 4 more cleared (together, two regions have open cells only in columns A and B), then D8, D9 and D10 cleared (every open cell left in this region sits in column D, so its marker has to land there), and then region 8 would be left with no open cell for its marker. So C3 can't be the marker there.
Following that, if J3 were the marker, it would force D1, E1, F1, and 2 more cleared (every open cell left in this region sits in row 1, so its marker has to land there), then F7, G7, G8, and 4 more cleared (every open cell in column H belongs to the same region, so that region's marker has to be somewhere in this column), then G2 (the last open cell in column G), and the chain continues, and column F would end up with no open cell for its marker, which can't happen. So it can't go there.
First, if B4 were the marker, it would force D8, D9 and D10 cleared (every open cell left in this region sits in column D, so its marker has to land there), then A6, A7 and A11 cleared (every open cell left in this region sits in column A, so its marker has to land there), then C11 (the last open cell in region 11), and region 8 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 C4 (the last open cell in region 2), then I3 cleared (every open cell in row 1 belongs to the same region, so that region's marker has to be somewhere in this row), then G5, H5, J5, and 4 more cleared (every open cell in row 3 belongs to the same region, so that region's marker has to be somewhere in this row), and the chain continues, and then region 8 would be left with no open cell for its marker. So E2 can't be the marker there.
Then, that one can't be the marker: at C4 it would force A5, A6, A7, and 2 more cleared (together, two regions have open cells only in columns A and B), then D8, D9 and D10 cleared (every open cell left in this region sits in column D, so its marker has to land there), and region 8 would be left with no open cell for its marker.
After that, that one can't be the marker: at K3 it would force F1, G1, H1, and 2 more cleared (together, two regions have open cells only in rows 1 and 2), then F7, H8, F9, and 3 more cleared (every open cell in column G belongs to the same region, so that region's marker has to be somewhere in this column), and column H would be left with no open cell for its marker.
Now, that one can't be the marker: at I1 it would force D4, D5, D6, and 4 more cleared (every open cell left in this region sits in column D, so its marker has to land there), then J6, J7, J8, and 2 more cleared (every open cell in column K belongs to the same region, so that region's marker has to be somewhere in this column), then G3, H3, G4, and 6 more cleared (every open cell in column J belongs to the same region, so that region's marker has to be somewhere in this column), and the chain continues, and column F would be left with no open cell for its marker.
From there, marking J1 would force D4, D5, D6, and 4 more cleared (every open cell left in this region sits in column D, so its marker has to land there), then I5, I6, I8, and 2 more cleared (every open cell in column K belongs to the same region, so that region's marker has to be somewhere in this column), then G3, H3, G4, and 6 more cleared (every open cell in column I belongs to the same region, so that region's marker has to be somewhere in this column), and the chain continues, and then column F would be left with no open cell for its marker. So J1 can't be the marker there.
Following that, if I2 were the marker, it would force J6, J7, J8, and 2 more cleared (every open cell in column K belongs to the same region, so that region's marker has to be somewhere in this column), then G3, G4, H4, and 5 more cleared (every open cell in column J belongs to the same region, so that region's marker has to be somewhere in this column), then F7, G7, G8, and 4 more 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 F would end up with no open cell for its marker, which can't happen. So it can't go there.
First, if G3 were the marker, it would force K4, K5, J6, and 9 more cleared (every open cell in column I belongs to the same region, so that region's marker has to be somewhere in this column), then J2 (the last open cell in column J), and column K would end up with no open cell for its marker, which can't happen. So it can't go there.
Next, that one can't be the marker: at H3 it would force K4, K5, J6, and 9 more cleared (every open cell in column I belongs to the same region, so that region's marker has to be somewhere in this column), then J2 (the last open cell in column J), and column K would be left with no open cell for its marker.
Then, marking D4 would force E1 (the last open cell in region 2), then I3 (the last open cell in row 3), and then row 2 would be left with no open cell for its marker. So D4 can't be the marker there.
After that, if E4 were the marker, it would force I3 (the last open cell in row 3), then D1 (the last open cell in row 1), and row 2 would end up with no open cell for its marker, which can't happen. So it can't go there.
Now, if H4 were the marker, it would force K5, J6, K6, and 8 more cleared (every open cell in column I belongs to the same region, so that region's marker has to be somewhere in this column), then J2 (the last open cell in column J), and column K would end up with no open cell for its marker, which can't happen. So it can't go there.
From there, that one can't be the marker: at A5 it would force B8 (the last open cell in region 10), then C11 (the last open cell in region 11), then D9 (the last open cell in region 8), and the chain continues, and region 9 would be left with no open cell for its marker.
Following that, marking B5 would force A7 and A11 cleared (every open cell left in this region sits in column A, so its marker has to land there), then C11 (the last open cell in region 11), then D9 (the last open cell in region 8), and the chain continues, and then column J would be left with no open cell for its marker. So B5 can't be the marker there.
First, if C5 were the marker, it would force A6, A7 and B7 cleared (together, two regions have open cells only in columns A and B), then D1, D2, D3, and 1 more cleared (every open cell left in this region sits in column D, so its marker has to land there), then E1 (the last open cell in region 2), and the chain continues, and column J would end up with no open cell for its marker, which can't happen. So it can't go there.
Next, marking D1 would force A6, A7, A11, and 2 more cleared (together, two regions have open cells only in columns A and B), then C11 (the last open cell in region 11), and then region 8 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 F1 it would force D5, D6, D7, and 3 more cleared (every open cell left in this region sits in column D, so its marker has to land there), then A6, A7, A11, and 2 more cleared (together, two regions have open cells only in columns A and B), then C11 (the last open cell in region 11), and region 8 would be left with no open cell for its marker.
After that, marking G1 would force D5, D6, D7, and 3 more cleared (every open cell left in this region sits in column D, so its marker has to land there), then F5, F6 and F8 cleared (every open cell in column E belongs to the same region, so that region's marker has to be somewhere in this column), then H8, H9 and H10 cleared (every open cell in column F belongs to the same region, so that region's marker has to be somewhere in this column), and the chain continues, and then region 8 would be left with no open cell for its marker. So G1 can't be the marker there.
Now, if H1 were the marker, it would force D5, D6, D7, and 3 more cleared (every open cell left in this region sits in column D, so its marker has to land there), then F5, F6 and F8 cleared (every open cell in column E belongs to the same region, so that region's marker has to be somewhere in this column), then G7, G8, G9, and 1 more cleared (every open cell in column F belongs to the same region, so that region's marker has to be somewhere in this column), and the chain continues, and region 8 would end up with no open cell for its marker, which can't happen. So it can't go there.
From there, if K1 were the marker, it would force D5, D6, D7, and 3 more cleared (every open cell left in this region sits in column D, so its marker has to land there), then A6, A7, A11, and 2 more cleared (together, two regions have open cells only in columns A and B), then C11 (the last open cell in region 11), and region 8 would end up with no open cell for its marker, which can't happen. So it can't go there.
Following that, place a marker at E1. Row 1 has exactly one open cell left. Placing here clears the rest of row 1, column E, and its region. Placing here also clears its 1 touching neighbour.
First, that one can't be the marker: at J2 it would force F3 (the last open cell in row 3), then D8 (the last open cell in region 7), then A4, A6, A7, 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 row 11 would be left with no open cell for its marker.
Next, marking K2 would force F3 (the last open cell in row 3), then D8 (the last open cell in region 7), then A4, A6, A7, 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 then row 11 would be left with no open cell for its marker. So K2 can't be the marker there.
Then, 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.
After that, place a marker at G2. Row 2 has exactly one open cell left. Placing here clears the rest of row 2, column G, and its region.
Now, every open cell in column K 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.
From there, place a marker at J5. Column 10 has exactly one open cell left. Placing here clears the rest of row 5, column J, and its region. Placing here also clears its 2 touching neighbours.
Following that, place a marker at A4. Row 4 has exactly one open cell left. Placing here clears the rest of row 4, column A, and its region.
First, place a marker at B8. Region has exactly one open cell left. Placing here clears the rest of row 8, column B, and its region. Placing here also clears its 2 touching neighbours.
Next, place a marker at F6. Region has exactly one open cell left. Placing here clears the rest of row 6, column F, and its region.
Then, place a marker at C11. Region has exactly one open cell left. Placing here clears the rest of row 11, column C, and its region. Placing here also clears its 1 touching neighbour.
After that, 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.
Now, place a marker at H10. Region has exactly one open cell left. Placing here clears the rest of row 10, column H, and its region.
Finally, place a marker at K7. Region has exactly one open cell left.

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