An Expert 11×11 Puzzle Solved with Forced Chain in 62 Steps

11×11 · expert · 62 steps

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This 11×11 board rates as expert (difficulty score 2804). Solving it from scratch takes 62 logical steps, using 6 techniques: Single cell, Row/column exclusion, Adjacency clear, Line confinement, Shape squeeze, Forced chain. Every step below is forced: nothing here is a guess.

First, that one can't be the marker: at I1 it would force K9, K10, K11, and 6 more cleared (together, two regions have open cells only in columns K and J), and region 11 would be left with no open cell for its marker.
Next, marking J1 would force K7, K8, K9, and 2 more cleared (every open cell left in this region sits in column K, so its marker has to land there), then I11 (the last open cell in region 11), and then region 8 would be left with no open cell for its marker. So J1 can't be the marker there.
Then, if K1 were the marker, it would force J6 (the last open cell in region 5), and region 8 would end up with no open cell for its marker, which can't happen. So it can't go there.
After that, marking E2 would force C1 (the last open cell in region 2), then J3 (the last open cell in region 3), then K7, K8, K9, and 2 more cleared (every open cell left in this region sits in column K, so its marker has to land there), 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.
Now, that one can't be the marker: at G2 it would force J3 (the last open cell in region 3), then K7, K8, K9, and 2 more cleared (every open cell left in this region sits in column K, so its marker has to land there), then I11 (the last open cell in region 11), and region 8 would be left with no open cell for its marker.
From there, marking H2 would force J3 (the last open cell in region 3), then K7, K8, K9, and 2 more cleared (every open cell left in this region sits in column K, so its marker has to land there), then I11 (the last open cell in region 11), and then region 8 would be left with no open cell for its marker. So H2 can't be the marker there.
Following that, if I2 were the marker, it would force K9, K10, K11, and 6 more cleared (together, two regions have open cells only in columns K and J), and region 11 would end up with no open cell for its marker, which can't happen. So it can't go there.
First, marking A1 would force J4, J5, J6, and 5 more cleared (every open cell left in this region sits in column J, so its marker has to land there), then K7, K8, K9, and 2 more cleared (every open cell left in this region sits in column K, so its marker has to land there), then I11 (the last open cell in region 11), and then region 8 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 J4, J5, J6, and 5 more cleared (every open cell left in this region sits in column J, so its marker has to land there), then K7, K8, K9, and 2 more cleared (every open cell left in this region sits in column K, so its marker has to land there), then I11 (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.
Then, that one can't be the marker: at C1 it would force J4, J5, J6, and 5 more cleared (every open cell left in this region sits in column J, so its marker has to land there), then K7, K8, K9, and 2 more cleared (every open cell left in this region sits in column K, so its marker has to land there), then I11 (the last open cell in region 11), and region 8 would be left with no open cell for its marker.
After that, marking D1 would force J4, J5, J6, and 5 more cleared (every open cell left in this region sits in column J, so its marker has to land there), then K7, K8, K9, and 2 more cleared (every open cell left in this region sits in column K, so its marker has to land there), then I11 (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.
Now, if E1 were the marker, it would force J4, J5, J6, and 5 more cleared (every open cell left in this region sits in column J, so its marker has to land there), then K7, K8, K9, and 2 more cleared (every open cell left in this region sits in column K, so its marker has to land there), then I11 (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.
From there, that one can't be the marker: at F1 it would force J4, J5, J6, and 5 more cleared (every open cell left in this region sits in column J, so its marker has to land there), then K7, K8, K9, and 2 more cleared (every open cell left in this region sits in column K, so its marker has to land there), then I11 (the last open cell in region 11), and region 8 would be left with no open cell for its marker.
Following that, every open cell in row 1 belongs to the same region, so that region's marker has to be somewhere in this row; that clears every other open cell in the region.
First, every remaining candidate in this region touches E3, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
Next, marking K2 would force F3 (the last open cell in region 2), then I4, I5, I8, and 4 more cleared (together, two regions have open cells only in columns I and J), then G8, G9, G10, and 6 more cleared (together, two regions have open cells only in columns G and H), and the chain continues, and then region 8 would be left with no open cell for its marker. So K2 can't be the marker there.
Then, marking I3 would force A2 and B2 cleared (every open cell left in this region sits in row 2, so its marker has to land there), then K9, K10, K11, and 5 more cleared (together, two regions have open cells only in columns K and J), and then region 11 would be left with no open cell for its marker. So I3 can't be the marker there.
After that, that one can't be the marker: at I4 it would force K9, K10, K11, and 4 more cleared (together, two regions have open cells only in columns K and J), and region 11 would be left with no open cell for its marker.
Now, marking J4 would force K6 (the last open cell in region 5), then I7 (the last open cell in region 8), and then region 11 would be left with no open cell for its marker. So J4 can't be the marker there.
From there, if C5 were the marker, it would force F3 cleared (every open cell in row 2 belongs to the same region, so that region's marker has to be somewhere in this row), then E8, B9, D9, and 12 more cleared (every open cell in column A belongs to the same region, so that region's marker has to be somewhere in this column), then B7 (the last open cell in column B), and the chain continues, and region 6 would end up with no open cell for its marker, which can't happen. So it can't go there.
Following that, that one can't be the marker: at D5 it would force F6, F7, F8, and 3 more cleared (every open cell left in this region sits in column F, so its marker has to land there), then A7, E7, G7, and 4 more cleared (every open cell left in this region sits in row 7, so its marker has to land there), then A9, B9, C9, and 8 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), and the chain continues, and column C would be left with no open cell for its marker.
First, if I5 were the marker, it would force K7, K8, K9, and 2 more cleared (every open cell left in this region sits in column K, so its marker has to land there), then J7 (the last open cell in region 8), and region 11 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 J5 it would force K3 (the last open cell in region 5), then I7 (the last open cell in region 8), and region 11 would be left with no open cell for its marker.
Then, marking G4 would force H1 (the last open cell in region 3), then F6 (the last open cell in region 6), then D2 (the last open cell in region 2), and the chain continues, and then region 8 would be left with no open cell for its marker. So G4 can't be the marker there.
After that, if A6 were the marker, it would force K7, K8, K9, and 2 more cleared (every open cell left in this region sits in column K, so its marker has to land there), then C7, D7, E7, and 3 more cleared (every open cell left in this region sits in row 7, so its marker has to land there), then D8 (the last open cell in region 7), and the chain continues, and column E 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 B6 it would force K7, K8, K9, and 2 more cleared (every open cell left in this region sits in column K, so its marker has to land there), then D2, D3, D4, and 3 more cleared (every open cell left in this region sits in column D, so its marker has to land there), then F4, F5, F7, and 4 more cleared (every open cell left in this region sits in column F, so its marker has to land there), and the chain continues, and column E would be left with no open cell for its marker.
From there, if D6 were the marker, it would force F4, F5, F7, and 4 more cleared (every open cell left in this region sits in column F, so its marker has to land there), then K7, K8, K9, and 2 more cleared (every open cell left in this region sits in column K, so its marker has to land there), then A7, G7 and H7 cleared (every open cell left in this region sits in row 7, so its marker has to land there), and the chain continues, and region 11 would end up with no open cell for its marker, which can't happen. So it can't go there.
Following that, that one can't be the marker: at E6 it would force K7, K8, K9, and 2 more cleared (every open cell left in this region sits in column K, so its marker has to land there), then A7, B7, C7, and 2 more cleared (every open cell left in this region sits in row 7, so its marker has to land there), then D8 (the last open cell in region 7), and the chain continues, and column B would be left with no open cell for its marker.
First, marking F6 would force D2 (the last open cell in region 2), then K7, K8, K9, and 2 more cleared (every open cell left in this region sits in column K, so its marker has to land there), then A7, H7, I7, and 1 more cleared (every open cell left in this region sits in row 7, so its marker has to land there), and then region 8 would be left with no open cell for its marker. So F6 can't be the marker there.
Next, if G6 were the marker, it would force H1 (the last open cell in region 3), then K7, K8, K9, and 2 more cleared (every open cell left in this region sits in column K, so its marker has to land there), then A7, B7, C7, and 2 more cleared (every open cell left in this region sits in row 7, so its marker has to land there), and the chain continues, and region 11 would end up with no open cell for its marker, which can't happen. So it can't go there.
Then, that one can't be the marker: at H6 it would force G1 (the last open cell in region 3), then K7, K8, K9, and 2 more cleared (every open cell left in this region sits in column K, so its marker has to land there), then J7 (the last open cell in region 8), and the chain continues, and region 9 would be left with no open cell for its marker.
After that, that one can't be the marker: at K6 it would force I7 (the last open cell in region 8), then D8 (the last open cell in region 7), then F4, F5, F9, and 2 more cleared (every open cell left in this region sits in column F, so its marker has to land there), and the chain continues, and region 9 would be left with no open cell for its marker.
Now, marking A7 would force E8, C9, D9, and 12 more cleared (every open cell in column B belongs to the same region, so that region's marker has to be somewhere in this column), then C2, C3, D3, and 5 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 C6 (the last open cell in column C), and the chain continues, and then region 5 would be left with no open cell for its marker. So A7 can't be the marker there.
From there, marking D7 would force F4, F5, F8, and 3 more cleared (every open cell left in this region sits in column F, so its marker has to land there), then G3, G8, G9, and 7 more cleared (together, two regions have open cells only in columns G and H), then I6 and I11 cleared (every open cell left in this region sits in column I, so its marker has to land there), and the chain continues, and then region 11 would be left with no open cell for its marker. So D7 can't be the marker there.
Following that, if E7 were the marker, it would force C6 (the last open cell in region 7), then K8 (the last open cell in region 8), and region 5 would end up with no open cell for its marker, which can't happen. So it can't go there.
First, that one can't be the marker: at A2 it would force F3 (the last open cell in region 2), then G8, G9, G10, and 6 more cleared (together, two regions have open cells only in columns G and H), then I6, I7 and I11 cleared (every open cell left in this region sits in column I, so its marker has to land there), and the chain continues, and region 11 would be left with no open cell for its marker.
Next, marking B2 would force F3 (the last open cell in region 2), then E8, C9, D9, and 9 more cleared (every open cell in column A belongs to the same region, so that region's marker has to be somewhere in this column), then E5 (the last open cell in column E), and the chain continues, and then column C would be left with no open cell for its marker. So B2 can't be the marker there.
Then, if C2 were the marker, it would force F3 (the last open cell in region 2), then A9, B9, D9, and 8 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 D8 (the last open cell in column D), and the chain continues, and column B would end up with no open cell for its marker, which can't happen. So it can't go there.
After that, every open cell in row 2 belongs to the same region, so that region's marker has to be somewhere in this row; that clears every other open cell in the region.
Now, if F2 were the marker, it would force H1 (the last open cell in region 3), then G8, G9, G10, and 1 more cleared (every open cell left in this region sits in column G, so its marker has to land there), then I6, I7 and I11 cleared (every open cell left in this region sits in column I, so its marker has to land there), and the chain continues, and region 11 would end up with no open cell for its marker, which can't happen. So it can't go there.
From there, place a marker at D2. Region has exactly one open cell left. Placing here clears the rest of row 2, column D, and its region. Placing here also clears its 1 touching neighbour.
Following that, that one can't be the marker: at B3 it would force C9, C10 and C11 cleared (every open cell left in this region sits in column C, so its marker has to land there), then E8, E9, F9, and 6 more cleared (every open cell in column A belongs to the same region, so that region's marker has to be somewhere in this column), then F4 cleared (every open cell in column E 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.
First, if G3 were the marker, it would force H1 (the last open cell in region 3), then F5 (the last open cell in region 6), then I6, I7 and I11 cleared (every open cell left in this region sits in column I, so its marker has to land there), and the chain continues, and column C 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 G1 (the last open cell in region 3), then F5 (the last open cell in region 6), then I6, I7 and I11 cleared (every open cell left in this region sits in column I, so its marker has to land there), and the chain continues, and column C would be left with no open cell for its marker.
Then, that one can't be the marker: at K3 it would force F7, G7 and H7 cleared (together, two regions have open cells only in rows 6 and 7), then A4, B4, A5, and 1 more cleared (together, two regions have open cells only in rows 4 and 5), then E8, F8, G8, and 3 more cleared (every open cell left in this region sits in row 8, so its marker has to land there), and the chain continues, and row 8 would be left with no open cell for its marker.
After that, place a marker at A3. Row 3 has exactly one open cell left. Placing here clears the rest of row 3, column A, and its region.
Now, that one can't be the marker: at F4 it would force B9, C9, G9, and 5 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 B7 (the last open cell in column B), and column C would be left with no open cell for its marker.
From there, if H4 were the marker, it would force G1 (the last open cell in region 3), then E5 (the last open cell in region 4), then J6 (the last open cell in region 5), 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.
Following that, if K4 were the marker, it would force E5 (the last open cell in region 4), then G7 (the last open cell in region 6), then H1 (the last open cell in region 3), 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.
First, every open cell in row 4 belongs to the same region, so that region's marker has to be somewhere in this row; that clears every other open cell in the region.
Next, that one can't be the marker: at G5 it would force H1 (the last open cell in region 3), then J6 (the last open cell in region 5), then K8 (the last open cell in region 8), and the chain continues, and region 7 would be left with no open cell for its marker.
Then, marking H5 would force G1 (the last open cell in region 3), then J6 (the last open cell in region 5), then K8 (the last open cell in region 8), and the chain continues, and then region 7 would be left with no open cell for its marker. So H5 can't be the marker there.
After that, if E4 were the marker, it would force G7 (the last open cell in region 6), then H1 (the last open cell in region 3), then C6 (the last open cell in region 7), 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.
Now, 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.
From there, 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.
Following that, 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.
First, place a marker at J6. Region has exactly one open cell left. Placing here clears the rest of row 6, column J, and its region.
Next, place a marker at K8. Region has exactly one open cell left. Placing here clears the rest of row 8, column K, and its region.
Then, place a marker at I11. Region has exactly one open cell left. Placing here clears the rest of row 11, column I, and its region. Placing here also clears its 1 touching neighbour.
After that, place a marker at H9. Region has exactly one open cell left. Placing here clears the rest of row 9, column H, and its region. Placing here also clears its 1 touching neighbour.
Now, place a marker at G1. Region has exactly one open cell left.
Finally, place a marker at E10. Region has exactly one open cell left.

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