An Expert 11×11 Puzzle Solved with Forced Chain in 52 Steps

11×11 · expert · 52 steps

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This 11×11 board rates as expert (difficulty score 2277). Solving it from scratch takes 52 logical steps, using 8 techniques: Single cell, Row/column exclusion, Adjacency clear, Region confinement, Line confinement, Paired regions, Shape squeeze, Forced chain. Every step below is forced: nothing here is a guess.

First, marking A1 would force J2 (the last open cell in region 3), then K4 (the last open cell in region 6), then B3 (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 J2 (the last open cell in region 3), then K4 (the last open cell in region 6), then A7, A8, A9, and 2 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 region 2 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 J2 (the last open cell in region 3), then K4 (the last open cell in region 6), then A3, B3, A5, and 2 more cleared (together, two regions have open cells only in rows 3 and 5), and the chain continues, and column D would be left with no open cell for its marker.
After that, if E1 were the marker, it would force J2 (the last open cell in region 3), then K4 (the last open cell in region 6), then A3, B3, A5, and 2 more cleared (together, two regions have open cells only in rows 3 and 5), and the chain continues, and column D 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 F1 it would force J2 (the last open cell in region 3), then K4 (the last open cell in region 6), then A3, B3, A5, and 3 more cleared (together, two regions have open cells only in rows 3 and 5), and the chain continues, and column D would be left with no open cell for its marker.
From there, marking B2 would force D1 cleared (every open cell left in this region sits in row 1, so its marker has to land there), then A7, A8, A9, and 2 more cleared (every open cell left in this region sits in column A, so its marker has to land there), then D6, D7, D8, and 4 more cleared (every open cell in column C belongs to the same region, so that region's marker has to be somewhere in this column), and the chain continues, and then column A 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 A7, A8, A9, and 2 more cleared (every open cell left in this region sits in column A, so its marker has to land there), then D6, D7, D8, and 4 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 D5 (the last open cell in column D), 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.
First, that one can't be the marker: at D2 it would force B6, A7, B7, and 9 more cleared (every open cell in column C belongs to the same region, so that region's marker has to be somewhere in this column), then B3 (the last open cell in column B), and column A would be left with no open cell for its marker.
Next, that one can't be the marker: at B3 it would force D4 (the last open cell in region 1), then J2 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 C6, C7, C8, and 4 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), and column C would be left with no open cell for its marker.
Then, every open cell left in this region sits in column A, so its marker has to land there; that clears every other open cell in the column.
After that, if D3 were the marker, it would force J2 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 C5, C6, C7, and 5 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), and column C 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 C4 it would force D6, D7, D8, and 4 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 D1 (the last open cell in column D), then J2 (the last open cell in region 3), and region 6 would be left with no open cell for its marker.
From there, if A3 were the marker, it would force B6, D6, B7, and 10 more cleared (every open cell in column C belongs to the same region, so that region's marker has to be somewhere in this column), then B5 (the last open cell in column B), then D1 (the last open cell in column D), and the chain continues, and region 5 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 E3 it would force B6, B7, B8, and 3 more cleared (every open cell left in this region sits in column B, so its marker has to land there), then J2 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 D6, D7, D8, and 3 more cleared (every open cell in column C belongs to the same region, so that region's marker has to be somewhere in this column), and the chain continues, and column A would be left with no open cell for its marker.
First, marking H2 would force D1 cleared (every open cell left in this region sits in row 1, so its marker has to land there), then J7, J8, J9, and 9 more cleared (together, two regions have open cells only in columns J and K), then I8, I9 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 column A would be left with no open cell for its marker. So H2 can't be the marker there.
Next, marking F3 would force J2 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 B6, D6, B7, and 10 more cleared (every open cell in column C belongs to the same region, so that region's marker has to be somewhere in this column), then D4 cleared (every open cell in column B belongs to the same region, so that region's marker has to be somewhere in this column), and the chain continues, and then column A would be left with no open cell for its marker. So F3 can't be the marker there.
Then, if G3 were the marker, it would force J2 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 B6, D6, B7, and 10 more cleared (every open cell in column C belongs to the same region, so that region's marker has to be somewhere in this column), then D4 cleared (every open cell in column B belongs to the same region, so that region's marker has to be somewhere in this column), 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.
After that, marking I3 would force D1 cleared (every open cell left in this region sits in row 1, so its marker has to land there), then B6, D6, B7, and 10 more cleared (every open cell in column C belongs to the same region, so that region's marker has to be somewhere in this column), then D4 cleared (every open cell in column B belongs to the same region, so that region's marker has to be somewhere in this column), and the chain continues, and then column A would be left with no open cell for its marker. So I3 can't be the marker there.
Now, if J3 were the marker, it would force D1 cleared (every open cell left in this region sits in row 1, so its marker has to land there), then B6, D6, B7, and 10 more cleared (every open cell in column C belongs to the same region, so that region's marker has to be somewhere in this column), then D4 cleared (every open cell in column B belongs to the same region, so that region's marker has to be somewhere in this column), and the chain continues, and region 5 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 K3 it would force D1 cleared (every open cell left in this region sits in row 1, so its marker has to land there), then B6, D6, B7, and 10 more cleared (every open cell in column C belongs to the same region, so that region's marker has to be somewhere in this column), then D4 cleared (every open cell in column B belongs to the same region, so that region's marker has to be somewhere in this column), and the chain continues, and region 2 would be left with no open cell for its marker.
Following that, if H1 were the marker, it would force C3 (the last open cell in row 3), then I6, I7, I8, and 3 more cleared (every open cell left in this region sits in column I, so its marker has to land there), then D6, D7, D8, and 4 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), 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.
First, marking J1 would force K4 (the last open cell in region 6), then A5, C5, D5, and 3 more cleared (together, two regions have open cells only in rows 3 and 5), then A2 cleared (every open cell left in this region sits in row 2, so its marker has to land there), and the chain continues, and then column D would be left with no open cell for its marker. So J1 can't be the marker there.
Next, that one can't be the marker: at G2 it would force C3 (the last open cell in row 3), then D6, D7, D8, and 4 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 D5 (the last open cell in column D), and the chain continues, and column A would be left with no open cell for its marker.
Then, if I2 were the marker, it would force C3 (the last open cell in row 3), then D1 cleared (every open cell left in this region sits in row 1, so its marker has to land there), then D6, D7, D8, and 4 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), 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.
After that, marking G1 would force E4, E5, E6, and 5 more cleared (together, two regions have open cells only in columns E and F), then H6, H7, I6, and 2 more cleared (together, two regions have open cells only in columns H and I), then D5 (the last open cell in region 7), and the chain continues, and then column A would be left with no open cell for its marker. So G1 can't be the marker there.
Now, marking K2 would force I1 (the last open cell in region 3), then H6, H7, H8, and 3 more cleared (every open cell left in this region sits in column H, so its marker has to land there), then F6, F7, F10, and 5 more cleared (together, two regions have open cells only in columns F and G), and the chain continues, and then column H would be left with no open cell for its marker. So K2 can't be the marker there.
From there, every remaining candidate in this region touches K5, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
Following that, marking A4 would force C3 (the last open cell in region 1), then J2, J7, J8, and 3 more cleared (every open cell left in this region sits in column J, so its marker has to land there), then D1 cleared (every open cell left in this region sits in row 1, so its marker has to land there), and the chain continues, and then column D would be left with no open cell for its marker. So A4 can't be the marker there.
First, if B4 were the marker, it would force H3 (the last open cell in row 3), then J2, J7, J8, and 3 more cleared (every open cell left in this region sits in column J, so its marker has to land there), then D1 cleared (every open cell left in this region sits in row 1, so its marker has to land there), and the chain continues, and region 2 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 B5 and B6 cleared (every remaining candidate in this region touches B5 and B6), then C5, C6, D6, and 10 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 C3 (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 E2 can't be the marker there.
Then, that one can't be the marker: at I1 it would force H6, H7, H8, and 3 more cleared (every open cell left in this region sits in column H, so its marker has to land there), then F6, F7, F10, and 5 more cleared (together, two regions have open cells only in columns F and G), then E4, E5, E6, and 1 more cleared (every open cell left in this region sits in column E, so its marker has to land there), and the chain continues, and column A would be left with no open cell for its marker.
After that, together, two regions have open cells only in columns K and J; 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.
Now, together, two regions have open cells only in columns H and I; 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.
From there, if K1 were the marker, it would force F6, F7, F10, and 5 more cleared (together, two regions have open cells only in columns F and G), then E4, E5, E6, and 1 more cleared (every open cell left in this region sits in column E, so its marker has to land there), then D5 (the last open cell in region 7), 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.
Following that, place a marker at J2. Region has exactly one open cell left. Placing here clears the rest of row 2, column J, and its region.
First, place a marker at K4. Region has exactly one open cell left. Placing here clears the rest of row 4, column K, and its region.
Next, place a marker at D1. Row 1 has exactly one open cell left. Placing here clears the rest of row 1, column D, and its region.
Then, together, two regions have open cells only in rows 3 and 5; 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.
After that, place a marker at A6. Region has exactly one open cell left. Placing here clears the rest of row 6, column A, and its region. Placing here also clears its 2 touching neighbours.
Now, place a marker at C3. Region has exactly one open cell left. Placing here clears the rest of row 3, column C, and its region.
From there, every open cell in column B 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.
Following that, every open cell in column E 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.
First, 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.
Next, if I5 were the marker, it would force H7 (the last open cell in region 9), and region 7 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 H5. Region has exactly one open cell left. Placing here clears the rest of row 5, column H, and its region.
After that, if E7 were the marker, it would force G8 (the last open cell in region 7), and column F would end up with no open cell for its marker, which can't happen. So it can't go there.
Now, every remaining candidate in this region touches F9, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
From there, place a marker at F7. Column 6 has exactly one open cell left. Placing here clears the rest of row 7, column F, and its region. Placing here also clears its 1 touching neighbour.
Following that, place a marker at I10. Region has exactly one open cell left. Placing here clears the rest of row 10, column I, and its region.
First, place a marker at E9. Region has exactly one open cell left. Placing here clears the rest of row 9, column E, and its region.
Next, place a marker at G11. Region has exactly one open cell left. Placing here clears the rest of row 11, column G, and its region.
Finally, place a marker at B8. Region has exactly one open cell left.

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