A Severe 11×11 Puzzle Solved with Forced Chain in 71 Steps

11×11 · severe · 71 steps

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This 11×11 board rates as severe (difficulty score 3167). Solving it from scratch takes 71 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, if B1 were the marker, it would force A11 (the last open cell in column A), then D2, D4, D5, and 2 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 D3 (the last open cell in column D), 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 C1 it would force A11 (the last open cell in column A), then B4 (the last open cell in column B), then D3 (the last open cell in column D), and the chain continues, and region 7 would be left with no open cell for its marker.
Then, marking D1 would force A11 (the last open cell in column A), then B4 (the last open cell in column B), and then column C would be left with no open cell for its marker. So D1 can't be the marker there.
After that, marking G1 would force I3, I4, I5, and 10 more cleared (together, two regions have open cells only in columns I and H), then A2, B2, C2, and 6 more cleared (together, two regions have open cells only in rows 2 and 3), then J2, J3, J9, and 6 more cleared (together, two regions have open cells only in columns J and K), 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, region 3 would end up with no open cell for its marker, which can't happen. So it can't go there.
From there, marking B2 would force A11 (the last open cell in column A), then C4 (the last open cell in column C), and then column D 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 B7, B8, E8, and 3 more cleared (every open cell in column D belongs to the same region, so that region's marker has to be somewhere in this column), then B5, B6, B9, and 1 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 B would end up with no open cell for its marker, which can't happen. So it can't go there.
First, if A3 were the marker, it would force C8, D8, E8, and 6 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 C4 (the last open cell in column C), and column D 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 B3 it would force A11 (the last open cell in column A), and column C would be left with no open cell for its marker.
Then, marking C3 would force B7, B8, E8, and 3 more cleared (every open cell in column D belongs to the same region, so that region's marker has to be somewhere in this column), then B5, B6, B9, and 1 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 then column B would be left with no open cell for its marker. So C3 can't be the marker there.
After that, marking I3 would force A1, E1 and F1 cleared (every open cell left in this region sits in row 1, so its marker has to land there), then J6, J7, J8, and 8 more cleared (together, two regions have open cells only in columns J and K), then A11, B11, C11, and 1 more cleared (every open cell left in this region sits in row 11, 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 I3 can't be the marker there.
Now, marking A4 would force D3, D8, D9, and 2 more cleared (every open cell left in this region sits in column D, so its marker has to land there), then C8, E8, C9, and 2 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 then column C would be left with no open cell for its marker. So A4 can't be the marker there.
From there, that one can't be the marker: at C4 it would force B7, B8, E8, and 3 more cleared (every open cell in column D belongs to the same region, so that region's marker has to be somewhere in this column), then B6, B9 and B10 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 B would be left with no open cell for its marker.
Following that, marking A1 would force B7, B8, D8, and 6 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 B4 (the last open cell in column B), then D3 (the last open cell in column D), and the chain continues, and then region 7 would be left with no open cell for its marker. So A1 can't be the marker there.
First, that one can't be the marker: at A5 it would force D3, D8, D9, and 2 more cleared (every open cell left in this region sits in column D, so its marker has to land there), then C8, E8, C9, and 2 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 be left with no open cell for its marker.
Next, marking B5 would force A11 (the last open cell in column A), and then column C would be left with no open cell for its marker. So B5 can't be the marker there.
Then, if C5 were the marker, it would force A11 (the last open cell in column A), 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, if A6 were the marker, it would force B8, D8, E8, and 5 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 B4 (the last open cell in column B), then D3 (the last open cell in column D), 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.
Now, that one can't be the marker: at B6 it would force A11 (the last open cell in column A), and column C would be left with no open cell for its marker.
From there, marking C6 would force A11 (the last open cell in column A), then B4 (the last open cell in column B), then D3 (the last open cell in column D), and the chain continues, and then column F would be left with no open cell for its marker. So C6 can't be the marker there.
Following that, marking A7 would force D8, E8, D9, 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), then B4 (the last open cell in column B), then D3 (the last open cell in column D), and the chain continues, and then region 7 would be left with no open cell for its marker. So A7 can't be the marker there.
First, if B7 were the marker, it would force C10 (the last open cell in column C), and column A 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 C7 it would force A11 (the last open cell in column A), then B4 (the last open cell in column B), then D3 (the last open cell in column D), and the chain continues, and column F would be left with no open cell for its marker.
Then, that one can't be the marker: at A8 it would force D9, E9, D10, and 2 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 B4 (the last open cell in column B), then D3 (the last open cell in column D), and the chain continues, and row 1 would be left with no open cell for its marker.
After that, marking B8 would force C10 (the last open cell in column C), and then column A would be left with no open cell for its marker. So B8 can't be the marker there.
Now, if C8 were the marker, it would force J9, K9, I10, and 2 more cleared (every open cell in row 11 belongs to the same region, so that region's marker has to be somewhere in this row), then B10 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 B4 (the last open cell in column B), and the chain continues, and row 11 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 D8 it would force B4 (the last open cell in region 4), then C10 (the last open cell in column C), and column A would be left with no open cell for its marker.
Following that, marking E8 would force C10 (the last open cell in column C), and then column A would be left with no open cell for its marker. So E8 can't be the marker there.
First, if A9 were the marker, it would force C11 (the last open cell in column C), then B4 (the last open cell in column B), then D3 (the last open cell in column D), 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.
Next, that one can't be the marker: at B9 it would force A11 (the last open cell in column A), and column C would be left with no open cell for its marker.
Then, that one can't be the marker: at D2 it would force E10, F10, G10, and 4 more cleared (every open cell left in this region sits in row 10, so its marker has to land there), then B11 cleared (every remaining candidate in this region touches B11), then B10 (the last open cell in column B), and column A would be left with no open cell for its marker.
After that, if E4 were the marker, it would force D9, D10 and D11 cleared (every open cell left in this region sits in column D, so its marker has to land there), then H2, I2 and H3 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 J2 cleared (every remaining candidate in this region touches J2), 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.
Now, marking G4 would force D3, D9, D10, and 1 more cleared (every open cell left in this region sits in column D, so its marker has to land there), then E9, E11 and F11 cleared (together, two regions have open cells only in columns E and F), then E6 cleared (every remaining candidate in this region touches E6), and the chain continues, and then column C would be left with no open cell for its marker. So G4 can't be the marker there.
From there, if J6 were the marker, it would force K1, 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 F8 (the last open cell in row 8), then D10, A11, B11, and 2 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 11 would end up with no open cell for its marker, which can't happen. So it can't go there.
Following that, if D9 were the marker, column C 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 E9 it would force C10 (the last open cell in column C), and column A would be left with no open cell for its marker.
Next, marking D4 would force B10 cleared (every remaining candidate in this region touches B10), then B11 (the last open cell in column B), and then column C would be left with no open cell for its marker. So D4 can't be the marker there.
Then, that one can't be the marker: at F4 it would force D10 and D11 cleared (every open cell left in this region sits in column D, so its marker has to land there), then H2, I2 and H3 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 J2 cleared (every remaining candidate in this region touches J2), and the chain continues, and column C would be left with no open cell for its marker.
After that, if J3 were the marker, it would force B4 (the last open cell in row 4), then C9, A11 and C11 cleared (every open cell in column D belongs to the same region, so that region's marker has to be somewhere in this column), then C10 (the last open cell in column C), 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 K3 (the last open cell in row 3), then D10, E10, F10, and 3 more cleared (every open cell left in this region sits in row 10, so its marker has to land there), then H8, H9, H11, and 4 more cleared (together, two regions have open cells only in columns H and I), and the chain continues, and region 9 would be left with no open cell for its marker.
From there, if H4 were the marker, it would force D3, D10 and D11 cleared (every open cell left in this region sits in column D, so its marker has to land there), then E6 cleared (every remaining candidate in this region touches E6), then B10 cleared (every remaining candidate in this region touches B10), 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.
Following that, that one can't be the marker: at I4 it would force D3, D10 and D11 cleared (every open cell left in this region sits in column D, so its marker has to land there), then K1, K7, K8, and 3 more cleared (every open cell left in this region sits in column K, so its marker has to land there), then E6 cleared (every remaining candidate in this region touches E6), and the chain continues, and column C would be left with no open cell for its marker.
First, if F2 were the marker, it would force H5 (the last open cell in region 5), then I7 (the last open cell in region 7), then K3 (the last open cell in row 3), and the chain continues, and row 8 would end up with no open cell for its marker, which can't happen. So it can't go there.
Next, marking H2 would force F3 (the last open cell in region 5), then E1 (the last open cell in row 1), then G7 (the last open cell in region 8), and the chain continues, and then region 6 would be left with no open cell for its marker. So H2 can't be the marker there.
Then, that one can't be the marker: at E3 it would force I2 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 J2 cleared (every remaining candidate in this region touches J2), then H5 (the last open cell in region 5), and the chain continues, and region 11 would be left with no open cell for its marker.
After that, marking F3 would force E6, E7, E10, and 1 more cleared (every open cell left in this region sits in column E, so its marker has to land there), then G7 (the last open cell in region 8), then I6 (the last open cell in region 7), and the chain continues, and then row 1 would be left with no open cell for its marker. So F3 can't be the marker there.
Now, that one can't be the marker: at I1 it would force G7, G8, G9, and 6 more cleared (together, two regions have open cells only in columns G and H), then K8, K9, K10, and 4 more cleared (together, two regions have open cells only in columns K and J), then A11, B11, C11, and 1 more cleared (every open cell left in this region sits in row 11, so its marker has to land there), and the chain continues, and column G would be left with no open cell for its marker.
From there, if I2 were the marker, it would force D10, E10, F10, and 4 more cleared (every open cell left in this region sits in row 10, so its marker has to land there), then D3, E5 and F5 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 G7, G8, G9, and 4 more cleared (together, two regions have open cells only in columns G and H), 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, marking K2 would force H3 (the last open cell in region 3), and then region 5 would be left with no open cell for its marker. So K2 can't be the marker there.
First, that one can't be the marker: at K3 it would force J1 (the last open cell in region 3), then H5 (the last open cell in region 5), then E2 (the last open cell in region 2), and the chain continues, and row 8 would be left with no open cell for its marker.
Next, marking J4 would force D3, D10 and D11 cleared (every open cell left in this region sits in column D, so its marker has to land there), then E6 cleared (every remaining candidate in this region touches E6), then B10 cleared (every remaining candidate in this region touches B10), and the chain continues, and then column C would be left with no open cell for its marker. So J4 can't be the marker there.
Then, if K1 were the marker, it would force B4 (the last open cell in row 4), then E5, F5, G5, and 1 more cleared (every open cell left in this region sits in row 5, so its marker has to land there), then G3 (the last open cell in region 5), 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.
After that, if K4 were the marker, it would force D3, D10 and D11 cleared (every open cell left in this region sits in column D, so its marker has to land there), then E6 cleared (every remaining candidate in this region touches E6), then B10 cleared (every remaining candidate in this region touches B10), 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.
Now, place a marker at B4. Row 4 has exactly one open cell left. Placing here clears the rest of row 4, column B, and its region.
From there, if E1 were the marker, it would force H3 (the last open cell in region 3), then J2 (the last open cell in region 5), then D10, F10, G10, and 2 more cleared (every open cell left in this region sits in row 10, 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.
Following that, that one can't be the marker: at F1 it would force H3 (the last open cell in region 3), then J2 (the last open cell in region 5), then D10, E10, G10, and 2 more cleared (every open cell left in this region sits in row 10, so its marker has to land there), and the chain continues, and column C would be left with no open cell for its marker.
First, place a marker at J1. Row 1 has exactly one open cell left. Placing here clears the rest of row 1, column J, and its region.
Next, marking E2 would force G3 (the last open cell in row 3), then D10, F10, H10, and 2 more cleared (every open cell left in this region sits in row 10, so its marker has to land there), then D11 (the last open cell in column D), and then column C would be left with no open cell for its marker. So E2 can't be the marker there.
Then, place a marker at A2. Row 2 has exactly one open cell left. Placing here clears the rest of row 2, column A, and its region.
After that, every open cell in column C 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.
Now, place a marker at D3. Column 4 has exactly one open cell left. Placing here clears the rest of row 3, column D, and its region.
From there, 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. Placing here also clears its 2 touching neighbours.
Following that, place a marker at K6. Region has exactly one open cell left. Placing here clears the rest of row 6, column K, and its region.
First, place a marker at I7. Region has exactly one open cell left. Placing here clears the rest of row 7, column I, and its region.
Next, every open cell left in this region sits in column G, so its marker has to land there; that clears every other open cell in the column.
Then, 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.
After that, place a marker at C9. Region has exactly one open cell left. Placing here clears the rest of row 9, column C, and its region.
Now, every remaining candidate in this region touches E10 and F10, so whichever candidate ends up marked will clear them, and they can be ruled out now regardless of which one wins.
From there, place a marker at F8. Region has exactly one open cell left. Placing here clears the rest of row 8, column F, and its region.
Following that, place a marker at G10. Region has exactly one open cell left.
Finally, place a marker at E11. Region has exactly one open cell left.

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