An Expert 11×11 Puzzle Solved with Forced Chain in 80 Steps

11×11 · expert · 80 steps

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This 11×11 board rates as expert (difficulty score 3435). Solving it from scratch takes 80 logical steps, using 6 techniques: Single cell, Row/column exclusion, Region confinement, Line confinement, Shape squeeze, Forced chain. Every step below is forced: nothing here is a guess.

First, every remaining candidate in this region touches K10, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
Next, marking J1 would force K11 (the last open cell in region 11), then A9, D9, E9, and 4 more cleared (together, two regions have open cells only in rows 9 and 10), then F8 (the last open cell in region 8), and then region 7 would be left with no open cell for its marker. So J1 can't be the marker there.
Then, that one can't be the marker: at J2 it would force K11 (the last open cell in region 11), then A3, B3, D3, and 3 more 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 C4, D4, E4, and 6 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 row 8 would be left with no open cell for its marker.
After that, if J3 were the marker, it would force K11 (the last open cell in region 11), then A2, B2, C2, and 5 more 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 H2 (the last open cell in row 2), 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, marking A4 would immediately leave region 5 with no open cell for its marker, so it can't be the marker there.
From there, marking J4 would force K11 (the last open cell in region 11), then A2, B2, C2, and 9 more 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 C3, F3, H3, and 3 more cleared (every open cell in row 2 belongs to the same region, so that region's marker has to be somewhere in this row), and then row 3 would be left with no open cell for its marker. So J4 can't be the marker there.
Following that, that one can't be the marker: at J5 it would force K11 (the last open cell in region 11), then A1, B1, C1, and 3 more cleared (every open cell left in this region sits in row 1, so its marker has to land there), then A9, D9, E9, and 4 more cleared (together, two regions have open cells only in rows 9 and 10), and the chain continues, and region 7 would be left with no open cell for its marker.
First, if J6 were the marker, it would force K11 (the last open cell in region 11), then A9, D9, E9, and 4 more cleared (together, two regions have open cells only in rows 9 and 10), then F8 (the last open cell in region 8), and region 7 would end up with no open cell for its marker, which can't happen. So it can't go there.
Next, marking D7 would force C10 (the last open cell in region 7), then A11 (the last open cell in region 10), and then region 11 would be left with no open cell for its marker. So D7 can't be the marker there.
Then, marking J7 would force K11 (the last open cell in region 11), then A9, D9, E9, and 4 more cleared (together, two regions have open cells only in rows 9 and 10), then F8 (the last open cell in region 8), and then region 7 would be left with no open cell for its marker. So J7 can't be the marker there.
After that, marking B8 would force A1, A2, A3, 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 10), then J10 (the last open cell in region 11), and the chain continues, and then region 8 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 B1, B2, B3, and 4 more cleared (together, two regions have open cells only in columns B and A), then E10, F10, G10, and 8 more cleared (together, two regions have open cells only in rows 10 and 11), then A9, E9 and F9 cleared (every open cell left in this region sits in row 9, 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.
From there, marking E8 would force A11, B11, C11, and 4 more cleared (every open cell left in this region sits in row 11, so its marker has to land there), then J10 (the last open cell in region 11), then A9, B9 and C9 cleared (every open cell left in this region sits in row 9, so its marker has to land there), and then region 10 would be left with no open cell for its marker. So E8 can't be the marker there.
Following that, if D6 were the marker, it would force C10 (the last open cell in region 7), then A11 (the last open cell in region 10), and region 11 would end up with no open cell for its marker, which can't happen. So it can't go there.
First, if B7 were the marker, it would force A1, A2, A3, and 2 more cleared (every open cell left in this region sits in column A, so its marker has to land there), then C1, C2, C3, and 2 more cleared (every open cell left in this region sits in column C, so its marker has to land there), then D1, D2, D3, and 3 more cleared (every open cell left in this region sits in column D, so its marker has to land there), 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 J8 it would force K11 (the last open cell in region 11), then A9, D9, C10, and 1 more cleared (together, two regions have open cells only in rows 9 and 10), then E7 (the last open cell in region 7), and the chain continues, and column D would be left with no open cell for its marker.
Then, if A9 were the marker, it would force D11, E11, F11, and 5 more cleared (every open cell left in this region sits in row 11, so its marker has to land there), then J10 (the last open cell in region 11), then K8 (the last open cell in region 9), 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, if D9 were the marker, it would force K11 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 F10, G10, H10, and 6 more cleared (together, two regions have open cells only in rows 10 and 11), then K8 (the last open cell in region 9), and region 8 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 A8 (the last open cell in region 5), then C1, C2, C3, and 2 more cleared (every open cell left in this region sits in column C, so its marker has to land there), then D10 cleared (every remaining candidate in this region touches D10), and the chain continues, and region 9 would be left with no open cell for its marker.
From there, that one can't be the marker: at C7 it would force D10 (the last open cell in region 7), then A11, B11, F11, and 3 more cleared (every open cell left in this region sits in row 11, so its marker has to land there), then B9 (the last open cell in region 10), and the chain continues, and region 8 would be left with no open cell for its marker.
Following that, if F8 were the marker, it would force A10, B10, G10, and 3 more cleared (every open cell left in this region sits in row 10, so its marker has to land there), then A11, B11, C11, and 2 more cleared (every open cell left in this region sits in row 11, so its marker has to land there), then B9 and C9 cleared (every open cell left in this region sits in row 9, so its marker has to land there), and region 10 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 B9 it would force A1, A2 and A3 cleared (every open cell left in this region sits in column A, so its marker has to land there), then K11 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 D10, F10, G10, and 4 more cleared (together, two regions have open cells only in rows 10 and 11), and the chain continues, and region 6 would be left with no open cell for its marker.
Next, that one can't be the marker: at C1 it would force I4 (the last open cell in region 2), then D10, E10, F10, and 7 more cleared (together, two regions have open cells only in rows 10 and 11), then G9, H9, J9, and 1 more cleared (every open cell left in this region sits in row 9, so its marker has to land there), and the chain continues, and row 2 would be left with no open cell for its marker.
Then, if E1 were the marker, it would force I4 (the last open cell in region 2), then C9 cleared (every remaining candidate in this region touches C9), then C10, D10, F10, and 6 more cleared (together, two regions have open cells only in rows 10 and 11), and the chain continues, and region 9 would end up with no open cell for its marker, which can't happen. So it can't go there.
After that, if C2 were the marker, it would force I4 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 D10, E10, F10, and 9 more cleared (together, two regions have open cells only in rows 10 and 11), then G9, H9, I9, and 2 more cleared (every open cell left in this region sits in row 9, so its marker has to land there), 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, marking E2 would force I4 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 C9 cleared (every remaining candidate in this region touches C9), then C10, D10, F10, and 8 more cleared (together, two regions have open cells only in rows 10 and 11), and the chain continues, and then region 9 would be left with no open cell for its marker. So E2 can't be the marker there.
From there, marking C3 would force A1, A2 and B1 cleared (together, two regions have open cells only in columns A and B), then D10, E10, F10, and 9 more cleared (together, two regions have open cells only in rows 10 and 11), then G9, H9, I9, and 2 more cleared (every open cell left in this region sits in row 9, 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 C3 can't be the marker there.
Following that, that one can't be the marker: at E3 it would force I4 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 C9 cleared (every remaining candidate in this region touches C9), then C10, D10, F10, and 8 more cleared (together, two regions have open cells only in rows 10 and 11), and the chain continues, and region 9 would be left with no open cell for its marker.
First, that one can't be the marker: at C4 it would force A1, B1, D1, and 1 more cleared (every open cell left in this region sits in row 1, so its marker has to land there), then A2, A3, A10, and 1 more cleared (every open cell left in this region sits in column A, so its marker has to land there), then B2 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 2 would be left with no open cell for its marker.
Next, if E4 were the marker, it would force A1, B1, D1, and 1 more cleared (every open cell left in this region sits in row 1, so its marker has to land there), then C9 cleared (every remaining candidate in this region touches C9), then C10, D10, F10, and 8 more cleared (together, two regions have open cells only in rows 10 and 11), and the chain continues, and region 9 would end up with no open cell for its marker, which can't happen. So it can't go there.
Then, if C5 were the marker, it would force A1, A2, A3, and 2 more cleared (every open cell left in this region sits in column A, so its marker has to land there), then B1, B2 and B3 cleared (every open cell left in this region sits in column B, so its marker has to land there), then D10, E10, F10, and 9 more cleared (together, two regions have open cells only in rows 10 and 11), and the chain continues, and region 4 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 F1 it would force I4 (the last open cell in region 2), then K3 (the last open cell in row 3), then H2 (the last open cell in row 2), and the chain continues, and row 8 would be left with no open cell for its marker.
Now, marking E5 would force A6, H6, I6, and 1 more cleared (every open cell left in this region sits in row 6, so its marker has to land there), then C9 cleared (every remaining candidate in this region touches C9), then C10, D10, F10, and 8 more cleared (together, two regions have open cells only in rows 10 and 11), and the chain continues, and then region 9 would be left with no open cell for its marker. So E5 can't be the marker there.
From there, if F5 were the marker, it would force C6 (the last open cell in region 6), then B1, B2, B3, and 3 more cleared (together, two regions have open cells only in columns B and A), then D10, E10, G10, and 7 more cleared (together, two regions have open cells only in rows 10 and 11), 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, marking C6 would force B1, B2, B3, and 3 more cleared (together, two regions have open cells only in columns B and A), then D10, E10, F10, and 9 more cleared (together, two regions have open cells only in rows 10 and 11), then G9, H9, I9, and 2 more cleared (every open cell left in this region sits in row 9, so its marker has to land there), and the chain continues, and then region 4 would be left with no open cell for its marker. So C6 can't be the marker there.
First, that one can't be the marker: at E6 it would force C9 cleared (every remaining candidate in this region touches C9), then C10, D10, F10, and 8 more cleared (together, two regions have open cells only in rows 10 and 11), then D8 (the last open cell in region 7), and the chain continues, and region 9 would be left with no open cell for its marker.
Next, marking F6 would force D11 and G11 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 C9 cleared (every remaining candidate in this region touches C9), then C10, D10, E10, and 6 more cleared (together, two regions have open cells only in rows 10 and 11), and the chain continues, and then column E would be left with no open cell for its marker. So F6 can't be the marker there.
Then, every remaining candidate in this region touches H6, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
After that, if H4 were the marker, it would force G6 (the last open cell in region 6), then A1, B1 and D1 cleared (every open cell left in this region sits in row 1, so its marker has to land there), then I9, I10, I11, and 3 more cleared (together, two regions have open cells only in columns I and K), 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, marking A7 would force F9, D11, F11, 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 C9 cleared (every remaining candidate in this region touches C9), then C10, D10, E10, and 7 more cleared (together, two regions have open cells only in rows 10 and 11), and the chain continues, and then column E would be left with no open cell for its marker. So A7 can't be the marker there.
From there, marking K2 would force E7 (the last open cell in row 7), then A8 (the last open cell in row 8), then G6 (the last open cell in row 6), and the chain continues, and then row 1 would be left with no open cell for its marker. So K2 can't be the marker there.
Following that, marking A1 would force I4 (the last open cell in region 2), then B5 (the last open cell in region 5), then G6 (the last open cell in region 6), and the chain continues, and then row 8 would be left with no open cell for its marker. So A1 can't be the marker there.
First, if B1 were the marker, it would force I4 (the last open cell in region 2), then H2 (the last open cell in row 2), then K3 (the last open cell in row 3), and the chain continues, and row 6 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 I4 (the last open cell in region 2), then H2 (the last open cell in row 2), then K3 (the last open cell in row 3), and the chain continues, and then row 6 would be left with no open cell for its marker. So D1 can't be the marker there.
Then, 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.
After that, marking H2 would force K1 (the last open cell in region 2), then B4 (the last open cell in row 4), then D3 (the last open cell in region 1), and the chain continues, and then row 8 would be left with no open cell for its marker. So H2 can't be the marker there.
Now, if A3 were the marker, it would force B5 (the last open cell in region 5), then G6 (the last open cell in region 6), then I2 (the last open cell in row 2), and the chain continues, and row 4 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 B3 it would force I2 (the last open cell in row 2), then K4 (the last open cell in row 4), then G1 (the last open cell in region 2), and the chain continues, and row 8 would be left with no open cell for its marker.
Following that, if G3 were the marker, it would force I2 (the last open cell in row 2), then K1 (the last open cell in region 2), then H5 (the last open cell in region 6), 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.
First, that one can't be the marker: at K3 it would force E7 (the last open cell in row 7), then A8 (the last open cell in row 8), then G6 (the last open cell in row 6), and the chain continues, and row 4 would be left with no open cell for its marker.
Next, marking B2 would force A10 and A11 cleared (every open cell left in this region sits in column A, so its marker has to land there), then C10 cleared (every open cell left in this region sits in column C, so its marker has to land there), then D4, F4, G4, and 1 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 row 5 would be left with no open cell for its marker. So B2 can't be the marker there.
Then, that one can't be the marker: at G2 it would force I3 (the last open cell in row 3), then K1 (the last open cell in region 2), then H5 (the last open cell in region 6), and the chain continues, and region 11 would be left with no open cell for its marker.
After that, marking D4 would immediately leave row 3 with no open cell for its marker, so it can't be the marker there.
Now, that one can't be the marker: at F4 it would force D3 (the last open cell in row 3), and row 2 would be left with no open cell for its marker.
From there, marking G4 would force I5 (the last open cell in region 6), then D3 (the last open cell in row 3), and then row 2 would be left with no open cell for its marker. So G4 can't be the marker there.
Following that, if K4 were the marker, it would force E7 (the last open cell in row 7), then A8 (the last open cell in row 8), then G6 (the last open cell in row 6), and the chain continues, and row 3 would end up with no open cell for its marker, which can't happen. So it can't go there.
First, 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.
Next, marking G1 would force D5 and K5 cleared (every open cell left in this region sits in row 5, so its marker has to land there), then F7, H7, I7, and 3 more cleared (every open cell in row 6 belongs to the same region, so that region's marker has to be somewhere in this row), then E7 (the last open cell in row 7), and the chain continues, and then row 5 would be left with no open cell for its marker. So G1 can't be the marker there.
Then, that one can't be the marker: at D2 it would force C9 and C11 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 C10 (the last open cell in column C), then A11 (the last open cell in region 10), and region 11 would be left with no open cell for its marker.
After that, if F2 were the marker, it would force D5 cleared (every open cell in row 3 belongs to the same region, so that region's marker has to be somewhere in this row), then C9 and C11 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 C10 (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.
Now, if I2 were the marker, it would force K1 (the last open cell in region 2), then D3 (the last open cell in region 1), then G6 (the last open cell in row 6), and row 5 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 A2. Row 2 has exactly one open cell left. Placing here clears the rest of row 2, column A, and its region.
Following that, every open cell left in this region sits in column C, so its marker has to land there; that clears every other open cell in the column.
First, every open cell in row 3 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, every remaining candidate in this region touches D10, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
Then, if K1 were the marker, it would force J9 cleared (every open cell left in this region sits in column J, so its marker has to land there), then G6 cleared (every open cell in row 5 belongs to the same region, so that region's marker has to be somewhere in this row), then I6 (the last open cell in row 6), 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.
After that, marking H5 would force I1 (the last open cell in region 2), then F3 (the last open cell in region 3), then K6 (the last open cell in row 6), and the chain continues, and then row 8 would be left with no open cell for its marker. So H5 can't be the marker there.
Now, if I5 were the marker, it would force H1 (the last open cell in region 2), then F3 (the last open cell in region 3), then K6 (the last open cell in row 6), 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.
From there, 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.
Following that, marking K5 would force G6 (the last open cell in region 6), then E7 (the last open cell in row 7), and then row 8 would be left with no open cell for its marker. So K5 can't be the marker there.
First, place a marker at G5. Row 5 has exactly one open cell left. Placing here clears the rest of row 5, column G, and its region.
Next, every open cell in row 6 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.
Then, place a marker at E7. Row 7 has exactly one open cell left. Placing here clears the rest of row 7, column E, and its region.
After that, place a marker at K8. Row 8 has exactly one open cell left. Placing here clears the rest of row 8, column K, and its region.
Now, place a marker at I6. Region has exactly one open cell left. Placing here clears the rest of row 6, column I, and its region.
From there, place a marker at H1. Region has exactly one open cell left. Placing here clears the rest of row 1, column H, and its region.
Following that, place a marker at F3. Region has exactly one open cell left. Placing here clears the rest of row 3, column F, and its region.
First, place a marker at D11. Region has exactly one open cell left. Placing here clears the rest of row 11, column D, and its region.
Next, place a marker at C9. Region has exactly one open cell left.
Finally, place a marker at J10. Region has exactly one open cell left.

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