An Expert 10×10 Puzzle Solved with Forced Chain in 44 Steps

10×10 · expert · 44 steps

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This 10×10 board rates as expert (difficulty score 1992). Solving it from scratch takes 44 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 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.
Next, if B1 were the marker, it would force C10 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 D3, D4, D5, and 6 more cleared (together, two regions have open cells only in columns D and C), then E4, E5, E6, and 4 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 region 3 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: place it at C1, and region 2 would be left with no open cell for its marker.
After that, marking B2 would force A8, A9 and A10 cleared (every open cell left in this region sits in column A, so its marker has to land there), then C10 (the last open cell in region 10), then D3, D4, D5, and 1 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 then region 6 would be left with no open cell for its marker. So B2 can't be the marker there.
Now, if C2 were the marker, it would force B4, B6, B7, and 1 more cleared (together, two regions have open cells only in columns A and B), then D4, D5, D8, and 2 more cleared (every open cell left in this region sits in column D, so its marker has to land there), then E3 (the last open cell in region 5), and region 6 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 D2 it would force B5, B10 and C10 cleared (together, two regions have open cells only in columns B and C), then A8, A9 and A10 cleared (every open cell left in this region sits in column A, so its marker has to land there), and region 10 would be left with no open cell for its marker.
Following that, marking A1 would force C3 (the last open cell in region 2), then E2 (the last open cell in region 5), then G4 (the last open cell in region 3), and the chain continues, and then region 4 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 F1 it would force C3 (the last open cell in region 2), and region 5 would be left with no open cell for its marker.
Next, marking G1 would force C3 (the last open cell in region 2), then E2 (the last open cell in region 5), then A4, A5, B5, and 2 more cleared (together, two regions have open cells only in rows 4 and 5), and the chain continues, and then column D would be left with no open cell for its marker. So G1 can't be the marker there.
Then, if H1 were the marker, it would force C3 (the last open cell in region 2), then E2 (the last open cell in region 5), then A4, A5, B5, and 3 more cleared (together, two regions have open cells only in rows 4 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.
After that, that one can't be the marker: at I1 it would force C3 (the last open cell in region 2), then E2 (the last open cell in region 5), then A4, A5, B5, and 3 more cleared (together, two regions have open cells only in rows 4 and 5), and the chain continues, and region 7 would be left with no open cell for its marker.
Now, marking J1 would force C3 (the last open cell in region 2), then E2 (the last open cell in region 5), then G4 (the last open cell in region 3), and the chain continues, and then region 9 would be left with no open cell for its marker. So J1 can't be the marker there.
From there, 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.
Following that, every remaining candidate in this region touches E2, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
First, if F2 were the marker, it would force D1 (the last open cell in region 2), then E7, E8, E9, and 1 more cleared (every open cell left in this region sits in column E, so its marker has to land there), then B5, B10 and C10 cleared (together, two regions have open cells only in columns B and C), and the chain continues, and region 10 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 J2 it would force F4 cleared (every remaining candidate in this region touches F4), then D3, D4, D6, and 9 more cleared (together, two regions have open cells only in columns D and E), then B5, B10 and C10 cleared (together, two regions have open cells only in columns B and C), and the chain continues, and region 10 would be left with no open cell for its marker.
Then, that one can't be the marker: at B3 it would force A8, A9 and A10 cleared (every open cell left in this region sits in column A, so its marker has to land there), then C10 (the last open cell in region 10), then D1, D5 and D8 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 D would be left with no open cell for its marker.
After that, that one can't be the marker: at E3 it would force D1 (the last open cell in region 2), and region 6 would be left with no open cell for its marker.
Now, marking D1 would force B5, B10 and C10 cleared (together, two regions have open cells only in columns B and C), then A8, A9 and A10 cleared (every open cell left in this region sits in column A, so its marker has to land there), and then region 10 would be left with no open cell for its marker. So D1 can't be the marker there.
From there, place a marker at E1. Region has exactly one open cell left. Placing here clears the rest of row 1, column E, and its region.
Following that, if A3 were the marker, it would force D10, F10, G10, and 3 more cleared (every open cell left in this region sits in row 10, so its marker has to land there), then G4 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 C4, D4, C5, and 3 more cleared (together, two regions have open cells only in rows 4 and 5), and the chain continues, and row 10 would end up with no open cell for its marker, which can't happen. So it can't go there.
First, if I2 were the marker, it would force D3 (the last open cell in row 3), then F4 (the last open cell in region 6), then H5 (the last open cell in region 4), 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 F3 would force D5 (the last open cell in region 6), then B4 (the last open cell in region 5), and then region 4 would be left with no open cell for its marker. So F3 can't be the marker there.
Then, every remaining candidate in this region touches H3, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
After that, if G3 were the marker, it would force D5 (the last open cell in region 6), then B4 (the last open cell in region 5), and region 4 would end up with no open cell for its marker, which can't happen. So it can't go there.
Now, marking I3 would force H5 (the last open cell in region 4), then F4 (the last open cell in region 6), then C6 (the last open cell in region 5), and the chain continues, and then region 10 would be left with no open cell for its marker. So I3 can't be the marker there.
From there, that one can't be the marker: at A2 it would force G4 (the last open cell in region 3), then D5 (the last open cell in region 6), and region 5 would be left with no open cell for its marker.
Following 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.
First, marking A4 would force D5 (the last open cell in region 6), and then region 5 would be left with no open cell for its marker. So A4 can't be the marker there.
Next, every remaining candidate in this region touches B6, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
Then, if B4 were the marker, it would force D5 (the last open cell in region 6), then J3 (the last open cell in region 4), then A8, A9 and A10 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 9 would end up with no open cell for its marker, which can't happen. So it can't go there.
After that, if J3 were the marker, it would force D5 cleared (every remaining candidate in this region touches D5), then F4 (the last open cell in region 6), then C7, C8, C9, and 1 more cleared (every open cell left in this region sits in column C, 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, place a marker at D3. Row 3 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 F4. Region has exactly one open cell left. Placing here clears the rest of row 4, column F, and its region. Placing here also clears its 1 touching neighbour.
Following that, every open cell left in this region sits in row 5, so its marker has to land there; that clears every other open cell in the row.
First, every open cell left in this region sits in row 6, so its marker has to land there; that clears every other open cell in the row.
Next, place a marker at A7. Region has exactly one open cell left. Placing here clears the rest of row 7, column A, and its region. Placing here also clears its 1 touching neighbour.
Then, place a marker at B10. Column 2 has exactly one open cell left. Placing here clears the rest of row 10, column B, and its region. Placing here also clears its 1 touching neighbour.
After that, place a marker at C8. Region has exactly one open cell left. Placing here clears the rest of row 8, column C, and its region.
Now, place a marker at J9. Column 10 has exactly one open cell left. Placing here clears the rest of row 9, column J, and its region.
From there, every remaining candidate in this region touches H6 and I6, so whichever candidate ends up marked will clear them, and they can be ruled out now regardless of which one wins.
Following that, place a marker at G6. Region has exactly one open cell left. Placing here clears the rest of row 6, column G, and its region. Placing here also clears its 1 touching neighbour.
First, place a marker at H2. Region has exactly one open cell left.
Finally, place a marker at I5. Region has exactly one open cell left.

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