An Expert 10×10 Puzzle Solved with Forced Chain in 49 Steps

10×10 · expert · 49 steps

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This 10×10 board rates as expert (difficulty score 2324). Solving it from scratch takes 49 logical steps, using 5 techniques: Single cell, Row/column exclusion, Adjacency clear, Region confinement, Forced chain. Every step below is forced: nothing here is a guess.

First, marking A1 would force B9 (the last open cell in region 8), and then region 10 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 A3, A4, A5, and 2 more cleared (every open cell left in this region sits in column A, so its marker has to land there), then C9, C10, D4, and 3 more cleared (together, two regions have open cells only in columns C and D), and region 10 would end up with no open cell for its marker, which can't happen. So it can't go there.
Then, marking D1 would force B9 cleared (every remaining candidate in this region touches B9), then A2, A3, A4, and 3 more cleared (every open cell left in this region sits in column A, so its marker has to land there), then B10, C9 and C10 cleared (together, two regions have open cells only in columns B and C), and then region 10 would be left with no open cell for its marker. So D1 can't be the marker there.
After that, that one can't be the marker: at F1 it would force D4, D5, D6, and 4 more cleared (every open cell left in this region sits in column D, so its marker has to land there), then G6, G7 and G8 cleared (together, two regions have open cells only in columns E and G), then C2, C3 and E3 cleared (every remaining candidate in this region touches C2 and C3 and E3), and the chain continues, and region 10 would be left with no open cell for its marker.
Now, that one can't be the marker: at A2 it would force B9 (the last open cell in region 8), and region 10 would be left with no open cell for its marker.
From there, marking B2 would force A4, A5, A6, and 1 more cleared (every open cell left in this region sits in column A, so its marker has to land there), then D3, D4, D6, and 1 more cleared (together, two regions have open cells only in columns C and D), then E1 (the last open cell in region 2), 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.
Following that, that one can't be the marker: at D2 it would force F3, G3, H3, and 1 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 B9 cleared (every remaining candidate in this region touches B9), then A3, A4, A5, 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 10 would be left with no open cell for its marker.
First, marking H2 would force F6, G6, G7, and 3 more cleared (every open cell in column I belongs to the same region, so that region's marker has to be somewhere in this column), then J1 (the last open cell in column J), then D3 (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 H2 can't be the marker there.
Next, if I2 were the marker, it would force F6, G6, G7, and 3 more cleared (every open cell in column H belongs to the same region, so that region's marker has to be somewhere in this column), and column J would end up with no open cell for its marker, which can't happen. So it can't go there.
Then, if A3 were the marker, it would force B9 (the last open cell in region 8), and region 10 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 C1 it would force B5, B7, B8, and 6 more cleared (together, two regions have open cells only in columns B and D), 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 8 would be left with no open cell for its marker.
Now, if F2 were the marker, row 1 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 G2 it would force E1 (the last open cell in row 1), then D5, D7, D8, and 2 more cleared (together, two regions have open cells only in columns D and F), then B5, B7, B8, and 3 more cleared (together, two regions have open cells only in columns B and C), and the chain continues, and region 8 would be left with no open cell for its marker.
Following that, that one can't be the marker: at B3 it would force 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), then A5, A6 and A7 cleared (every open cell left in this region sits in column A, so its marker has to land there), then D4, D6 and D10 cleared (together, two regions have open cells only in columns C and D), and the chain continues, and row 5 would be left with no open cell for its marker.
First, if D3 were the marker, it would force J2 (the last open cell in row 2), then G4 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 F6, G6, G7, and 5 more cleared (every open cell in column I belongs to the same region, so that region's marker has to be somewhere in this column), 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.
Next, every open cell left in this region sits in column E, so its marker has to land there; that clears every other open cell in the column.
Then, marking G1 would force E2 (the last open cell in region 2), then D5, D7, D8, and 2 more cleared (together, two regions have open cells only in columns D and F), then C7, C8, B5, and 3 more cleared (together, two regions have open cells only in columns C and B), 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.
After that, marking F3 would force E1 (the last open cell in region 2), then D5, D7, D8, and 4 more cleared (together, two regions have open cells only in columns D and G), then C7, C8, B5, and 3 more cleared (together, two regions have open cells only in columns C and B), and the chain continues, and then region 8 would be left with no open cell for its marker. So F3 can't be the marker there.
Now, if G3 were the marker, it would force E1 (the last open cell in row 1), then D5, D7, D8, and 2 more cleared (together, two regions have open cells only in columns D and F), then C7, C8, B5, and 3 more cleared (together, two regions have open cells only in columns C and B), 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.
From there, marking A4 would force B9 (the last open cell in region 8), and then region 10 would be left with no open cell for its marker. So A4 can't be the marker there.
Following that, if B4 were the marker, it would force A6 and A7 cleared (every open cell left in this region sits in column A, so its marker has to land there), then D6 and D10 cleared (together, two regions have open cells only in columns C and D), then F6, G6, G7, and 1 more cleared (together, two regions have open cells only in columns F and G), and the chain continues, and row 1 would end up with no open cell for its marker, which can't happen. So it can't go there.
First, marking D4 would force C7, C8, B5, and 3 more cleared (together, two regions have open cells only in columns C and B), 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 8 would be left with no open cell for its marker. So D4 can't be the marker there.
Next, marking G4 would force E1 (the last open cell in row 1), then D5, D7, D8, and 2 more cleared (together, two regions have open cells only in columns D and F), then C7, C8, B5, and 3 more cleared (together, two regions have open cells only in columns C and B), 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.
Then, that one can't be the marker: at A5 it would force B9 (the last open cell in region 8), and region 10 would be left with no open cell for its marker.
After that, if C5 were the marker, it would force F6, F8, F9, and 1 more cleared (every open cell left in this region sits in column F, so its marker has to land there), then A9, B9, H9, and 6 more cleared (together, two regions have open cells only in rows 9 and 10), then A8 (the last open cell in region 8), and the chain continues, and row 9 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 D5 it would force F6, F8, F9, and 1 more cleared (every open cell left in this region sits in column F, so its marker has to land there), then G6, G7 and G8 cleared (every open cell left in this region sits in column G, so its marker has to land there), then A9, B9, H9, and 6 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.
From there, that one can't be the marker: at J2 it would force E1 (the last open cell in region 2), then H3 (the last open cell in region 3), then F6, G6, G7, and 1 more cleared (every open cell in column I belongs to the same region, so that region's marker has to be somewhere in this column), and the chain continues, and row 7 would be left with no open cell for its marker.
Following that, marking I3 would force E1 cleared (every open cell left in this region sits in row 1, so its marker has to land there), then E2 (the last open cell in region 2), then B5 cleared (every remaining candidate in this region touches B5), 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.
First, if J3 were the marker, it would force E1 cleared (every open cell left in this region sits in row 1, so its marker has to land there), then E2 (the last open cell in region 2), then B5 cleared (every remaining candidate in this region touches B5), 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 C4 it would force E2 (the last open cell in row 2), then H3 (the last open cell in row 3), and row 1 would be left with no open cell for its marker.
Then, if F5 were the marker, it would force H6, I6, J6, and 3 more cleared (every open cell in row 4 belongs to the same region, so that region's marker has to be somewhere in this row), then A9, B9, H9, and 6 more cleared (together, two regions have open cells only in rows 9 and 10), then A8 (the last open cell in region 8), 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.
After that, if H4 were the marker, it would force C3 (the last open cell in row 3), then E2 (the last open cell in row 2), then B5 (the last open cell in row 5), 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.
Now, marking H5 would force C3 (the last open cell in row 3), then E2 (the last open cell in row 2), then F4 (the last open cell in row 4), and the chain continues, and then region 10 would be left with no open cell for its marker. So H5 can't be the marker there.
From there, if I5 were the marker, it would force F4 (the last open cell in row 4), then A9, B9, H9, and 4 more cleared (together, two regions have open cells only in rows 9 and 10), then A8 (the last open cell in region 8), and the chain continues, and column H 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 J5 it would force F4 (the last open cell in row 4), then A9, B9, H9, and 4 more cleared (together, two regions have open cells only in rows 9 and 10), then A8 (the last open cell in region 8), and the chain continues, and column H would be left with no open cell for its marker.
First, if A6 were the marker, it would force B9 (the last open cell in region 8), 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 G5 it would force I6, J6, I7, and 2 more cleared (every open cell in row 4 belongs to the same region, so that region's marker has to be somewhere in this row), then C2 and C3 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 E2 (the last open cell in row 2), and the chain continues, and row 1 would be left with no open cell for its marker.
Then, place a marker at B5. Row 5 has exactly one open cell left. Placing here clears the rest of row 5, column B, and its region. Placing here also clears its 1 touching neighbour.
After 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.
Now, place a marker at D9. Region has exactly one open cell left. Placing here clears the rest of row 9, column D, and its region.
From there, every open cell left in this region sits in column F, so its marker has to land there; that clears every other open cell in the column.
Following that, place a marker at G10. Region has exactly one open cell left. Placing here clears the rest of row 10, column G, and its region.
First, place a marker at A8. Region has exactly one open cell left. Placing here clears the rest of row 8, column A, and its region.
Next, place a marker at H7. Region has exactly one open cell left. Placing here clears the rest of row 7, column H, and its region. Placing here also clears its 1 touching neighbour.
Then, 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.
After that, 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.
Now, place a marker at I1. Region has exactly one open cell left. Placing here clears the rest of row 1, column I, and its region.
From there, place a marker at E2. Region has exactly one open cell left. Placing here clears the rest of row 2, column E, and its region.
Finally, place a marker at C3. Region has exactly one open cell left.

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