An Expert 10×10 Puzzle Solved with Forced Chain in 26 Steps
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This 10×10 board rates as expert (difficulty score 1286). Solving it from scratch takes 26 logical steps, using 7 techniques: Single cell, Row/column exclusion, Adjacency clear, Region confinement, Paired regions, Shape squeeze, Forced chain. Every step below is forced: nothing here is a guess.
- First, 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.
- Next, marking A1 would force B5, F5, G5, and 5 more cleared (together, two regions have open cells only in rows 5 and 6), then F3, G3, J3, and 3 more cleared (together, two regions have open cells only in rows 3 and 4), then G2 (the last open cell in region 2), and then region 3 would be left with no open cell for its marker. So A1 can't be the marker there.
- Then, if B1 were the marker, it would force I3 cleared (every remaining candidate in this region touches I3), then A4, C4, D4, and 3 more cleared (every open cell left in this region sits in row 4, so its marker has to land there), then D3 and E3 cleared (together, two regions have open cells only in rows 2 and 3), and the chain continues, and region 6 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 I3 cleared (every remaining candidate in this region touches I3), then A4, B4, D4, and 3 more cleared (every open cell left in this region sits in row 4, so its marker has to land there), then D3 and E3 cleared (together, two regions have open cells only in rows 2 and 3), and the chain continues, and row 4 would be left with no open cell for its marker.
- Now, marking D1 would force I3 cleared (every remaining candidate in this region touches I3), then A4, B4, C4, and 3 more cleared (every open cell left in this region sits in row 4, so its marker has to land there), then E3 cleared (together, two regions have open cells only in rows 2 and 3), and the chain continues, and then row 4 would be left with no open cell for its marker. So D1 can't be the marker there.
- From there, if E1 were the marker, it would force I3 cleared (every remaining candidate in this region touches I3), then A4, B4, C4, and 3 more cleared (every open cell left in this region sits in row 4, so its marker has to land there), then D3 cleared (together, two regions have open cells only in rows 2 and 3), 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.
- Following that, marking G1 would force I3 cleared (every remaining candidate in this region touches I3), then A4, B4, C4, and 2 more cleared (every open cell left in this region sits in row 4, so its marker has to land there), then D3 and E3 cleared (together, two regions have open cells only in rows 2 and 3), and the chain continues, and then region 3 would be left with no open cell for its marker. So G1 can't be the marker there.
- First, if H1 were the marker, it would force J5, J6, J7, and 3 more cleared (every open cell left in this region sits in column J, so its marker has to land there), then I3, I4, I7, and 3 more cleared (every open cell left in this region sits in column I, so its marker has to land there), and region 9 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 I1 it would force A4, B4, C4, and 3 more cleared (every open cell left in this region sits in row 4, so its marker has to land there), then A5, B5, C5, and 4 more cleared (every open cell left in this region sits in row 5, so its marker has to land there), then A3, B3, C3, and 2 more cleared (every open cell left in this region sits in row 3, so its marker has to land there), and the chain continues, and region 1 would be left with no open cell for its marker.
- Then, marking J1 would force H4, H6, H7, and 5 more cleared (together, two regions have open cells only in columns H and I), then G4 (the last open cell in region 5), and then region 2 would be left with no open cell for its marker. So J1 can't be the marker there.
- After that, place a marker at F1. Row 1 has exactly one open cell left. Placing here clears the rest of row 1, column F, and its region. Placing here also clears its 1 touching neighbour.
- Now, every remaining candidate in this region touches I3, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
- From there, every open cell left in this region sits in row 4, so its marker has to land there; that clears every other open cell in the row.
- Following that, together, two regions have open cells only in rows 2 and 3; 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.
- First, 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.
- Next, 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. Placing here also clears its 2 touching neighbours.
- Then, 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.
- After that, place a marker at B5. Region has exactly one open cell left. Placing here clears the rest of row 5, column B, and its region.
- Now, every open cell left in this region sits in row 8, so its marker has to land there; that clears every other open cell in the row.
- From there, every open cell left in this region sits in column J, so its marker has to land there; that clears every other open cell in the column.
- Following that, place a marker at H2. Region has exactly one open cell left. Placing here clears the rest of row 2, column H, and its region.
- First, 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.
- Next, place a marker at G4. Region has exactly one open cell left. Placing here clears the rest of row 4, column G, and its region.
- Then, place a marker at E8. Region has exactly one open cell left. Placing here clears the rest of row 8, column E, and its region. Placing here also clears its 1 touching neighbour.
- After that, place a marker at D10. Region has exactly one open cell left. Placing here clears the rest of row 10, column D, and its region.
- Finally, place a marker at J9. Region has exactly one open cell left.
The key move here was Forced chain. Once you can spot that, this board falls quickly. ∎