An Expert 9×9 Puzzle Solved with Forced Chain in 22 Steps
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This 9×9 board rates as expert (difficulty score 1197). Solving it from scratch takes 22 logical steps, using 7 techniques: Single cell, Row/column exclusion, Adjacency clear, Line confinement, Paired regions, Shape squeeze, Forced chain. Every step below is forced: nothing here is a guess.
- First, together, two regions have open cells only in rows 8 and 9; 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.
- Next, place a marker at F7. Region has exactly one open cell left. Placing here clears the rest of row 7, column F, and its region. Placing here also clears its 2 touching neighbours.
- Then, every remaining candidate in this region touches C5, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
- After that, marking A1 would force I4, I5 and I6 cleared (every open cell left in this region sits in column I, so its marker has to land there), then D9 and E9 cleared (every open cell left in this region sits in row 9, so its marker has to land there), then C2, D2, E2, and 2 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 the chain continues, and then column G would be left with no open cell for its marker. So A1 can't be the marker there.
- Now, if B1 were the marker, it would force I4, I5 and I6 cleared (every open cell left in this region sits in column I, so its marker has to land there), then D2, E2, G2, and 1 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), then E5 cleared (every open cell in column G belongs to the same region, so that region's marker has to be somewhere in this column), 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.
- From there, that one can't be the marker: place it at C1, and column H would be left with no open cell for its marker.
- Following that, 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.
- First, marking D1 would force C6 (the last open cell in region 5), then E9 (the last open cell in region 9), then A8 (the last open cell in region 8), and the chain continues, and then region 1 would be left with no open cell for its marker. So D1 can't be the marker there.
- Next, if E1 were the marker, it would force G2 and G3 cleared (every open cell left in this region sits in column G, so its marker has to land there), then C2, I4, I5, and 1 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 I would end up with no open cell for its marker, which can't happen. So it can't go there.
- Then, every remaining candidate in this region touches H2, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
- After that, marking G1 would force E5 (the last open cell in region 6), then C6 (the last open cell in region 5), then I4 (the last open cell in column I), and then column H would be left with no open cell for its marker. So G1 can't be the marker there.
- Now, if H1 were the marker, it would force C2, D2, E2, 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), then A3, B3, C3, 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 row 3 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 I1. Region has exactly one open cell left. Placing here clears the rest of row 1, column I, and its region.
- Following that, every open cell in column H 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.
- First, 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.
- Next, place a marker at H3. Row 3 has exactly one open cell left. Placing here clears the rest of row 3, column H, and its region. Placing here also clears its 1 touching neighbour.
- Then, place a marker at G5. Column 7 has exactly one open cell left. Placing here clears the rest of row 5, column G, and its region.
- After that, place a marker at E9. Column 5 has exactly one open cell left. Placing here clears the rest of row 9, column E, and its region.
- Now, 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.
- From there, place a marker at B2. Region has exactly one open cell left. Placing here clears the rest of row 2, column B, and its region.
- Following that, place a marker at C4. Region has exactly one open cell left. Placing here clears the rest of row 4, column C, and its region.
- Finally, place a marker at D6. Region has exactly one open cell left.
The key move here was Forced chain. Once you can spot that, this board falls quickly. ∎