An Expert 9×9 Puzzle Solved with Forced Chain in 25 Steps
Play this board yourself →A
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This 9×9 board rates as expert (difficulty score 1348). Solving it from scratch takes 25 logical steps, using 8 techniques: Single cell, Row/column exclusion, Adjacency clear, Region confinement, Line confinement, Paired regions, Shape squeeze, Forced chain. Every step below is forced: nothing here is a guess.
- First, every open cell left in this region sits in column A, so its marker has to land there; that clears every other open cell in the column.
- Next, place a marker at B6. Region has exactly one open cell left. Placing here clears the rest of row 6, column B, and its region. Placing here also clears its 3 touching neighbours.
- Then, place a marker at A4. Region has exactly one open cell left. Placing here clears the rest of row 4, column A, and its region.
- After that, every remaining candidate in this region touches H8, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
- Now, marking D1 would force F5, F7, F8, and 1 more cleared (every open cell left in this region sits in column F, so its marker has to land there), then G5 and I5 cleared (every open cell left in this region sits in row 5, so its marker has to land there), then F3, G3 and I3 cleared (every open cell left in this region sits in row 3, so its marker has to land there), and the chain continues, and then region 3 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 F3 (the last open cell in region 2), then G5 (the last open cell in region 4), and region 6 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 F1 it would force G5 and I5 cleared (every open cell left in this region sits in row 5, so its marker has to land there), then D3, E3, G3, and 1 more cleared (every open cell left in this region sits in row 3, so its marker has to land there), then H2 and I2 cleared (every open cell left in this region sits in row 2, so its marker has to land there), and region 3 would be left with no open cell for its marker.
- First, marking G1 would force I7, I8 and I9 cleared (every open cell left in this region sits in column I, so its marker has to land there), then D3, E3 and I3 cleared (every open cell left in this region sits in row 3, so its marker has to land there), then I2 cleared (every open cell left in this region sits in row 2, so its marker has to land there), and the chain continues, and then column E would be left with no open cell for its marker. So G1 can't be the marker there.
- Next, if H1 were the marker, it would force D3 and E3 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 D5, E5 and F5 cleared (together, two regions have open cells only in rows 3 and 5), then F7 (the last open cell in region 6), and the chain continues, and row 5 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 E3, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
- After that, that one can't be the marker: at I1 it would force D7, E7, C9, and 4 more cleared (every open cell in row 8 belongs to the same region, so that region's marker has to be somewhere in this row), then H9 (the last open cell in row 9), then F7 (the last open cell in row 7), and the chain continues, and region 1 would be left with no open cell for its marker.
- Now, place a marker at C1. Row 1 has exactly one open cell left. Placing here clears the rest of row 1, column C, and its region.
- From there, if F2 were the marker, it would force I7, I8 and I9 cleared (every open cell left in this region sits in column I, so its marker has to land there), then G5 and I5 cleared (every open cell left in this region sits in row 5, so its marker has to land there), then I3 (the last open cell in region 3), and region 4 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: place it at G2, and row 3 would be left with no open cell for its marker.
- First, marking H2 would force F3 (the last open cell in region 2), then G5 (the last open cell in region 4), and then region 6 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 F3 (the last open cell in region 2), then G5 (the last open cell in region 4), and region 6 would end up with no open cell for its marker, which can't happen. So it can't go there.
- Then, place a marker at E2. Row 2 has exactly one open cell left. Placing here clears the rest of row 2, column E, and its region.
- After that, together, two regions have open cells only in rows 3 and 5; 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.
- Now, 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 1 touching neighbour.
- From there, every open cell in column D 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.
- Following that, if G3 were the marker, row 5 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 G5. Column 7 has exactly one open cell left. Placing here clears the rest of row 5, column G, and its region.
- Next, place a marker at I3. Region has exactly one open cell left. Placing here clears the rest of row 3, column I, and its region.
- Then, place a marker at H9. Region has exactly one open cell left. Placing here clears the rest of row 9, column H, and its region.
- Finally, place a marker at D8. Region has exactly one open cell left.
The key move here was Forced chain. Once you can spot that, this board falls quickly. ∎