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 1214). 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 row 1, so its marker has to land there; that clears every other open cell in the row.
- Next, every remaining candidate in this region touches G2 and H2, so whichever candidate ends up marked will clear them, and they can be ruled out now regardless of which one wins.
- Then, every remaining candidate in this region touches F3, 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 immediately leave region 3 with no open cell for its marker, so it can't be the marker there.
- Now, place a marker at H1. Region has exactly one open cell left. Placing here clears the rest of row 1, column H, and its region. Placing here also clears its 1 touching neighbour.
- From there, that one can't be the marker: at A2 it would force G5, G6, G7, and 2 more cleared (every open cell left in this region sits in column G, so its marker has to land there), then I5, I6, I7, and 2 more cleared (every open cell left in this region sits in column I, so its marker has to land there), then F4, F5, F6, and 1 more cleared (every open cell left in this region sits in column F, so its marker has to land there), and the chain continues, and region 2 would be left with no open cell for its marker.
- Following that, marking B2 would force G5, G6, G7, and 2 more cleared (every open cell left in this region sits in column G, so its marker has to land there), then I5, I6, I7, and 2 more cleared (every open cell left in this region sits in column I, so its marker has to land there), then F4, F5, F6, and 1 more cleared (every open cell left in this region sits in column F, so its marker has to land there), and the chain continues, and then row 6 would be left with no open cell for its marker. So B2 can't be the marker there.
- First, if C2 were the marker, it would force G5, G6, G7, and 2 more cleared (every open cell left in this region sits in column G, so its marker has to land there), then I5, I6, I7, and 2 more cleared (every open cell left in this region sits in column I, so its marker has to land there), then F4, F5, F6, and 1 more cleared (every open cell left in this region sits in column F, so its marker has to land there), and the chain continues, and row 7 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: place it at D2, and region 2 would be left with no open cell for its marker.
- Then, marking E2 would force G5, G6, G7, and 2 more cleared (every open cell left in this region sits in column G, so its marker has to land there), then I5, I6, I7, and 2 more cleared (every open cell left in this region sits in column I, so its marker has to land there), then F8 and F9 cleared (every open cell left in this region sits in column F, so its marker has to land there), and then region 8 would be left with no open cell for its marker. So E2 can't be the marker there.
- After that, place a marker at F2. Row 2 has exactly one open cell left. Placing here clears the rest of row 2, column F, and its region. Placing here also clears its 1 touching neighbour.
- Now, every open cell left in this region sits in column D, so its marker has to land there; that clears every other open cell in the column.
- From there, together, two regions have open cells only in columns I and G; with one marker each, those two markers have to fill exactly those two columns, so every other region's open cells there can be cleared.
- Following that, 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.
- First, every remaining candidate in this region touches C3 and C4 and E4, so whichever candidate ends up marked will clear them, and they can be ruled out now regardless of which one wins.
- Next, place a marker at E5. Region has exactly one open cell left. Placing here clears the rest of row 5, column E, and its region. Placing here also clears its 1 touching neighbour.
- Then, place a marker at D3. Region has exactly one open cell left. Placing here clears the rest of row 3, column D, and its region.
- After that, place a marker at I4. Region has exactly one open cell left. Placing here clears the rest of row 4, column I, and its region.
- Now, 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.
- From there, every open cell in row 7 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 B7 and C7, so whichever candidate ends up marked will clear them, and they can be ruled out now regardless of which one wins.
- First, 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.
- Next, place a marker at C6. Region has exactly one open cell left. Placing here clears the rest of row 6, column C, and its region.
- Then, place a marker at B9. Region has exactly one open cell left. Placing here clears the rest of row 9, column B, and its region.
- Finally, place a marker at G8. Region has exactly one open cell left.
The key move here was Forced chain. Once you can spot that, this board falls quickly. ∎