QED Logic quod erat demonstrandum

A Hard 9×9 Puzzle Solved with Forced Chain in 17 Steps

9×9 · hard · 17 steps

Play this board yourself →
A
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I
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This 9×9 board rates as hard (difficulty score 1093). Solving it from scratch takes 17 logical steps, using 6 techniques: Single cell, Row/column exclusion, Adjacency clear, Region confinement, Paired regions, Forced chain. Every step below is forced: nothing here is a guess.

  1. First, marking A1 would force H2 (the last open cell in region 3), then I4 (the last open cell in region 4), then G6 (the last open cell in region 7), and then region 5 would be left with no open cell for its marker. So A1 can't be the marker there.
  2. Next, if B1 were the marker, it would force H2 (the last open cell in region 3), then I4 (the last open cell in region 4), then G6 (the last open cell in region 7), 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.
  3. Then, that one can't be the marker: at C1 it would force H2 (the last open cell in region 3), then I4 (the last open cell in region 4), then G6 (the last open cell in region 7), and the chain continues, and region 8 would be left with no open cell for its marker.
  4. After that, marking D1 would force H2 (the last open cell in region 3), then I4 (the last open cell in region 4), then G6 (the last open cell in region 7), and the chain continues, and then region 8 would be left with no open cell for its marker. So D1 can't be the marker there.
  5. Now, that one can't be the marker: at F1 it would force H2 (the last open cell in region 3), then I4 (the last open cell in region 4), then G6 (the last open cell in region 7), and region 8 would be left with no open cell for its marker.
  6. From there, marking G1 would force I7, I8, I9, and 2 more cleared (together, two regions have open cells only in columns I and H), then F9 (the last open cell in region 8), and then region 2 would be left with no open cell for its marker. So G1 can't be the marker there.
  7. Following that, together, two regions have open cells only in columns H and I; 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.
  8. First, place a marker at G6. Region has exactly one open cell left. Placing here clears the rest of row 6, column G, and its region. Placing here also clears its 3 touching neighbours.
  9. Next, place a marker at F9. Region has exactly one open cell left. Placing here clears the rest of row 9, column F, and its region. Placing here also clears its 1 touching neighbour.
  10. Then, place a marker at E1. Region has exactly one open cell left. Placing here clears the rest of row 1, column E, and its region. Placing here also clears its 1 touching neighbour.
  11. After 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. Placing here also clears its 1 touching neighbour.
  12. Now, 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.
  13. From there, 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.
  14. Following that, place a marker at B8. Region has exactly one open cell left. Placing here clears the rest of row 8, column B, and its region. Placing here also clears its 2 touching neighbours.
  15. First, place a marker at D7. Row 7 has exactly one open cell left. Placing here clears the rest of row 7, column D, and its region.
  16. Next, 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.
  17. Finally, place a marker at A5. Region has exactly one open cell left.

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