QED Logic quod erat demonstrandum

An Expert 10×10 Puzzle Solved with Forced Chain in 23 Steps

10×10 · expert · 23 steps

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A
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This 10×10 board rates as expert (difficulty score 1243). Solving it from scratch takes 23 logical steps, using 7 techniques: Single cell, Row/column exclusion, Adjacency clear, Region confinement, Line confinement, Paired regions, Forced chain. Every step below is forced: nothing here is a guess.

  1. First, together, two regions have open cells only in rows 1 and 2; 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.
  2. 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. Placing here also clears its 2 touching neighbours.
  3. 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. Placing here also clears its 1 touching neighbour.
  4. After that, every open cell in row 10 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.
  5. Now, together, two regions have open cells only in rows 5 and 6; 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.
  6. From there, place a marker at B7. Region has exactly one open cell left. Placing here clears the rest of row 7, column B, and its region.
  7. Following that, 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.
  8. First, that one can't be the marker: at F1 it would force E10 (the last open cell in column E), then D8 (the last open cell in region 9), then J9 (the last open cell in region 8), and the chain continues, and row 6 would be left with no open cell for its marker.
  9. Next, marking G1 would force J2 (the last open cell in region 3), then I5 (the last open cell in region 5), and then row 6 would be left with no open cell for its marker. So G1 can't be the marker there.
  10. Then, if H1 were the marker, it would force F2 (the last open cell in region 2), then G5 (the last open cell in region 6), then J6 (the last open cell in region 5), and region 8 would end up with no open cell for its marker, which can't happen. So it can't go there.
  11. After that, that one can't be the marker: at I1 it would force F2 (the last open cell in region 2), then E10 (the last open cell in column E), then D8 (the last open cell in region 9), and the chain continues, and region 5 would be left with no open cell for its marker.
  12. Now, marking J1 would force F2 (the last open cell in region 2), then I5 (the last open cell in region 5), and then row 6 would be left with no open cell for its marker. So J1 can't be the marker there.
  13. From there, place a marker at E1. Row 1 has exactly one open cell left. Placing here clears the rest of row 1, column E, and its region.
  14. Following that, marking H2 would force J6 (the last open cell in row 6), and then region 8 would be left with no open cell for its marker. So H2 can't be the marker there.
  15. First, that one can't be the marker: at J2 it would force I5 (the last open cell in region 5), and row 6 would be left with no open cell for its marker.
  16. Next, place a marker at G2. Region has exactly one open cell left. Placing here clears the rest of row 2, column G, and its region.
  17. Then, marking H5 would force J6 (the last open cell in region 5), and then region 8 would be left with no open cell for its marker. So H5 can't be the marker there.
  18. After that, if I5 were the marker, row 6 would end up with no open cell for its marker, which can't happen. So it can't go there.
  19. Now, place a marker at I10. Column 9 has exactly one open cell left. Placing here clears the rest of row 10, column I, and its region. Placing here also clears its 1 touching neighbour.
  20. From there, place a marker at D9. Row 9 has exactly one open cell left. Placing here clears the rest of row 9, column D, and its region.
  21. Following that, place a marker at F5. Column 6 has exactly one open cell left. Placing here clears the rest of row 5, column F, and its region.
  22. First, place a marker at J6. Region has exactly one open cell left. Placing here clears the rest of row 6, column J, and its region.
  23. Finally, place a marker at H8. Region has exactly one open cell left.

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