QED Logic quod erat demonstrandum

An Expert 10×10 Puzzle Solved with Forced Chain in 24 Steps

10×10 · expert · 24 steps

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This 10×10 board rates as expert (difficulty score 1134). Solving it from scratch takes 24 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.

  1. First, every open cell in column A 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.
  2. Next, every remaining candidate in this region touches I2, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  3. Then, every open cell left in this region sits in column J, so its marker has to land there; that clears every other open cell in the column.
  4. After 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.
  5. Now, 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.
  6. From there, marking A1 would force E8, B10, C10, and 4 more cleared (every open cell in row 9 belongs to the same region, so that region's marker has to be somewhere in this row), and then row 10 would be left with no open cell for its marker. So A1 can't be the marker there.
  7. Following that, that one can't be the marker: at I1 it would force J3 (the last open cell in region 3), then H5 (the last open cell in region 8), then A2, F2 and G2 cleared (every open cell left in this region sits in row 2, so its marker has to land there), and region 5 would be left with no open cell for its marker.
  8. First, marking J1 would force A2, F2, G2, and 4 more cleared (together, two regions have open cells only in rows 2 and 3), then G4 (the last open cell in region 5), then I5 (the last open cell in region 8), and the chain continues, and then column C would be left with no open cell for its marker. So J1 can't be the marker there.
  9. Next, place a marker at H1. Row 1 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.
  10. Then, together, two regions have open cells only in rows 2 and 3; 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.
  11. After that, place a marker at G4. Region has exactly one open cell left. Placing here clears the rest of row 4, column G, and its region. Placing here also clears its 1 touching neighbour.
  12. Now, place a marker at I5. Region has exactly one open cell left. Placing here clears the rest of row 5, column I, and its region.
  13. From there, place a marker at F6. Region has exactly one open cell left. Placing here clears the rest of row 6, column F, and its region. Placing here also clears its 1 touching neighbour.
  14. Following that, every open cell left in this region sits in column B, so its marker has to land there; that clears every other open cell in the column.
  15. First, together, two regions have open cells only in rows 7 and 8; 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.
  16. Next, every remaining candidate in this region touches C7 and C8, so whichever candidate ends up marked will clear them, and they can be ruled out now regardless of which one wins.
  17. Then, 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.
  18. After that, place a marker at E2. Region has exactly one open cell left. Placing here clears the rest of row 2, column E, and its region.
  19. Now, place a marker at J3. Region has exactly one open cell left.
  20. From there, marking D7 would force B8 (the last open cell in region 6), and then row 9 would be left with no open cell for its marker. So D7 can't be the marker there.
  21. Following that, place a marker at D8. Region has exactly one open cell left. Placing here clears the rest of row 8, column D, and its region. Placing here also clears its 1 touching neighbour.
  22. First, place a marker at B7. Region has exactly one open cell left.
  23. Next, place a marker at C10. Region has exactly one open cell left. Placing here clears the rest of row 10, column C, and its region.
  24. Finally, place a marker at A9. Region has exactly one open cell left.

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