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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This 10×10 board rates as expert (difficulty score 1178). Solving it from scratch takes 23 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, marking A1 would force J2 (the last open cell in region 3), then H3 (the last open cell in region 2), then B4 (the last open cell in region 4), and then region 6 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 J2 (the last open cell in region 3), then H3 (the last open cell in region 2), then A4 (the last open cell in region 4), 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 J2 (the last open cell in region 3), then H3 (the last open cell in region 2), then D4, E4 and F4 cleared (every open cell left in this region sits in row 4, so its marker has to land there), and the chain continues, and region 8 would be left with no open cell for its marker.
  4. After that, marking D1 would force J2 (the last open cell in region 3), then H3 (the last open cell in region 2), then C4, E4 and F4 cleared (every open cell left in this region sits in row 4, so its marker has to land there), 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, if E1 were the marker, it would force J2 (the last open cell in region 3), then H3 (the last open cell in region 2), then C4, D4 and F4 cleared (every open cell left in this region sits in row 4, so its marker has to land there), 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.
  6. From there, that one can't be the marker: at F1 it would force J2 (the last open cell in region 3), then B3, C3 and G3 cleared (every open cell left in this region sits in row 3, so its marker has to land there), then C4, D4, E4, and 3 more cleared (every open cell left in this region sits in row 4, so its marker has to land there), and the chain continues, and region 7 would be left with no open cell for its marker.
  7. Following that, every open cell in row 1 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.
  8. First, 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.
  9. Next, every open cell left in this region sits in row 4, so its marker has to land there; that clears every other open cell in the row.
  10. Then, 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.
  11. After that, 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 2 touching neighbours.
  12. Now, together, two regions have open cells only in columns A and B; 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.
  13. From there, place a marker at C5. Region has exactly one open cell left. Placing here clears the rest of row 5, column C, and its region. Placing here also clears its 1 touching neighbour.
  14. Following that, 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.
  15. First, 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.
  16. Next, place a marker at J9. Region has exactly one open cell left. Placing here clears the rest of row 9, column J, and its region. Placing here also clears its 1 touching neighbour.
  17. Then, place a marker at I6. Region has exactly one open cell left. Placing here clears the rest of row 6, column I, and its region.
  18. After that, 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.
  19. Now, every remaining candidate in this region touches E2, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  20. From there, place a marker at H3. Region has exactly one open cell left. Placing here clears the rest of row 3, column H, and its region.
  21. Following that, place a marker at G1. Region has exactly one open cell left. Placing here clears the rest of row 1, column G, and its region.
  22. First, place a marker at D2. Region has exactly one open cell left. Placing here clears the rest of row 2, column D, and its region.
  23. Finally, place a marker at E10. Region has exactly one open cell left.

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