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 1129). 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 remaining candidate in this region touches C2, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  2. Next, marking A1 would force B6 (the last open cell in region 8), then E3, F3, G3, and 3 more cleared (every open cell left in this region sits in row 3, so its marker has to land there), then F2, G2, H2, and 2 more cleared (every open cell left in this region sits in row 2, so its marker has to land there), and the chain continues, and then region 3 would be left with no open cell for its marker. So A1 can't be the marker there.
  3. Then, if B1 were the marker, it would force E3, F3, G3, and 3 more cleared (every open cell left in this region sits in row 3, so its marker has to land there), then F2, G2, H2, and 2 more cleared (every open cell left in this region sits in row 2, so its marker has to land there), then A10 cleared (every open cell left in this region sits in column A, so its marker has to land there), and the chain continues, and region 3 would end up with no open cell for its marker, which can't happen. So it can't go there.
  4. After that, that one can't be the marker: at C1 it would force E4, E5, E6, and 4 more cleared (every open cell left in this region sits in column E, so its marker has to land there), then E3, F3, G3, and 3 more cleared (every open cell left in this region sits in row 3, so its marker has to land there), then E2 (the last open cell in region 2), and the chain continues, and region 3 would be left with no open cell for its marker.
  5. Now, marking D1 would force E3 (the last open cell in region 2), then B2 (the last open cell in region 4), then C8, C9 and C10 cleared (every open cell left in this region sits in column C, 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.
  6. 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.
  7. Following that, place a marker at B6. Region has exactly one open cell left. Placing here clears the rest of row 6, column B, and its region. Placing here also clears its 3 touching neighbours.
  8. First, every open cell left in this region sits in row 3, so its marker has to land there; that clears every other open cell in the row.
  9. Next, together, two regions have open cells only in columns C and D; 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.
  10. Then, together, two regions have open cells only in rows 2 and 4; 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, 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.
  12. Now, place a marker at G5. Region has exactly one open cell left. Placing here clears the rest of row 5, column G, and its region. Placing here also clears its 1 touching neighbour.
  13. From there, every open cell in column J 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.
  14. Following that, every open cell in column I 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.
  15. First, place a marker at H7. Column 8 has exactly one open cell left. Placing here clears the rest of row 7, column H, and its region.
  16. Next, every remaining candidate in this region touches F2, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  17. Then, place a marker at E4. Region has exactly one open cell left. Placing here clears the rest of row 4, column E, and its region. Placing here also clears its 1 touching neighbour.
  18. After that, place a marker at F1. Region has exactly one open cell left.
  19. Now, place a marker at A2. Region has exactly one open cell left.
  20. From there, 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.
  21. Following that, every remaining candidate in this region touches I9, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  22. First, place a marker at I10. Region 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.
  23. Next, place a marker at J8. Region has exactly one open cell left. Placing here clears the rest of row 8, column J, and its region.
  24. Finally, place a marker at D9. Region has exactly one open cell left.

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