QED Logic quod erat demonstrandum

A Hard 10×10 Puzzle Solved with Forced Chain in 22 Steps

10×10 · hard · 22 steps

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This 10×10 board rates as hard (difficulty score 1112). Solving it from scratch takes 22 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, 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 A3. Region has exactly one open cell left. Placing here clears the rest of row 3, column A, and its region. Placing here also clears its 1 touching neighbour.
  3. Then, 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.
  4. After that, place a marker at C4. Region has exactly one open cell left. Placing here clears the rest of row 4, column C, and its region. Placing here also clears its 1 touching neighbour.
  5. Now, every open cell left in this region sits in row 5, so its marker has to land there; that clears every other open cell in the row.
  6. From there, every remaining candidate in this region touches F6 and G6, so whichever candidate ends up marked will clear them, and they can be ruled out now regardless of which one wins.
  7. Following that, if E1 were the marker, it would force F7 (the last open cell in region 7), then G5 (the last open cell in region 6), then B9 (the last open cell in region 10), 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.
  8. First, that one can't be the marker: at F1 it would force G5 (the last open cell in region 6), then E8, E9 and E10 cleared (every open cell left in this region sits in column E, so its marker has to land there), then D8 and D9 cleared (every open cell left in this region sits in column D, so its marker has to land there), and the chain continues, and row 8 would be left with no open cell for its marker.
  9. Next, every open cell left in this region sits in row 2, so its marker has to land there; that clears every other open cell in the row.
  10. Then, marking G1 would force F5 (the last open cell in region 6), then E7 (the last open cell in region 7), then D2 (the last open cell in region 2), and then region 8 would be left with no open cell for its marker. So G1 can't be the marker there.
  11. After that, if H1 were the marker, it would force J6, J7, J8, and 4 more cleared (every open cell in column I belongs to the same region, so that region's marker has to be somewhere in this column), and column J would end up with no open cell for its marker, which can't happen. So it can't go there.
  12. Now, marking J1 would force H8, H9, F10, and 2 more cleared (every open cell in column I belongs to the same region, so that region's marker has to be somewhere in this column), then G7, D8, E8, and 5 more cleared (every open cell in column H belongs to the same region, so that region's marker has to be somewhere in this column), then I8 (the last open cell in row 8), and the chain continues, and then column G 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 I1. Region has exactly one open cell left. Placing here clears the rest of row 1, column I, and its region.
  14. Following that, 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.
  15. First, every open cell in column H 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.
  16. Next, place a marker at J8. Row 8 has exactly one open cell left. Placing here clears the rest of row 8, column J, and its region.
  17. Then, place a marker at G5. Column 7 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.
  18. After that, place a marker at H7. Region has exactly one open cell left. Placing here clears the rest of row 7, column H, and its region.
  19. Now, place a marker at E6. Region has exactly one open cell left. Placing here clears the rest of row 6, column E, and its region.
  20. From there, place a marker at B9. Region has exactly one open cell left.
  21. Following that, place a marker at D10. Region has exactly one open cell left. Placing here clears the rest of row 10, column D, and its region.
  22. Finally, place a marker at F2. Region has exactly one open cell left.

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