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 1169). Solving it from scratch takes 23 logical steps, using 7 techniques: Single cell, Row/column exclusion, Adjacency clear, Region confinement, Paired regions, Shape squeeze, Forced chain. Every step below is forced: nothing here is a guess.

  1. First, 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.
  2. Next, place a marker at J2. Region has exactly one open cell left. Placing here clears the rest of row 2, column J, and its region. Placing here also clears its 2 touching neighbours.
  3. Then, place a marker at H1. Region has exactly one open cell left. Placing here clears the rest of row 1, column H, and its region.
  4. After that, every remaining candidate in this region touches D9, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  5. Now, if A3 were the marker, it would force I6, I7, I8, and 2 more cleared (every open cell left in this region sits in column I, so its marker has to land there), then G7 (the last open cell in region 8), then F10 (the last open cell in region 9), and the chain continues, and region 5 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 B3 it would force I6, I7, I8, and 2 more cleared (every open cell left in this region sits in column I, so its marker has to land there), then G7 (the last open cell in region 8), then F10 (the last open cell in region 9), and the chain continues, and region 5 would be left with no open cell for its marker.
  7. Following that, marking C3 would force I6, I7, I8, and 2 more cleared (every open cell left in this region sits in column I, so its marker has to land there), then G7 (the last open cell in region 8), then F10 (the last open cell in region 9), and then region 10 would be left with no open cell for its marker. So C3 can't be the marker there.
  8. First, if D3 were the marker, it would force I6, I7, I8, and 2 more cleared (every open cell left in this region sits in column I, so its marker has to land there), then G7 (the last open cell in region 8), then F4 (the last open cell in region 1), and region 9 would end up with no open cell for its marker, which can't happen. So it can't go there.
  9. Next, marking F3 would force G5 (the last open cell in region 1), then I9 and I10 cleared (every open cell left in this region sits in column I, so its marker has to land there), and then region 9 would be left with no open cell for its marker. So F3 can't be the marker there.
  10. Then, together, two regions have open cells only in columns G 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.
  11. After that, every open cell left in this region sits in column F, so its marker has to land there; that clears every other open cell in the column.
  12. Now, every open cell left in this region sits in column E, so its marker has to land there; that clears every other open cell in the column.
  13. From there, 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.
  14. Following that, every remaining candidate in this region touches D4, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  15. First, if G3 were the marker, it would force E4 (the last open cell in region 1), and region 5 would end up with no open cell for its marker, which can't happen. So it can't go there.
  16. Next, place a marker at E3. Row 3 has exactly one open cell left. Placing here clears the rest of row 3, column E, and its region.
  17. Then, place a marker at G7. Column 7 has exactly one open cell left. Placing here clears the rest of row 7, column G, and its region. Placing here also clears its 1 touching neighbour.
  18. After that, place a marker at F10. Region has exactly one open cell left. Placing here clears the rest of row 10, column F, and its region.
  19. Now, place a marker at C9. Region has exactly one open cell left. Placing here clears the rest of row 9, column C, and its region. Placing here also clears its 1 touching neighbour.
  20. From there, place a marker at D5. Region has exactly one open cell left. Placing here clears the rest of row 5, column D, and its region.
  21. Following that, place a marker at I4. Region has exactly one open cell left. Placing here clears the rest of row 4, column I, and its region.
  22. First, place a marker at A8. Region has exactly one open cell left. Placing here clears the rest of row 8, column A, and its region.
  23. Finally, place a marker at B6. Region has exactly one open cell left.

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