QED Logic quod erat demonstrandum

A Hard 10×10 Puzzle Solved with Forced Chain in 18 Steps

10×10 · hard · 18 steps

Play this board yourself →
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This 10×10 board rates as hard (difficulty score 963). Solving it from scratch takes 18 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 2, so its marker has to land there; that clears every other open cell in the row.
  2. Next, 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.
  3. Then, 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.
  4. After that, every remaining candidate in this region touches H1, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  5. Now, together, two regions have open cells only in columns I and J; 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.
  6. From there, 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.
  7. Following that, marking H2 would force J1 (the last open cell in region 3), then D5, E5 and F5 cleared (together, two regions have open cells only in rows 4 and 5), then F8, F9, F10, and 2 more cleared (together, two regions have open cells only in columns F and G), and then region 10 would be left with no open cell for its marker. So H2 can't be the marker there.
  8. First, place a marker at G2. Region has exactly one open cell left. Placing here clears the rest of row 2, column G, and its region.
  9. Next, together, two regions have open cells only in columns F and H; 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, place a marker at E5. Region has exactly one open cell left. Placing here clears the rest of row 5, column E, and its region. Placing here also clears its 2 touching neighbours.
  11. After that, 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. Placing here also clears its 1 touching neighbour.
  12. Now, 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.
  13. From there, place a marker at H6. Region has exactly one open cell left. Placing here clears the rest of row 6, column H, and its region. Placing here also clears its 1 touching neighbour.
  14. Following that, place a marker at J7. Region has exactly one open cell left. Placing here clears the rest of row 7, column J, and its region.
  15. First, place a marker at I1. Region has exactly one open cell left.
  16. Next, 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.
  17. Then, 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.
  18. 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. ∎