QED Logic quod erat demonstrandum

An Expert 10×10 Puzzle Solved with Forced Chain in 26 Steps

10×10 · expert · 26 steps

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This 10×10 board rates as expert (difficulty score 1233). Solving it from scratch takes 26 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 remaining candidate in this region touches J2, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  2. Next, 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.
  3. Then, every remaining candidate in this region touches I2 and I4, so whichever candidate ends up marked will clear them, and they can be ruled out now regardless of which one wins.
  4. 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.
  5. Now, marking B2 would force C4, J4, A5, and 3 more cleared (together, two regions have open cells only in rows 4 and 5), then A6, C6, D6, and 1 more cleared (every open cell left in this region sits in row 6, so its marker has to land there), then C7 (the last open cell in region 6), and then region 8 would be left with no open cell for its marker. So B2 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, if C2 were the marker, it would force A4 (the last open cell in region 1), then H5, I5 and J5 cleared (every open cell left in this region sits in row 5, so its marker has to land there), then B6, D6 and J6 cleared (every open cell left in this region sits in row 6, so its marker has to land there), and the chain continues, and region 10 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 D2 it would force A4 (the last open cell in region 1), then H5, I5 and J5 cleared (every open cell left in this region sits in row 5, so its marker has to land there), then B6, C6 and J6 cleared (every open cell left in this region sits in row 6, so its marker has to land there), and the chain continues, and region 8 would be left with no open cell for its marker.
  9. Next, marking E2 would force A4 (the last open cell in region 1), then H5, I5 and J5 cleared (every open cell left in this region sits in row 5, so its marker has to land there), then B6, C6, D6, and 1 more cleared (every open cell left in this region sits in row 6, so its marker has to land there), and the chain continues, and then region 10 would be left with no open cell for its marker. So E2 can't be the marker there.
  10. Then, if F2 were the marker, it would force A4 (the last open cell in region 1), then H5, I5 and J5 cleared (every open cell left in this region sits in row 5, so its marker has to land there), then B6, C6, D6, and 1 more cleared (every open cell left in this region sits in row 6, so its marker has to land there), and the chain continues, and region 10 would end up with no open cell for its marker, which can't happen. So it can't go there.
  11. After that, that one can't be the marker: at G2 it would force A4 (the last open cell in region 1), then H5, I5 and J5 cleared (every open cell left in this region sits in row 5, so its marker has to land there), then B6, C6, D6, and 1 more cleared (every open cell left in this region sits in row 6, so its marker has to land there), and the chain continues, and region 10 would be left with no open cell for its marker.
  12. Now, that one can't be the marker: at I1 it would force A2 (the last open cell in row 2), then B4, C4, J4, and 3 more cleared (together, two regions have open cells only in rows 4 and 5), then B6, C6, D6, and 1 more cleared (every open cell left in this region sits in row 6, so its marker has to land there), and the chain continues, and row 6 would be left with no open cell for its marker.
  13. From there, place a marker at J1. Region has exactly one open cell left. Placing here clears the rest of row 1, column J, and its region.
  14. Following that, every remaining candidate in this region touches H2 and H4, so whichever candidate ends up marked will clear them, and they can be ruled out now regardless of which one wins.
  15. First, place a marker at A2. Row 2 has exactly one open cell left. Placing here clears the rest of row 2, column A, and its region.
  16. Next, together, two regions have open cells only in rows 4 and 5; 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.
  17. Then, every open cell left in this region sits in row 6, so its marker has to land there; that clears every other open cell in the row.
  18. After that, every open cell left in this region sits in row 7, so its marker has to land there; that clears every other open cell in the row.
  19. Now, place a marker at D8. Region has exactly one open cell left. Placing here clears the rest of row 8, column D, and its region. Placing here also clears its 3 touching neighbours.
  20. From there, place a marker at B7. Region has exactly one open cell left. Placing here clears the rest of row 7, column B, and its region.
  21. Following that, place a marker at C10. Region has exactly one open cell left. Placing here clears the rest of row 10, column C, and its region.
  22. First, 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.
  23. Next, 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 2 touching neighbours.
  24. 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.
  25. After that, 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.
  26. Finally, place a marker at F9. Region has exactly one open cell left.

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