QED Logic quod erat demonstrandum

An Expert 10×10 Puzzle Solved with Forced Chain in 22 Steps

10×10 · expert · 22 steps

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This 10×10 board rates as expert (difficulty score 1156). Solving it from scratch takes 22 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, place a marker at I3. Region has exactly one open cell left. Placing here clears the rest of row 3, column I, and its region. Placing here also clears its 2 touching neighbours.
  3. Then, 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.
  4. After that, together, two regions have open cells only in columns A and B; 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.
  5. Now, place a marker at D4. Region has exactly one open cell left. Placing here clears the rest of row 4, column D, and its region. Placing here also clears its 2 touching neighbours.
  6. From there, every remaining candidate in this region touches F1 and G1, so whichever candidate ends up marked will clear them, and they can be ruled out now regardless of which one wins.
  7. Following that, together, two regions have open cells only in columns H 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.
  8. First, marking J1 would force H8 (the last open cell in region 7), then B7 (the last open cell in column B), then G6 cleared (every open cell in row 5 belongs to the same region, so that region's marker has to be somewhere in this row), and the chain continues, and then column C would be left with no open cell for its marker. So J1 can't be the marker there.
  9. Next, 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. Placing here also clears its 1 touching neighbour.
  10. Then, place a marker at F2. Region has exactly one open cell left. Placing here clears the rest of row 2, column F, and its region.
  11. After that, that one can't be the marker: at G5 it would force E10 (the last open cell in region 10), then B8 cleared (every remaining candidate in this region touches B8), then B7 (the last open cell in column B), and row 8 would be left with no open cell for its marker.
  12. Now, if A6 were the marker, it would force B8 (the last open cell in region 9), then G7 (the last open cell in region 6), then J5 (the last open cell in region 7), and the chain continues, and row 9 would end up with no open cell for its marker, which can't happen. So it can't go there.
  13. From there, that one can't be the marker: at J5 it would force C7, E7 and G7 cleared (every open cell left in this region sits in row 7, so its marker has to land there), then E6 cleared (every open cell left in this region sits in row 6, so its marker has to land there), then A8 and B8 cleared (every remaining candidate in this region touches A8 and B8), and the chain continues, and row 8 would be left with no open cell for its marker.
  14. Following that, place a marker at A5. Row 5 has exactly one open cell left. Placing here clears the rest of row 5, column A, and its region.
  15. First, 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. Placing here also clears its 2 touching neighbours.
  16. Next, together, two regions have open cells only in rows 6 and 7; 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, if G6 were the marker, it would force J7 (the last open cell in region 7), then E10 (the last open cell in region 10), and row 9 would end up with no open cell for its marker, which can't happen. So it can't go there.
  18. After that, if J6 were the marker, it would force G7 (the last open cell in region 6), then E10 (the last open cell in region 10), and row 9 would end up with no open cell for its marker, which can't happen. So it can't go there.
  19. Now, 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.
  20. From there, place a marker at C6. Region has exactly one open cell left. Placing here clears the rest of row 6, column C, and its region.
  21. Following that, place a marker at E9. Region has exactly one open cell left. Placing here clears the rest of row 9, column E, and its region.
  22. Finally, place a marker at G10. Region has exactly one open cell left.

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