QED Logic quod erat demonstrandum

An Expert 11×11 Puzzle Solved with Forced Chain in 26 Steps

11×11 · expert · 26 steps

Play this board yourself →
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This 11×11 board rates as expert (difficulty score 1177). Solving it from scratch takes 26 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, every open cell left in this region sits in column K, so its marker has to land there; that clears every other open cell in the column.
  2. Next, every open cell left in this region sits in column J, so its marker has to land there; that clears every other open cell in the column.
  3. Then, every open cell left in this region sits in row 9, 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 rows 10 and 11; 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.
  5. Now, together, two regions have open cells only in rows 7 and 8; 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.
  6. From there, place a marker at J6. Region has exactly one open cell left. Placing here clears the rest of row 6, column J, and its region. Placing here also clears its 2 touching neighbours.
  7. Following that, place a marker at C5. Region has exactly one open cell left. Placing here clears the rest of row 5, column C, and its region. Placing here also clears its 2 touching neighbours.
  8. First, 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.
  9. Next, every remaining candidate in this region touches E3, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  10. Then, every remaining candidate in this region touches A10 and B10, so whichever candidate ends up marked will clear them, and they can be ruled out now regardless of which one wins.
  11. After that, place a marker at A11. Region has exactly one open cell left. Placing here clears the rest of row 11, column A, and its region.
  12. Now, place a marker at B9. Region has exactly one open cell left.
  13. From there, marking D1 would force E4 (the last open cell in region 3), then F8 (the last open cell in region 7), and then row 7 would be left with no open cell for its marker. So D1 can't be the marker there.
  14. Following that, if E1 were the marker, it would force D3 (the last open cell in region 3), then K2 (the last open cell in region 4), then F8 (the last open cell in region 7), and row 7 would end up with no open cell for its marker, which can't happen. So it can't go there.
  15. First, that one can't be the marker: at F1 it would force D10 and E10 cleared (together, two regions have open cells only in columns D and E), and row 10 would be left with no open cell for its marker.
  16. Next, every open cell in row 1 belongs to the same region, so that region's marker has to be somewhere in this row; that clears every other open cell in the region.
  17. Then, every remaining candidate in this region touches H2, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  18. After that, marking G1 would force H8 (the last open cell in column H), and then column I would be left with no open cell for its marker. So G1 can't be the marker there.
  19. Now, if H1 were the marker, it would force I8 (the last open cell in column I), then D7 (the last open cell in region 7), then E4 (the last open cell in region 3), and column G would end up with no open cell for its marker, which can't happen. So it can't go there.
  20. 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.
  21. Following that, place a marker at H8. Column 8 has exactly one open cell left. Placing here clears the rest of row 8, column H, and its region.
  22. First, place a marker at D7. Region has exactly one open cell left. Placing here clears the rest of row 7, column D, and its region.
  23. Next, 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.
  24. 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.
  25. After that, 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.
  26. Finally, place a marker at K3. Region has exactly one open cell left.

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