QED Logic quod erat demonstrandum

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

11×11 · expert · 26 steps

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This 11×11 board rates as expert (difficulty score 1310). 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 remaining candidate in this region touches F2, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  2. Next, marking A1 would force G2 (the last open cell in region 2), and then region 3 would be left with no open cell for its marker. So A1 can't be the marker there.
  3. Then, if B1 were the marker, it would force G2 (the last open cell in region 2), and region 3 would end up with no open cell for its marker, which can't happen. So it can't go there.
  4. After that, that one can't be the marker: at C1 it would force G2 (the last open cell in region 2), and region 3 would be left with no open cell for its marker.
  5. Now, marking D1 would force G2 (the last open cell in region 2), and then region 3 would be left with no open cell for its marker. So D1 can't be the marker there.
  6. From there, if E1 were the marker, it would force G2 (the last open cell in region 2), and region 3 would end up with no open cell for its marker, which can't happen. So it can't go there.
  7. Following that, marking G1 would force H3, H4, H7, and 4 more cleared (every open cell left in this region sits in column H, so its marker has to land there), then J2 (the last open cell in region 3), then A3 (the last open cell in region 1), and the chain continues, and then column H would be left with no open cell for its marker. So G1 can't be the marker there.
  8. First, if H1 were the marker, region 2 would end up with no open cell for its marker, which can't happen. So it can't go there.
  9. Next, that one can't be the marker: at I1 it would force G2 (the last open cell in region 2), then A3 (the last open cell in region 1), then H4, H7, H8, and 3 more cleared (every open cell left in this region sits in column H, so its marker has to land there), and the chain continues, and region 6 would be left with no open cell for its marker.
  10. Then, marking J1 would force G2 (the last open cell in region 2), then A3 (the last open cell in region 1), then H4, H7, H8, and 3 more cleared (every open cell left in this region sits in column H, so its marker has to land there), and the chain continues, and then column H would be left with no open cell for its marker. So J1 can't be the marker there.
  11. After that, if K1 were the marker, it would force G2 (the last open cell in region 2), then A3 (the last open cell in region 1), then H4, H7, H8, and 3 more cleared (every open cell left in this region sits in column H, so its marker has to land there), and the chain continues, and column H would end up with no open cell for its marker, which can't happen. So it can't go there.
  12. Now, place a marker at F1. Row 1 has exactly one open cell left. Placing here clears the rest of row 1, column F, and its region. Placing here also clears its 1 touching neighbour.
  13. From there, together, two regions have open cells only in rows 2 and 3; 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.
  14. Following that, 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.
  15. First, 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. Placing here also clears its 2 touching neighbours.
  16. Next, place a marker at G7. Region 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.
  17. Then, every open cell in column K belongs to the same region, so that region's marker has to be somewhere in this column; that clears every other open cell in the region.
  18. After that, place a marker at J2. Column 10 has exactly one open cell left. Placing here clears the rest of row 2, column J, and its region.
  19. Now, place a marker at A3. Region has exactly one open cell left. Placing here clears the rest of row 3, column A, and its region. Placing here also clears its 1 touching neighbour.
  20. From there, every open cell left in this region sits in row 10, so its marker has to land there; that clears every other open cell in the row.
  21. Following that, place a marker at H4. Column 8 has exactly one open cell left. Placing here clears the rest of row 4, column H, and its region.
  22. First, place a marker at B5. Region has exactly one open cell left. Placing here clears the rest of row 5, column B, and its region.
  23. Next, 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. Placing here also clears its 2 touching neighbours.
  24. Then, 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 1 touching neighbour.
  25. After that, place a marker at E11. Region has exactly one open cell left. Placing here clears the rest of row 11, column E, and its region.
  26. Finally, place a marker at K9. Region has exactly one open cell left.

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