QED Logic quod erat demonstrandum

A Severe 8×8 Puzzle Solved with Forced Chain in 21 Steps

8×8 · severe · 21 steps

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A
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C
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H
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This 8×8 board rates as severe (difficulty score 1140). Solving it from scratch takes 21 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, 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.
  2. Next, marking A1 would force C2, D2, E2, and 1 more cleared (every open cell left in this region sits in row 2, so its marker has to land there), then F3 and G3 cleared (every remaining candidate in this region touches F3 and G3), then H6, H7 and H8 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 row 6 would be left with no open cell for its marker. So A1 can't be the marker there.
  3. Then, that one can't be the marker: at C1 it would force A4, A5 and A6 cleared (every open cell left in this region sits in column A, so its marker has to land there), then A2 and H2 cleared (every open cell left in this region sits in row 2, so its marker has to land there), then A3 (the last open cell in region 1), and the chain continues, and region 5 would be left with no open cell for its marker.
  4. After that, marking D1 would force A4, A5 and A6 cleared (every open cell left in this region sits in column A, so its marker has to land there), then A2 and H2 cleared (every open cell left in this region sits in row 2, so its marker has to land there), then A3 (the last open cell in region 1), and the chain continues, and then region 5 would be left with no open cell for its marker. So D1 can't be the marker there.
  5. Now, if E1 were the marker, it would force A4, A5 and A6 cleared (every open cell left in this region sits in column A, so its marker has to land there), then C4, G4, H4, and 4 more cleared (together, two regions have open cells only in rows 4 and 5), then H2, F3, G3, and 1 more cleared (together, two regions have open cells only in rows 2 and 3), and region 4 would end up with no open cell for its marker, which can't happen. So it can't go there.
  6. From there, every remaining candidate in this region touches G1, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  7. Following that, that one can't be the marker: at F1 it would force A4, A5 and A6 cleared (every open cell left in this region sits in column A, so its marker has to land there), then G8 cleared (together, two regions have open cells only in columns H and G), then D3 cleared (every remaining candidate in this region touches D3), and the chain continues, and column B would be left with no open cell for its marker.
  8. 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.
  9. Next, every remaining candidate in this region touches F3 and G3, so whichever candidate ends up marked will clear them, and they can be ruled out now regardless of which one wins.
  10. Then, every open cell left in this region sits in column H, so its marker has to land there; that clears every other open cell in the column.
  11. After that, every open cell in row 8 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.
  12. Now, place a marker at G7. Row 7 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.
  13. From there, 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. Placing here also clears its 1 touching neighbour.
  14. Following that, place a marker at A6. Row 6 has exactly one open cell left. Placing here clears the rest of row 6, column A, and its region.
  15. First, place a marker at B1. Region has exactly one open cell left. Placing here clears the rest of row 1, column B, and its region.
  16. Next, place a marker at H5. Row 5 has exactly one open cell left. Placing here clears the rest of row 5, column H, and its region.
  17. Then, every open cell left in this region sits in row 4, so its marker has to land there; that clears every other open cell in the row.
  18. After that, 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.
  19. Now, 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.
  20. From there, 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.
  21. Finally, place a marker at D8. Region has exactly one open cell left.

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