QED Logic quod erat demonstrandum

An Expert 7×7 Puzzle Solved with Forced Chain in 21 Steps

7×7 · expert · 21 steps

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A
B
C
D
E
F
G
1
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5
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7
1×
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3×
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4×
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5×
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6×
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7×
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This 7×7 board rates as expert (difficulty score 1171). Solving it from scratch takes 21 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 D1, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  2. Next, every remaining candidate in this region touches B2, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  3. Then, marking A1 would force F2 and G2 cleared (every open cell left in this region sits in row 2, so its marker has to land there), then F3 (the last open cell in region 3), then D4 cleared (every open cell left in this region sits in row 4, so its marker has to land there), and the chain continues, and then region 7 would be left with no open cell for its marker. So A1 can't be the marker there.
  4. After that, if B1 were the marker, it would force F2 and G2 cleared (every open cell left in this region sits in row 2, so its marker has to land there), then F3 (the last open cell in region 3), then D2 (the last open cell in region 2), and region 5 would end up with no open cell for its marker, which can't happen. So it can't go there.
  5. Now, that one can't be the marker: at C1 it would force E2 (the last open cell in region 2), and region 3 would be left with no open cell for its marker.
  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, together, two regions have open cells only in columns B and C; 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, together, two regions have open cells only in columns E and D; 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 B3, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  10. 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.
  11. After that, every remaining candidate in this region touches B5 and C5, so whichever candidate ends up marked will clear them, and they can be ruled out now regardless of which one wins.
  12. Now, that one can't be the marker: at F1 it would force D2 (the last open cell in region 2), and column E would be left with no open cell for its marker.
  13. From there, marking G1 would force A2 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), then C4 (the last open cell in region 4), and then region 5 would be left with no open cell for its marker. So G1 can't be the marker there.
  14. Following that, place a marker at E1. Row 1 has exactly one open cell left. Placing here clears the rest of row 1, column E, and its region. Placing here also clears its 1 touching neighbour.
  15. First, 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.
  16. Next, place a marker at D5. Region has exactly one open cell left. Placing here clears the rest of row 5, column D, and its region. Placing here also clears its 2 touching neighbours.
  17. Then, place a marker at B4. Region has exactly one open cell left. Placing here clears the rest of row 4, column B, and its region. Placing here also clears its 1 touching neighbour.
  18. After that, place a marker at A2. Region has exactly one open cell left. Placing here clears the rest of row 2, column A, and its region.
  19. Now, place a marker at F3. Region has exactly one open cell left. Placing here clears the rest of row 3, column F, and its region.
  20. From there, place a marker at C7. Region has exactly one open cell left. Placing here clears the rest of row 7, column C, and its region.
  21. Finally, place a marker at G6. Region has exactly one open cell left.

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