QED Logic quod erat demonstrandum

A Gentle 9×9 Puzzle Solved with Paired Regions in 13 Steps

9×9 · gentle · 13 steps

Play this board yourself →
A
B
C
D
E
F
G
H
I
1
2
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8
9
1×
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7×
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9×
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This 9×9 board rates as gentle (difficulty score 664). Solving it from scratch takes 13 logical steps, using 6 techniques: Single cell, Row/column exclusion, Adjacency clear, Region confinement, Line confinement, Paired regions. Every step below is forced: nothing here is a guess.

  1. First, 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.
  2. Next, every open cell in row 9 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.
  3. Then, place a marker at A8. Row 8 has exactly one open cell left. Placing here clears the rest of row 8, column A, and its region. Placing here also clears its 1 touching neighbour.
  4. After that, every open cell left in this region sits in column B, so its marker has to land there; that clears every other open cell in the column.
  5. Now, place a marker at C2. Region has exactly one open cell left. Placing here clears the rest of row 2, column C, and its region. Placing here also clears its 1 touching neighbour.
  6. From there, together, two regions have open cells only in columns H and I; 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.
  7. Following that, place a marker at G3. Region has exactly one open cell left. Placing here clears the rest of row 3, column G, and its region. Placing here also clears its 1 touching neighbour.
  8. First, 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 2 touching neighbours.
  9. Next, place a marker at D6. Region has exactly one open cell left. Placing here clears the rest of row 6, column D, and its region.
  10. Then, place a marker at B5. Region has exactly one open cell left.
  11. After that, place a marker at I7. Region has exactly one open cell left. Placing here clears the rest of row 7, column I, and its region.
  12. Now, place a marker at H1. Region has exactly one open cell left.
  13. Finally, place a marker at F9. Region has exactly one open cell left.

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