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SOLO-001 · Classic Sudoku

EXPERT

The Ninth Lock

Complete the grid completely. Only one solution respects all constraints.

This page contains the solution.

The answer

423715689 / 615893247 / 987624513 / 876532194 / 132469875 / 549187362 / 758246931 / 364951728 / 291378456

Coordinates: lines from top to bottom, columns from left to right.
L / C123456789
1423715689
2615893247
3987624513
4876532194
5132469875
6549187362
7758246931
8364951728
9291378456

The reasoning

  1. Note the candidates in each box by crossing the line, column and region.
  2. In the certificate below, units 1 to 9 are lines, 10 to 18 columns, 19 to 27 regions. Units 28 and 29, if present, are diagonals.
  3. A nude subset of two or three cells reserves as many numbers: these figures are excluded from the other cells of the unit. An interaction limits one digit in the intersection of two units and excludes from the rest of the second.
  4. Each hypothesis opens a separate branch. Never keep in the neighbouring branch a value found under an abandoned hypothesis. The contradictions and all the figures deduced appear in the tree of evidence.
  5. The search finds exactly a complete grid. It runs through 3 states; the 1 contradictory branches are closed. Finally check each unit and each starting point.

Common mistakes and false leads

Show the full search proof

Each branch is independent. The tree records the deductions and contradictions found with the method used here.

  1. L3C3: only place for 7 in unit 3 → 7
  2. L5C8: only place for 7 in unit 5 → 7
  3. L6C6: only place for 7 in unit 6 → 7
  4. L4C2: only place for 7 in unit 4 → 7
  5. L7C8: only place for 3 in unit 7 → 3
  6. L3C9: only place for 3 in unit 3 → 3
  7. L1C3: only place for 3 in unit 12 → 3
  8. L4C6: only place for 2 in unit 15 → 2
  9. L4C5: only place for 3 in unit 4 → 3
  10. L2C6: only place for 3 in unit 2 → 3
  11. L4C7: only place for 1 in unit 4 → 1
  12. L4C9: only place for 4 in unit 4 → 4
  13. L1C6: only place for 5 in unit 15 → 5
  14. L6C1: only place for 5 in unit 22 → 5
  15. L6C3: only place for 9 in unit 6 → 9
  16. L6C9: only place for 2 in unit 6 → 2
  17. L5C3: only place for 2 in unit 5 → 2
  18. L2C7: only place for 2 in unit 16 → 2
  19. Nude subset in Unit 5: L5C7, L5C9 reserve [5, 8]; remove these numbers from L5C4, L5C5.
  20. Nude subset in unit 26: L7C6, L8C6, L9C6 reserve [1, 6, 8]; remove these numbers from L7C5, L8C4.
  21. Units 12 and 25 overlap. In unit 12, 1 must lie in the overlap. Remove it from L8C2, L9C2.
L1C9 = 8

Assumption: L1C9 = 8

  1. L5C9: only candidate → 5
  2. L5C7: only candidate → 8
  3. L3C7: only place for 9 in unit 21 → 9
  4. L3C1: only candidate → 6
  5. L3C4: only candidate → 1
  6. L1C5: only candidate → 9
  7. L3C8: only candidate → 5
  8. L4C1: only candidate → 8
  9. L4C3: only candidate → 6
  10. L6C4: only candidate → 8
  11. L2C4: only candidate → 6
  12. L2C5: only candidate → 8
  13. L5C4: only candidate → 4
  14. L5C5: only candidate → 6
  15. L6C5: only candidate → 1
  16. L7C5: only candidate → 4
  17. L7C3: only candidate → 8
  18. L7C6: only candidate → 6
  19. L7C7: only candidate → 5
  20. L7C2: only candidate → 9
  21. L2C2: only candidate → 1
  22. L1C2: only candidate → 2
  23. L1C1: only candidate → 4
  24. L1C8: only candidate → 1
  25. L2C1: only candidate → 9
  26. L2C8: only candidate → 4
  27. L8C2: only candidate → 6
  28. L8C4: only candidate → 9

contradiction: a box no longer has no candidate

L1C9 = 9

Assumption: L1C9 = 9

  1. L3C7: only candidate → 5
  2. L3C8: only candidate → 1
  3. L5C7: only candidate → 8
  4. L5C9: only candidate → 5
  5. L7C7: only candidate → 9
  6. L7C2: only place for 5 in unit 7 → 5
  7. L9C8: only place for 5 in unit 9 → 5
  8. L2C5: only place for 9 in unit 14 → 9
  9. L3C4: only candidate → 6
  10. L3C1: only candidate → 9
  11. L5C4: only candidate → 4
  12. L5C5: only candidate → 6
  13. L8C3: only place for 4 in unit 8 → 4
  14. L7C5: only place for 4 in unit 7 → 4
  15. L9C2: only place for 9 in unit 9 → 9
  16. L8C4: only place for 9 in unit 8 → 9
  17. L9C1: only place for 2 in unit 9 → 2
  18. L1C1: only candidate → 4
  19. L1C8: only candidate → 8
  20. L1C5: only candidate → 1
  21. L1C2: only candidate → 2
  22. L2C1: only candidate → 6
  23. L2C2: only candidate → 1
  24. L2C4: only candidate → 8
  25. L2C8: only candidate → 4
  26. L4C1: only candidate → 8
  27. L4C3: only candidate → 6
  28. L6C4: only candidate → 1
  29. L6C5: only candidate → 8
  30. L7C3: only candidate → 8
  31. L7C6: only candidate → 6
  32. L8C2: only candidate → 6
  33. L8C8: only candidate → 2
  34. L8C9: only candidate → 8
  35. L8C6: only candidate → 1
  36. L9C3: only candidate → 1
  37. L9C6: only candidate → 8
  38. L9C9: only candidate → 6

valid complete solution

all the possibilities of this bifurcation have been examined

How we verified this answer

Exact enumeration, shutdown after two solutions, terminal verification of all units and cages.

{
  "nodes": 3,
  "branches": 1,
  "contradictions": 1,
  "depth": 1,
  "pairs": 2,
  "locked": 1
}

After single candidates, single places, bare pairs/triples and interactions between units, the solver still reaches bifurcations. The exhaustive evidence reaches 1 intertwined hypotheses, examines 3 states and closes 1 branches by contradiction. The Expert level asks to keep a chain of possibilities in addition to the interactions. Editorial evaluation according to the protocol of the collection, without calibration on a panel of players; another logical path can be shorter.

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