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SOLO-005 · Diagonal Sudoku

EXPERT

The Two Crossings

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

This page contains the solution.

The answer

716854392 / 893612754 / 245973168 / 371489625 / 624135879 / 589726413 / 138597246 / 962341587 / 457268931

Coordinates: lines from top to bottom, columns from left to right.
L / C123456789
1716854392
2893612754
3245973168
4371489625
5624135879
6589726413
7138597246
8962341587
9457268931

The reasoning

  1. Note candidates in each box by crossing lines, columns and regions, diagonals.
  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 one complete grid. It visits 5 states and rules out the 2 contradictory branches. Check each unit and starting clue.

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. L9C1: only place for 4 in unit 9 → 4
  2. L9C7: only place for 9 in unit 9 → 9
  3. L9C2: only place for 5 in unit 9 → 5
  4. L5C5: only place for 3 in unit 28 → 3
  5. L4C1: only place for 3 in unit 4 → 3
  6. L7C2: only place for 3 in unit 11 → 3
  7. L1C7: only place for 3 in unit 16 → 3
  8. L1C6: only place for 4 in unit 1 → 4
  9. L2C9: only place for 4 in unit 2 → 4
  10. L2C8: only place for 5 in unit 2 → 5
  11. L4C9: only place for 5 in unit 4 → 5
  12. L6C1: only place for 5 in unit 6 → 5
  13. L5C6: only place for 5 in unit 5 → 5
  14. L8C7: only place for 5 in unit 8 → 5
  15. L6C4: only place for 7 in unit 29 → 7
  16. L1C1: only place for 7 in unit 1 → 7
  17. L3C5: only place for 7 in unit 14 → 7
  18. L5C8: only place for 7 in unit 17 → 7
  19. L4C6: only place for 9 in unit 23 → 9
  20. L5C9: only place for 9 in unit 18 → 9
  21. L6C3: only place for 9 in unit 6 → 9
  22. L6C7: only place for 4 in unit 6 → 4
  23. L5C3: only place for 4 in unit 5 → 4
  24. L8C1: only place for 9 in unit 8 → 9
  25. Units 5 and 22 overlap. In unit 5, 6 must lie in the overlap. Remove it from L6C2.
  26. Units 9 and 26 overlap. In unit 9, 8 must lie in the overlap. Remove it from L7C6, L8C4, L8C6.
  27. Units 24 and 17 overlap. In unit 24, 1 must lie in the overlap. Remove it from L1C8, L3C8, L8C8.
  28. Units 21 and 29 overlap. In unit 21, 1 must lie in the overlap. Remove it from L7C3, L8C2.
  29. Units 27 and 28 overlap. In unit 27, 8 must lie in the overlap. Remove it from L6C6.
  30. Units 23 and 14 overlap. In unit 23, 8 must lie in the overlap. Remove it from L2C5, L9C5.
  31. Units 28 and 27 overlap. In unit 28, 1 must lie in the overlap. Remove it from L7C9, L8C9.
  32. Units 29 and 25 overlap. In unit 29, 8 must lie in the overlap. Remove it from L7C1, L8C3.
L3C7 = 1

Assumption: L3C7 = 1

  1. Units 1 and 19 overlap. In unit 1, 1 must lie in the overlap. Remove it from L2C1.
  2. L7C1: only place for 1 in unit 10 → 1
  3. Units 5 and 22 overlap. In unit 5, 6 must lie in the overlap. Remove it from L6C2.
  4. Units 9 and 26 overlap. In unit 9, 8 must lie in the overlap. Remove it from L7C6, L8C4, L8C6.
  5. Units 18 and 27 overlap. In unit 18, 1 must lie in the overlap. Remove it from L8C8.
  6. L9C9: only place for 1 in unit 28 → 1
  7. L8C6: only place for 1 in unit 8 → 1
  8. L2C5: only place for 1 in unit 2 → 1
  9. L8C4: only place for 3 in unit 8 → 3
  10. L3C6: only place for 3 in unit 3 → 3
  11. L8C9: only place for 7 in unit 8 → 7
  12. L7C6: only place for 7 in unit 7 → 7
  13. Units 27 and 28 overlap. In unit 27, 8 must lie in the overlap. Remove it from L6C6.
  14. Units 23 and 14 overlap. In unit 23, 8 must lie in the overlap. Remove it from L9C5.
  15. Units 29 and 25 overlap. In unit 29, 8 must lie in the overlap. Remove it from L8C3.
L1C9 = 2

Assumption: L1C9 = 2

  1. L7C9: only candidate → 6
  2. L7C3: only candidate → 8
  3. L7C7: only candidate → 2
  4. L5C7: only candidate → 8
  5. L8C2: only candidate → 6
  6. L5C2: only candidate → 2
  7. L4C3: only candidate → 1
  8. L1C3: only candidate → 6
  9. L1C8: only candidate → 9
  10. L1C4: only candidate → 8
  11. L1C2: only candidate → 1
  12. L3C1: only candidate → 2
  13. L2C1: only candidate → 8
  14. L3C8: only candidate → 6
  15. L3C4: only candidate → 9
  16. L4C8: only candidate → 2
  17. L4C5: only candidate → 8
  18. L5C1: only candidate → 6
  19. L6C2: only candidate → 8
  20. L6C6: only candidate → 6
  21. L2C6: only candidate → 2
  22. L2C4: only candidate → 6
  23. L6C5: only candidate → 2
  24. L6C8: only candidate → 1
  25. L8C3: only candidate → 2
  26. L8C8: only candidate → 8
  27. L9C4: only candidate → 2
  28. L9C5: only candidate → 6
  29. L9C6: only candidate → 8

valid complete solution

L1C9 = 6

Assumption: L1C9 = 6

  1. L7C9: only candidate → 2
  2. L7C3: only candidate → 8

contradiction: a box no longer has no candidate

all the possibilities of this bifurcation have been examined

L3C7 = 2

Assumption: L3C7 = 2

  1. L5C7: only candidate → 8
  2. L7C7: only candidate → 1
  3. L8C8: only place for 8 in unit 17 → 8
  4. L1C9: only place for 1 in unit 18 → 1
  5. L8C2: only candidate → 6
  6. L5C2: only candidate → 2
  7. L1C2: only candidate → 8
  8. L5C1: only candidate → 6
  9. L3C1: only candidate → 1
  10. L2C1: only candidate → 2
  11. L1C3: only candidate → 6
  12. L1C8: only candidate → 9
  13. L1C4: only candidate → 2
  14. L3C8: only candidate → 6
  15. L3C6: only candidate → 3
  16. L3C4: only candidate → 9
  17. L6C2: only candidate → 1
  18. L4C3: only candidate → 8

contradiction: a box no longer has no candidate

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": 5,
  "branches": 2,
  "contradictions": 2,
  "depth": 2,
  "pairs": 0,
  "locked": 15
}

After unique candidates, unique places, bare pairs/triples and interactions between units, the solver still reaches bifurcations. The exhaustive evidence reaches 2 intertwined hypotheses, examines 5 states and closes 2 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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