The Big BookOF GAMES
← All 40 challenges

SOLO-009 · Killer Sudoku

EXPERT

Captive Sums

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

This page contains the solution.

The answer

861472395 / 537916428 / 294835176 / 415283769 / 678149532 / 923567841 / 782654913 / 146398257 / 359721684

Coordinates: lines from top to bottom, columns from left to right.
L / C123456789
120861134721239225
2518324791116428
32948163517176
4124116522183371469
51567814913532
69102351867841
715788218651241791143
8191463998257
9335972151684

Thick stroke: region. Dotted inside the cells: cage and its total. A tint on a diagonal reminds the diagonal constraint, if it applies.

Exact list of cages (coordinates and totals)

The reasoning

  1. Note the candidates in each box by crossing the line, column and region and amounts of cages.
  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 9 states; the 4 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. L4C6: only candidate → 3
  2. L8C5: only candidate → 9
  3. L9C1: only candidate → 3
  4. Nude sub-assembly in Unit 10: L5C1, L6C1 reserve [6, 9]; remove these numbers from L2C1, L3C1, L7C1, L8C1.
  5. L7C1: only candidate → 7
  6. L7C2: only candidate → 8
  7. Nude subset in Unit 12: L7C3, L8C3 reserve [2, 6]; remove these numbers from L1C3, L3C3, L4C3, L5C3, L6C3.
  8. Nude sub-assembly in unit 22: L5C1, L6C1 reserve [6, 9]; remove these numbers from L4C2, L4C3, L5C2, L5C3, L6C2.
  9. L6C2: only place for 2 in unit 22 → 2
  10. Units 4 and 24 overlap. In unit 4, 9 must lie in the overlap. Remove it from L5C7, L5C9, L6C7.
  11. Units 7 and 27 overlap. In unit 7, 9 must lie in the overlap. Remove it from L9C7, L9C8, L9C9.
  12. Units 13 and 20 overlap. In unit 13, 9 must lie in the overlap. Remove it from L1C6, L3C6.
  13. Units 23 and 13 overlap. In unit 23, 1 must lie in the overlap. Remove it from L1C4, L7C4, L8C4, L9C4.
  14. Units 23 and 13 overlap. In unit 23, 2 must lie in the overlap. Remove it from L1C4, L7C4, L8C4, L9C4.
L2C3 = 7

Assumption: L2C3 = 7

  1. Nude subset in Unit 10: L5C1, L6C1 reserve [6, 9]; remove these numbers from L2C1, L3C1.
  2. Nude subset in Unit 12: L7C3, L8C3 reserve [2, 6]; remove these numbers from L1C3, L3C3, L4C3, L5C3.
  3. Nude subset in Unit 13: L2C4, L3C4 reserve [8, 9]; remove these numbers from L1C4, L4C4, L5C4, L8C4, L9C4.
  4. Nude subset in unit 20: L2C4, L3C4 reserve [8, 9]; remove these numbers from L1C6, L2C5, L2C6, L3C5, L3C6.
  5. Nude subassembly in unit 22: L5C1, L6C1 reserve [6, 9]; remove these numbers from L4C3, L5C3.
  6. Units 4 and 24 overlap. In unit 4, 9 must lie in the overlap. Remove it from L5C7, L5C9, L6C7.
  7. Units 7 and 27 overlap. In unit 7, 9 must lie in the overlap. Remove it from L9C7, L9C8, L9C9.
  8. Units 23 and 13 overlap. In unit 23, 1 must lie in the overlap. Remove it from L1C4, L7C4, L8C4, L9C4.
  9. Units 23 and 13 overlap. In unit 23, 2 must lie in the overlap. Remove it from L1C4, L7C4, L8C4, L9C4.
  10. Units 14 and 23 overlap. In unit 14, 8 must lie in the overlap. Remove it from L5C6, L6C6.
L2C4 = 8

Assumption: L2C4 = 8

  1. L3C4: only candidate → 9
  2. Nude subset in Unit 10: L5C1, L6C1 reserve [6, 9]; remove these numbers from L2C1, L3C1.
  3. Nude subset in Unit 12: L7C3, L8C3 reserve [2, 6]; remove these numbers from L1C3, L3C3, L4C3, L5C3.
  4. Nude subassembly in unit 22: L5C1, L6C1 reserve [6, 9]; remove these numbers from L4C3, L5C3.
  5. Units 4 and 24 overlap. In unit 4, 9 must lie in the overlap. Remove it from L5C7, L5C9, L6C7.
  6. Units 7 and 27 overlap. In unit 7, 9 must lie in the overlap. Remove it from L9C7, L9C8, L9C9.
  7. Units 23 and 13 overlap. In unit 23, 1 must lie in the overlap. Remove it from L1C4, L7C4, L8C4, L9C4.
  8. Units 23 and 13 overlap. In unit 23, 2 must lie in the overlap. Remove it from L1C4, L7C4, L8C4, L9C4.
  9. Units 14 and 23 overlap. In unit 14, 8 must lie in the overlap. Remove it from L5C6, L6C6.
L5C1 = 6

Assumption: L5C1 = 6

  1. L6C1: only candidate → 9
  2. L5C6: only place for 9 in unit 15 → 9
  3. Nude sub-assembly in unit 12: L7C3, L8C3 reserve [2, 6]; remove these numbers from L1C3, L3C3.
  4. Units 7 and 27 overlap. In unit 7, 9 must lie in the overlap. Remove it from L9C7, L9C8, L9C9.
  5. Units 23 and 13 overlap. In unit 23, 1 must lie in the overlap. Remove it from L1C4, L7C4, L8C4, L9C4.
  6. Units 7 and 27 overlap. In unit 7, 1 must lie in the overlap. Remove it from L8C7, L8C8, L8C9, L9C7, L9C8, L9C9.
  7. Units 8 and 25 overlap. In unit 8, 1 must lie in the overlap. Remove it from L9C2, L9C3.
  8. Units 23 and 13 overlap. In unit 23, 2 must lie in the overlap. Remove it from L1C4, L7C4, L8C4, L9C4.
  9. Units 14 and 23 overlap. In unit 14, 8 must lie in the overlap. Remove it from L6C6.
  10. Units 26 and 9 overlap. In unit 26, 2 must lie in the overlap. Remove it from L9C7, L9C8, L9C9.
  11. L8C6: only place for 8 in unit 15 → 8
  12. L7C6: only candidate → 4
  13. L9C5: only candidate → 2
  14. L9C6: only place for 1 in unit 9 → 1
  15. L7C5: only place for 5 in unit 26 → 5
  16. L4C5: only candidate → 8
  17. L5C5: only candidate → 4
  18. L5C3: only place for 8 in unit 12 → 8
  19. L1C4: only place for 4 in unit 13 → 4
  20. L3C6: only place for 5 in unit 15 → 5
  21. L2C2: only candidate → 9
  22. L1C2: only candidate → 6
  23. L1C5: only candidate → 7
  24. L1C6: only candidate → 2
  25. L2C6: only candidate → 6
  26. L1C9: only candidate → 9

contradiction: a box no longer has no candidate

L5C1 = 9

Assumption: L5C1 = 9

  1. L6C1: only candidate → 6
  2. L6C6: only place for 9 in unit 15 → 9
  3. Nude sub-assembly in unit 12: L7C3, L8C3 reserve [2, 6]; remove these numbers from L1C3, L3C3.
  4. Nude sub-assembly in Unit 23: L4C5, L5C5, L5C6 reserve [6, 7, 8]; remove these numbers from L4C4, L5C4, L6C4, L6C5.
  5. L6C3: only candidate → 3
  6. L6C4: only candidate → 5
  7. L6C5: only candidate → 4
  8. L4C8: only candidate → 2
  9. L4C4: only candidate → 1
  10. L4C2: only candidate → 7
  11. L5C2: only candidate → 1
  12. L5C4: only candidate → 2
  13. L7C5: only candidate → 5
  14. L7C6: only candidate → 4
  15. L8C6: only candidate → 8
  16. L9C5: only candidate → 2
  17. L9C3: only place for 9 in unit 12 → 9
  18. L8C1: only candidate → 1
  19. L1C4: only place for 4 in unit 13 → 4
  20. L1C3: only candidate → 1
  21. L3C3: only candidate → 4
  22. L4C9: only place for 9 in unit 4 → 9
  23. L5C8: only place for 4 in unit 5 → 4
  24. L5C9: only place for 3 in unit 5 → 3
  25. L3C9: only candidate → 5
  26. L2C2: only candidate → 9
  27. L1C2: only candidate → 6
  28. L1C5: only candidate → 7

contradiction: a box no longer has no candidate

all the possibilities of this bifurcation have been examined

L2C4 = 9

Assumption: L2C4 = 9

  1. L3C4: only candidate → 8
  2. Nude subset in Unit 10: L5C1, L6C1 reserve [6, 9]; remove these numbers from L2C1, L3C1.
  3. Nude subset in Unit 12: L7C3, L8C3 reserve [2, 6]; remove these numbers from L1C3, L3C3, L4C3, L5C3.
  4. Nude subassembly in unit 22: L5C1, L6C1 reserve [6, 9]; remove these numbers from L4C3, L5C3.
  5. Units 4 and 24 overlap. In unit 4, 9 must lie in the overlap. Remove it from L5C7, L5C9, L6C7.
  6. Units 7 and 27 overlap. In unit 7, 9 must lie in the overlap. Remove it from L9C7, L9C8, L9C9.
  7. Units 23 and 13 overlap. In unit 23, 1 must lie in the overlap. Remove it from L1C4, L7C4, L8C4, L9C4.
  8. Units 23 and 13 overlap. In unit 23, 2 must lie in the overlap. Remove it from L1C4, L7C4, L8C4, L9C4.
  9. Units 14 and 23 overlap. In unit 14, 8 must lie in the overlap. Remove it from L5C6, L6C6.
L5C1 = 6

Assumption: L5C1 = 6

  1. L6C1: only candidate → 9
  2. L5C6: only place for 9 in unit 15 → 9
  3. Nude sub-assembly in unit 12: L7C3, L8C3 reserve [2, 6]; remove these numbers from L1C3, L3C3.
  4. Units 7 and 27 overlap. In unit 7, 9 must lie in the overlap. Remove it from L9C7, L9C8, L9C9.
  5. Units 23 and 13 overlap. In unit 23, 1 must lie in the overlap. Remove it from L1C4, L7C4, L8C4, L9C4.
  6. Units 7 and 27 overlap. In unit 7, 1 must lie in the overlap. Remove it from L8C7, L8C8, L8C9, L9C7, L9C8, L9C9.
  7. Units 8 and 25 overlap. In unit 8, 1 must lie in the overlap. Remove it from L9C2, L9C3.
  8. Units 23 and 13 overlap. In unit 23, 2 must lie in the overlap. Remove it from L1C4, L7C4, L8C4, L9C4.
  9. Units 14 and 23 overlap. In unit 14, 8 must lie in the overlap. Remove it from L6C6.
  10. Units 26 and 9 overlap. In unit 26, 2 must lie in the overlap. Remove it from L9C7, L9C8, L9C9.
  11. L8C6: only place for 8 in unit 15 → 8
  12. L7C6: only candidate → 4
  13. L9C5: only candidate → 2
  14. L9C6: only place for 1 in unit 9 → 1
  15. L7C5: only place for 5 in unit 26 → 5
  16. L4C5: only candidate → 8
  17. L5C5: only candidate → 4
  18. L5C3: only place for 8 in unit 12 → 8
  19. L1C4: only place for 4 in unit 13 → 4
  20. L3C6: only place for 5 in unit 15 → 5
  21. L4C9: only place for 9 in unit 4 → 9
  22. L6C6: only place for 7 in unit 15 → 7
  23. L6C3: only candidate → 3
  24. L6C4: only candidate → 5
  25. L4C3: only candidate → 5
  26. L6C5: only candidate → 6
  27. L5C7: only candidate → 5
  28. L6C7: only candidate → 8
  29. L9C7: only candidate → 6
  30. L2C6: only candidate → 6
  31. L1C5: only candidate → 7
  32. L1C6: only candidate → 2
  33. L9C4: only candidate → 7
  34. L9C8: only candidate → 8
  35. L8C9: only candidate → 7
  36. L3C9: only candidate → 6
  37. L2C9: only candidate → 8
  38. L5C9: only candidate → 2
  39. L4C4: only candidate → 2
  40. L4C8: only candidate → 6
  41. L5C4: only candidate → 1
  42. L4C2: only candidate → 1
  43. L1C3: only candidate → 1
  44. L3C1: only candidate → 2
  45. L1C2: only candidate → 6
  46. L2C1: only candidate → 5
  47. L4C7: only candidate → 7
  48. L5C2: only candidate → 7
  49. L5C8: only candidate → 3
  50. L1C7: only candidate → 3
  51. L1C8: only candidate → 9
  52. L1C9: only candidate → 5
  53. L2C8: only candidate → 2
  54. L2C5: only candidate → 1
  55. L2C7: only candidate → 4
  56. L2C2: only candidate → 3
  57. L3C5: only candidate → 3
  58. L3C7: only candidate → 1
  59. L3C8: only candidate → 7
  60. L7C7: only candidate → 9
  61. L7C8: only candidate → 1
  62. L6C8: only candidate → 4
  63. L6C9: only candidate → 1
  64. L7C9: only candidate → 3
  65. L7C4: only candidate → 6
  66. L7C3: only candidate → 2
  67. L8C1: only candidate → 1
  68. L8C3: only candidate → 6
  69. L8C4: only candidate → 3
  70. L8C7: only candidate → 2
  71. L8C8: only candidate → 5
  72. L8C2: only candidate → 4
  73. L3C2: only candidate → 9
  74. L3C3: only candidate → 4
  75. L9C2: only candidate → 5
  76. L9C3: only candidate → 9
  77. L9C9: only candidate → 4

valid complete solution

L5C1 = 9

Assumption: L5C1 = 9

  1. L6C1: only candidate → 6
  2. L6C6: only place for 9 in unit 15 → 9
  3. Nude sub-assembly in unit 12: L7C3, L8C3 reserve [2, 6]; remove these numbers from L1C3, L3C3.
  4. Nude sub-assembly in Unit 23: L4C5, L5C5, L5C6 reserve [6, 7, 8]; remove these numbers from L4C4, L5C4, L6C4, L6C5.
  5. L6C3: only candidate → 3
  6. L6C4: only candidate → 5
  7. L6C5: only candidate → 4
  8. L4C8: only candidate → 2
  9. L4C4: only candidate → 1
  10. L4C2: only candidate → 7
  11. L5C2: only candidate → 1
  12. L5C4: only candidate → 2
  13. L7C5: only candidate → 5
  14. L7C6: only candidate → 4
  15. L8C6: only candidate → 8
  16. L9C5: only candidate → 2
  17. L1C4: only place for 4 in unit 13 → 4
  18. L3C6: only place for 5 in unit 15 → 5
  19. L4C9: only place for 9 in unit 4 → 9
  20. L3C8: only place for 6 in unit 3 → 6
  21. L1C3: only place for 1 in unit 1 → 1
  22. L3C1: only candidate → 2
  23. L1C2: only candidate → 6
  24. L1C5: only candidate → 7
  25. L1C6: only candidate → 2
  26. L2C1: only candidate → 5
  27. L1C9: only candidate → 5
  28. L3C5: only candidate → 1

contradiction: a box no longer has no candidate

all the possibilities of this bifurcation have been examined

all the possibilities of this bifurcation have been examined

L2C3 = 9

Assumption: L2C3 = 9

  1. L9C2: only candidate → 9
  2. L1C4: only place for 9 in unit 13 → 9
  3. L1C5: only candidate → 3
  4. L1C6: only candidate → 1
  5. Nude subassembly in Unit 1: L1C7, L1C8 reserve [5, 7]; remove these numbers from L1C2, L1C3, L1C9.
  6. Nude subset in Unit 10: L5C1, L6C1 reserve [6, 9]; remove these numbers from L2C1, L3C1.
  7. L2C1: only candidate → 2
  8. L2C7: only candidate → 1
  9. L3C7: only candidate → 3
  10. L1C9: only place for 2 in unit 1 → 2
  11. L2C2: only place for 3 in unit 2 → 3
  12. L3C1: only candidate → 5
  13. L8C1: only candidate → 1

contradiction: the number 1 can no longer appear in unit 3

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": 9,
  "branches": 4,
  "contradictions": 4,
  "depth": 3,
  "pairs": 22,
  "locked": 34
}

After unique candidates, single places, bare pairs/triples and interactions between units, the solver still reaches bifurcations. The exhaustive proof reaches 3 intertwined hypotheses, examines 9 states and closes 4 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.

← Return to puzzle SOLO-009