SOLO-011 · Advanced Mastermind

The Eight Inks

Find the only ordered code of 5 digits.

This page contains the solution.

The answer

Code: 70730.

The reasoning

1. Separates two constraints for each line: the number of equal positions per position, and then the total of common occurrences. This total is well placed + wrong placed; for each number, the minimum of the two multiplicities is taken.
2. To cut down the work, ignore the positions for now. Let n₀,…,n₇ be the counts of the eight digits. Their sum is 5.
3. Proposal 02553 requires min(n0, 1) + min(n2, 1) + min(n3, 1) + min(n5, 2) = 2.
4. Proposal 62532 requires min(n2, 2) + min(n3, 1) + min(n5, 1) + min(n6, 1) = 1.
5. Proposal 05062 requires min(n0, 2) + min(n2, 1) + min(n5, 1) + min(n6, 1) = 2.
6. Proposal 73375 requires min(n3, 2) + min(n5, 1) + min(n7, 2) = 3.
7. These limitations of occurrences alone leave 5 multi-unit(s).Here they are, written in ascending order: 00337; 00377; 01577; 04577; 05777.
8. Multiassembly 00337 : after successive application of the numbers of property placed, there remains 12 → 5 → 2 → 0 arrangement(s).
9. Multiassembly 00377 : after successive application of the numbers of property placed, there remains 15 → 3 → 1 → 1 arrangement(s).
10. Multi ensemble 01577 : after successive application of the numbers of property placed, there is left 30 → 0 → 0 → 0 arrangement(s).
11. Multi ensemble 04577 : after successive application of the numbers of property placed, there remains 30 → 0 → 0 → 0 arrangement(s).
12. Multiassembly 05777 : after successive application of the numbers of property placed, there remains 10 → 0 → 0 → 0 arrangement(s).
13. So we can make two separate tables, occurrences and then places. Only one line in the second table keeps an arrangement; that's the code announced. This separation avoids blindly browsing all the ordered words.
14. Starts with the 32.768 words ordered length 5 out of 0 to 7 and do not remove the initial zeros or the words with repetition.
15. Index 1, proposal 02553: retains exactly the configurations with 0 position equal(s) and 2 common occurrence(s). There are 7.051 candidates left.
16. Index 2, proposal 62532: keeps exactly the configurations with 1 position equal(s) and 1 common occurrence(s). There are 319 candidates left.
17. Index 3, proposal 05062: retains exactly the configurations with 0 position equal(s) and 2 common occurrence(s). There are 16 candidates left.
18. Index 4, proposal 73375: retains exactly the configurations with 1 position equal(s) and 3 common occurrence(s). There is 1 candidate remaining.
19. In index 1, candidate 00000 would give 1 well placed and 0 badly placed, instead of 0 and 2: it is eliminated.
20. In Index 2, candidate 10005 would give 0 well placed and 1 ill placed, instead of 1 and 0: it is eliminated.
21. In index 3, candidate 10002 would give 2 well placed and 1 ill placed, instead of 0 and 2: it is eliminated.
22. This filtering can be done on paper in branches: first fixes the possible occurrences, then affects their places and cuts a branch as soon as a line exceeds its number of places or can no longer reach its common total. You must not choose a candidate only because it satisfies the last line.
23. Final check against 02553: 0 well placed, 2 wrong placed.
24. Final check against 62532: 1 well placed, 0 wrong placed.
25. Final check against 05062: 0 well placed, 2 wrong placed.
26. Final check against 73375: 1 well placed, 2 badly placed.
27. The exhaustive check takes all possible words and has exactly one solution. The numbers are therefore determined at all places; it is not just a compatible code found at random.

Common mistakes and false leads

• Consolidate wrongly placed and figure absent.
• Count twice an occurrence of the code when the proposal repeats a number.
• Prohibit repetitions or the initial zero without the rules requiring it.

{
  "multisets_and_position_counts": [
    {
      "bag": "00337",
      "remaining_after_position_constraints": [
        12,
        5,
        2,
        0
      ]
    },
    {
      "bag": "00377",
      "remaining_after_position_constraints": [
        15,
        3,
        1,
        1
      ]
    },
    {
      "bag": "01577",
      "remaining_after_position_constraints": [
        30,
        0,
        0,
        0
      ]
    },
    {
      "bag": "04577",
      "remaining_after_position_constraints": [
        30,
        0,
        0,
        0
      ]
    },
    {
      "bag": "05777",
      "remaining_after_position_constraints": [
        10,
        0,
        0,
        0
      ]
    }
  ]
}

{
  "candidates": 32768,
  "solutions": 1,
  "clues": 4,
  "survivors_after_clues": [
    7051,
    319,
    16,
    1
  ]
}

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