SOLO-014 · Advanced Mastermind

The Chamber of Duplicates

Find the only orderly code of 6 digits.

This page contains the solution.

The answer

Code: 140633.

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 6.
3. Proposal 474563 requires min(n3, 1) + min(n4, 2) + min(n5, 1) + min(n6, 1) + min(n7, 1) = 3.
4. Proposal 013237 requires min(n0, 1) + min(n1, 1) + min(n2, 1) + min(n3, 2) + min(n7, 1) = 4.
5. Proposal 163003 requires min(n0, 2) + min(n1, 1) + min(n3, 2) + min(n6, 1) = 5.
6. Proposal 635050 requires min(n0, 2) + min(n3, 1) + min(n5, 2) + min(n6, 1) = 3.
7. Proposal 042546 requires min(n0, 1) + min(n2, 1) + min(n4, 2) + min(n5, 1) + min(n6, 1) = 3.
8. These limitations of occurrences alone leave 1 multiset(s).Here they are, written in ascending order: 013346.
9. Multi ensemble 013346 : after successive application of the numbers of property placed, there remains 144 → 64 → 19 → 5 → 1 arrangement(s).
10. 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.
11. Starts with the 262.144 ordered words of length 6 out of 0 to 7. The initial zeros or words with repetition are not removed.
12. Index 1, proposal 474563: retains exactly the configurations with 1 position equal(s) and 3 common occurrence(s). There are 42,344 candidates left.
13. Index 2, proposal 013237: retains exactly the configurations with 1 position equal(s) and 4 common occurrence(s). There are 3,996 candidates left.
14. Index 3, proposal 163003: retains exactly the configurations with 2 positions and 5 common occurrence(s). 58 candidates remain.
15. Index 4, proposal 635050: retains exactly the configurations with 0 position equal(s) and 3 common occurrence(s). There are 5 candidates left.
16. Index 5, proposal 042546: retains exactly the configurations with 1 position equal(s) and 3 common occurrence(s). There are 1 candidate(s) left.
17. In index 1, the candidate 000000 would give 0 well placed and 0 badly placed, instead of 1 and 2: it is eliminated.
18. In index 2, candidate 000364 would give 1 well placed and 1 ill placed, instead of 1 and 3: it is eliminated.
19. In Index 3, candidate 001473 would give 1 well placed and 3 badly placed, instead of 2 and 3: it is eliminated.
20. 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.
21. Final check against 474563: 1 well placed, 2 badly placed.
22. Final check against 013237: 1 well placed, 3 badly placed.
23. Final check against 163003: 2 well placed, 3 badly placed.
24. Final check against 635050: 0 well placed, 3 badly placed.
25. Final check against 042546: 1 well placed, 2 badly placed.
26. 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": "013346",
      "remaining_after_position_constraints": [
        144,
        64,
        19,
        5,
        1
      ]
    }
  ]
}

{
  "candidates": 262144,
  "solutions": 1,
  "clues": 5,
  "survivors_after_clues": [
    42344,
    3996,
    58,
    5,
    1
  ]
}

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