SOLO-012 · Advanced Mastermind

The Silent Lock

Find the only ordered code of 5 digits.

This page contains the solution.

The answer

Code: 32752.

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 22543 requires min(n2, 2) + min(n3, 1) + min(n4, 1) + min(n5, 1) = 4.
4. Proposal 41050 requires min(n0, 2) + min(n1, 1) + min(n4, 1) + min(n5, 1) = 1.
5. Proposal 70302 requires min(n0, 2) + min(n2, 1) + min(n3, 1) + min(n7, 1) = 3.
6. Proposal 37433 requires min(n3, 3) + min(n4, 1) + min(n7, 1) = 2.
7. These limitations of occurrences alone leave 1 multi set(s).Here they are, written in ascending order: 22357.
8. Multi ensemble 22357 : after successive application of the numbers of property placed, there remains 26 → 6 → 2 → 1 arrangement(s).
9. 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.
10. 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.
11. Index 1, proposal 22543: keeps exactly the configurations with 1 position equal(s) and 4 common occurrence(s). There are 652 candidates left.
12. Index 2, proposal 41050: retains exactly the configurations with 1 position equal(s) and 1 common occurrence(s). There are 39 candidates left.
13. Index 3, proposal 70302: keeps exactly the configurations with 1 position equal(s) and 3 common occurrence(s). There are 4 candidates left.
14. Index 4, proposal 37433: retains exactly the configurations with 1 position equal(s) and 2 common occurrence(s). There are 1 candidate(s).
15. In Index 1, candidate 00000 would give 0 well placed and 0 badly placed, instead of 1 and 3: it is eliminated.
16. In Index 2, candidate 02234 would give 0 well placed and 2 badly placed, instead of 1 and 0: it is eliminated.
17. In index 3, candidate 23252 would give 1 well placed and 1 ill placed, instead of 1 and 2: it is eliminated.
18. 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.
19. Final check against 22543: 1 well placed, 3 badly placed.
20. Final check against 41050: 1 well placed, 0 wrong placed.
21. Final check against 70302 : 1 well placed, 2 wrong placed.
22. Final check against 37433 : 1 well placed, 1 wrong placed.
23. 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": "22357",
      "remaining_after_position_constraints": [
        26,
        6,
        2,
        1
      ]
    }
  ]
}

{
  "candidates": 32768,
  "solutions": 1,
  "clues": 4,
  "survivors_after_clues": [
    652,
    39,
    4,
    1
  ]
}

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