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A locked safe (Posted on 2002-07-16) Difficulty: 5 of 5
(Puzzle by Raymond Smullyan)

There is a safe containing millions of dollars – unfortunately the combination is written on only one card, and that card has been accidentally locked inside the safe! If the wrong combination is used, the lock will jam and the only way to open the safe would be to blow it up, destroying the contents.

A combination is a string of digits from 0 through 9. It can be any length and contain any number of digits occurring any number of times; 90915 is a combination; so is 2133127; so is 5. Certain combinations will open the lock, certain combinations will jam the lock, and the remaining combinations will have no effect whatever (these last are called neutral).

The small letters x and y will represent arbitrary combinations, and by xy is meant the combination x followed by the combination y; for example, if x is 213 and y is 3812, then xy is 2133812. By the reverse of a combination is meant the combination written backwards; for example, the reverse of 3812 is 2183. By the repeat xx of a combination x is meant the combination followed by itself; for example, the repeat of 3182 is 31823182.

Now, some of the combinations are related to other combinations. There are five properties of this relation:

  • Property A: For any combination x, the combination 2x2 is related to x. (For example, 21452 is related to 145.)
  • Property B: If x is related to y, then 1x is related to 2y. (For example, since 21452 is related to 145, then 121452 is related to 2145.)
  • Property C: If x is related to y, then 5x is related to the reverse of y. (For example, since 21452 is related to 145, then 521452 is related to 541.)
  • Property D: If x is related to y, then 9x is related to yy (the repeat of y).(For example, since 21452 is related to 145, then 921452 is related to 145145. Also, 521452 is related to 541, so 9521452 is related to 541541.)
  • Property E: If x is related to y, then if x is neutral then y jams the lock, and if x jams the lock then y is neutral. (For example, if 521452 is neutral, then 541 will jam the lock.)

    Find the shortest possible combination that will open the lock.

    Notes/Clues:
    a) The relation is only one way. Think of it like mother and son. The mother is the parent of the son, but the son is not the parent of the mother.
    b) The first thing you need to do is to establish (just using property E) how to solve the puzzle (i.e. how do you know if a combination opens the lock?). Then use this information to solve the puzzle using properties A thru D.

  • See The Solution Submitted by levik    
    Rating: 4.1935 (31 votes)

    Comments: ( Back to comment list | You must be logged in to post comments.)
    Solution The Real Solution (Part I) | Comment 34 of 43 |
    I had 'proved' an 8-digit solution before, but my oversight was that the reverse of x is not necessarily x. So, I will redo the problem, with the invented notation of X being the reverse of x (if x=123, X=321). As before, I will attempt to prove that a number relates to itself, which by property E means that it must be a solution and open the lock. The shortest solution still presumably has the form of xx2 or x2x2, where x is a series of 1s, 5s, and 9s that enumerates the steps needed to find that very arrangement.
    First, for xx2, lets work backwards:
    xx2 - 2XX - XX - xx - x
    or
    xx2 - 2XX - XX - X - x.
    A consecutive reversal (5) and repetition (9) are interchangeable. This yields the following proof:
    2x2 ~ x
    92x2 ~ xx
    592x2 ~ XX
    1592x2 ~ 2XX
    51592x2 ~ xx2
    :. x must equal 51592, and this shows that 51592515922 relates to itself and therefore is a combination that opens the safe. Also (using the 5-9 commutability), 51952519522 opens the safe by similar proof.
      Posted by DJ on 2003-04-22 16:13:48
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