perplexus.info :: Logic : A locked safe

A locked safe (Posted on 2002-07-16)

(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.)

re(5): more hints

| Comment 18 of 43 |

(In reply to re(4): more hints by Cheradenine)

I actually looked at the website where this problem was originally found by levik, and their final solution is the same as the one we've created here. Personally, though, I have a problem with the certainty of the reasoning behind it. After all, all that has been done here is the smallest self-related number (10-digit) has been found, and then this has been reduced further to give an 8-digit solution. From what I could see no-one was able to show for certain that starting at some larger self-related number would never reduce better than to 8 digits - e.g. 'some' 12-digit self-related number ends up reducing down to 6 digits, say. It was just 'assumed'. Maybe it's correct - I don't know - I can't really think how to definitely prove it one way or the other, but the final stated solution seems a little unsatisfactory in my opinion...

Posted by Nick Reed on 2002-07-23 23:35:42

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