(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.

It's time for my random puzzle-solving!

I don't actually pick random puzzles to solve... but that's besides the point!

This one caught my interest and I plan to solve it in the near future.  I'd solve it now, but I don't have time.  I don't want to waste the time of anyone who bothered to read this, so I'll include the steps suggested by hint b) here.

It seems I must find a cycle of an odd number of related combinations.  For example x is related to y, y is related to z and x is related to z.  It doesn't matter whether y is related to z or vise versa.  Similarly, it would work with 5 combinations, or 7, or any other odd number.  Perhaps there is some other pattern of relations that forces the combinations to open the lock, but I have not yet thought of it.

It occurs to me that these combinations, though strings of digits, are put in all at once.  When I first read it, I thought maybe to input 1456 you would on the way have to input 1, 4, 14, 145, etc. on the way, but that's not the case.  If it were, then the solution would have to be a combination that didn't jam the lock before opening it... in other words, 1 number.  That seems unlikely.

That's all for now.  Hopefully I've brought this interesting problem to the attention of people who are interested in it.  Look for my solution later, perhaps tomorrow.

Posted by Tristan on 2005-06-14 01:46:50