(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.

Given property E, we know that in a transitive chain of relation, if the first combination is not a "valid" (unlocking) combination, then none of them are. We also notice a parity in the sequence of related combinations. If the odd members are "jammers," the even are neutral, and vice versa.

So if a combination relates to itself, either directly, or in a transitive sequennce with an even number of intermediate combinations, it can't be either neutral or "jamming," so it must be "valid."

Unfortunately, this won't work directly since all the rules for generating new combinations are non-decreasing in terms of length.

However if the same starting combination can produce the same ending combination in two different sequences whose lengths are of opposite parity, then the starting and ending combinations are both "valid." (In fact if the ending combination is "valid" the intermediate combinations must also be "valid", since a non-"valid" combination cannot give rise to a valid one.

Posted by TomM on 2002-07-16 14:35:51