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:

As far as i can tell, there is no number smaller than 4 digits which encodes a self relationship for less than 10 digits.
you can probably prove this exhaustively; i could find none
with 3. ie the 10 digit case is the smallest self relation.

you could conjecture that the relation 8-10 is the only
possible shortening, since 9 is the only reducer provided
there is a form xx in the target. since the 8 digit "solutions"
do not exhibit form xx, they cannot be reduced.

im not trying to prove it rigorously here, but it seems
like 8 is the minimum..

Posted by Cheradenine on 2002-07-23 23:22:47