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:

For the sake of completeness, this is what I said earlier:

"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."

When I have found such a cycle, not every combination included necessarily opens the lock.  For example, in the cycle x>y<z>x (x>y means x is related to y), z opens the lock, but x and y do not necessarily do that.  It seems that there are even some cycles where we know at least one opens the lock, but we don't know which.  For example, in the cycle a>b>c<d>e<a, we know that at least one of a and d opens the lock, but we don't know which.

Here is a list of the properties in short hand:
x<2x2
x>y => 1x>2y
x>y => 5x>-y (I'm notated the reverse of y as -y)
x>y => 9x>yy

What I'm going to do, is list a bunch of true statements in search of a cycle.  Afterwards, I'll delete all the irrelevant statements.

22x22>2x2>x
52x2>-x
152x2>2-x
5152x2>x2
95152x2>x2x2
substitute x for 9515

9515295152>9515295152

<o:p></o:p>It seems I’ve missed another way to find a opening combination.  If a combination is related to itself, it must open the lock.  9515295152 is the such a combination, though I’m not sure that it’s the shortest that we can know opens the lock for certain.

<o:p> </o:p>
Well, this certainly was an exciting puzzle!<o:p></o:p>