(Puzzle by Raymond Smullyan
There is a safe containing millions of dollars Ė 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.
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.
(In reply to I'll tackle this soon
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>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.
substitute x for 9515
<o:p></o:p>It seems Iíve 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 Iím not sure that itís the
shortest that we can know opens the lock for certain.
Well, this certainly was an exciting puzzle!<o:p></o:p>
Posted by Tristan
on 2005-06-14 16:17:10