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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.
regarding a greater than 10 self-relator reduced to less than 8:
any combination related to another combination is of the form
t2x2 where t={9,5,1}
from this it follows that self relators are
also of the form t2x2.
a self relator can be transformed via finding
another combination which relates to it.
in this way, we can transform a self relator
t2x2 : t'2x2
since the comb. tranformed to is also a comb.
that relates to another one. so, reducing
a greater than 10 combination is the process of
t2x2 (self relator) : t'2x2 : t''2x2 ..
until some t'''..2x2 is less than 8 digits.
the only tranformation which reduces is for
t2x2 = aa, which gives
t2x2 : 92t22 (t2 = x2 = a). because initially
we start at greater than 10 digits, at least two
such reductions must occur. so the problem is
redefined in terms of finding
92t22 : t'2x2 : t''2x2 ...
however, 92t22 cannot be further reduced since
it cannot be of the form xx. moreover, any
transformation 92t22 : t'2x2 will itself
not yield a comb. reducible via 9(). this is
because 92t22 contains three 2's, making it
impossible for it to be of form xx. transformations of 92t22 cannot eliminate this
problem..
basically im saying a greater that a self relating comb cannot be reduced (9) twice.
this would prove there is no valid less than 8.
what do you think?
forward..
