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In the 1960's, the grad students at the University of Chicago math
department worked on a series of astoundingly useless and
time-consuming puzzles. One of these follows. Consider the set of
all possible strings of the alphabet ('a' .. 'z'). Let us agree to
consider two strings "equivalent" if the following conditions hold:
1. They contain precisely the same letters, and
2. They both appear in Webster's Third International Dictionary.
In such a case, the two strings are considered interchangeable in all
contexts. For example, "am" and "ma" are equivalent, and this also
implies that "amount" and "maount" are equivalent, as are "grammar"
and "grmamar" and "gramamr" and "grmaamr".
Moreover, equal letters can be cancelled from the front and back end
of any string. For example, "abby" and "baby" are equivalent, and,
cancelling the trailing "by", this implies that "ab" and "ba" are also
equivalent, and can be exchanged anywhere. When two letters can be
exchanged in this way, we say that they "commute". The third floor of
the math building at UC had a huge 26x26 chart; the square in column i
and row j contained a proof that letters i and j would commute.
Sometimes these proofs can be rather elaborate. For example,
the dictionary has "dire" and "ride", so, by cancelling the trailing
"e"s, one has "dir" = "rid".
The dictionary also has
dirten = rident
(No, I don't know what those mean.) Since "dir" = "rid" we have:
rident = dirent
and since "rident" = "dirten",
dirten = dirent
even though "dirent" is not a dictionary word. Cancelling the leading
"dir" leaves:
ten = ent
but
ten = net
because "ten" and "net" are dictionary words, so
ent = net
and, cancelling the "t",
en = ne
and we've just proved that "en" and "ne" commute. This fact might be
useful in later proofs.
What's the point of all this? Well, the goal is to find out which
letters commute with *every* other letter; such letters are said to be
in the "center" of the system. As for the *point*, I'm not sure there
is one. Apparently the math grads at UC didn't have enough to occupy
their time.
The chart in the UC math building has since been lost, so your task is
to write a program whose input is a word list, with one word per line,
and which makes appropriate deductions and eventually computes the
center of the system. I don't have the headword list from Webster's
Third, but I do have the list from Webster's Second, so let's use
that. You can get a copy from
http://perl.plover.com/qotw/words/Web2.bz2
http://perl.plover.com/qotw/words/Web2.gz