white@brahms (Samuel P. White) (10/29/86)
I am looking for problems with elementary statements (understandable
by average college sophmores in mathematics) but whose solutions are very
deep and only understandable by a few knowledgeable mathematicians.
I will give a few examples.
One is Falting's theorem. One of the consequences of this theorem
is that the equation x^n + y^n = z^n has at most a finite number of
solutions in the integers for n>3(a first step in proving Fermat's
Last Theorem).
Another is Determinancy. Play the following game.
Let S be a subset of [0,1]. Let player X and Player Y take turns
bisecting and choosing halves of subintervals of [0,1] as follows.
To start let player X choose either [0,.5] or [.5,1].
If the other player chose [x,y], the current player (whose turn it is)
can chose either [x,(x+y)/2] or [(x+y)/2,y]. This will define
a sequence of closed subintervals. Does player X have a strategy to
guarantee that the intersection of all these subintervals is in S?
The answer is that it depends on S and which axioms you choose
for your Set theory.
And of course, there is the Four-Color Map Theorem. I do not know
if the existence of fake R^4s can be phrased in a suitable manner.
My last example is a corollary of the classification of finite simple groups.
One of the consequences is the following. Let S be a set of permutations
(e.g. one-to-one and onto functions) of a finite set X which is closed under
composition and inversion. We define n-transitivity as
follows: Let a_1,...,a_n be a set of distinct elements from X.
Let b_1,...,b_n be another set of distinct elements from X.
S is n-transitive if for every choice of a_1,...,a_n and
b_1,...,b_n there exists an element f of S such that f(a_i)=b_i.
Assume S is n-transitive but not n+1-transitive and n is greater or
equal to six. Then S is either the set of all permutations on
a set of n elements (e.g. S sub n) or the set of all
even permutations on the set of n+1 elements(e.g. A sub n+1).
I am looking for more examples like these. The model example would
be a problem easily (e.g. not very technical) understandable by
high school students but requiring such a deep understanding of
mathematics to solve that only a few had ever seen the solution. Note:
I am less inspired by examples whose answers are hard to understand
(as in example 2) or whose proofs are better known for their
length and not their depth (as in example 3).
If I get a sufficient number of examples I will post the results
to this newsgroup and will give my opinion on their relative merits.
My hope is to get a ready supply of easy to state problems
to baffle and impress the man-on-the-street mathematician.
ucbvax!brahms!white Samuel P. White/UCB Math Dept/Berkeley CA 94720