It is not often that we come across a problem whose solution can fit in the margins of a notebook. In fact, many of the problems I have worked on in the field of quantum many-body physics require proofs that often exceed 30 pages. And that is without taking into account the several references included as sources for results used as “elementary” tools in the proof (referees love these papers…) So it is natural to think that a proof is a proof, so long as it is correct, and once confirmed by the academic community it is time to move on to something new.
But… sometimes things get interesting. A 30-page proof collapses to a 3-page proof when a different point of view is adopted (see the famous Prime Number Theorem). Below, you will find two problems that may, or may not have a “one-line” proof. The challenge is for you to find the shortest, most elegant proof for each problem:
A unit triangle: A triangle with sides $latex ale b le c$ has area 1. Show that $latex b^2 ge 2$.
Fermat’s Lost Theorem: Show that $latex (x+y)^n = x^m+y^m$ has only one solution with positive integers $latex x > y > 0$ and $latex m,n > 1$.
In case you figure out the above problems (who knows, you guys are pretty good), you can try your luck with this one:
Magic Square Cubed: Show that there is no $latex 3times3$ array of distinct cubed numbers (e.g. $latex 8,27,125,ldots$) such that the sums of numbers in each row, each column and each of the two diagonals are all equal. That is, show that there is no magic square of cubes.
And if you are an alien supergenius, you can go for all the money:
Magic Square Superpower Challenge: Show that there is no $latex 3times3$ array of n-th powers of distinct numbers with $latex nge 4$, such that the sums of numbers in each row, each column and each of the two diagonals are all equal. That is, show that there is no magic square of n-th powers, for $latex n ge 4$.
The most elegant solution to the Superpower Challenge will receive a $100 prize. And in case you can show that there is no magic square of distinct squares… Nah, that’s impossible.
P.S.: To claim the prize, you and I should be able to understand your solution.