Two stories:

Budapest Semesters in Mathematics runs a four-hour, five-problem math contest for its participants, both for cash prizes and to select a team to send to the International Math Competition. The contest was two weeks ago; today I graded part of it. Although I didn’t grade it, I found out that a problem I submitted as a replacement for an originally proposed problem 1 that I’d covered in my problem solving class last semester, which, as a problem #1, was supposed to be the easiest on the test, was actually solved by no one. Oops.

The late great mathematician Pál Erdős was famous for, on meeting a potentially promising young mathematician, giving some short math problem to test their promise. These had to be solvable in just a few minutes (lest they delay the conversation too much) and without training (to test for promise, not accomplishment). This is actually a fairly common practice among mathematicians, apparently: when I met one of the first people to learn Set Mao’s father, a math professor, visiting Princeton two years ago, he gave me a little problem that’d come up in his research, which I solved in three or four minutes on a napkin. (I remembered the problem mostly because it was the first time in a while I’d actually done math on a napkin. Napkins are clearly the best writing surface to do math on.)

The unsolved BSM local contest problem and the quick napkin problem were actually the same one. So, are BSM students completely unable to solve such quick problems, or am I both smart enough to solve hard contest problems quickly on napkins and completely unable to judge when a problem’s actually hard? Actually, neither*; this is a case where the context in which the problem was given mattered a lot. It’s a true-or-false problem with a lot of conditions that seem useful if the answer is true but would be pretty arbitrary restrictions if the answer were false; such problems on math contests are usually true, and no one I talked with after the contest said they’d spent more than a minute or two assuming it was false. On the other hand, a quick problem given to test a new mathematician’s promise you’d expect might have some little trick like an answer of “false” when you’d expect “true,” so I started working on the right answer almost immediately. (Assuming the answer from the structure of the problem is a bad habit I’ll try to drill out of the students who come to my practices.)

The problem, for the mathy: Prove or disprove that if A is a square matrix with positive integer entries, and for all i \not = j, a_{i,i} > a_{i,j} and a_{i,i} > a_{j,i}, then det(A) \not = 0.

*Well, I am bad at judging whether a problem’s actually hard, but that’s a separate issue.