Please tell us something surprising or amusing that one of you has discovered. (The answer need not be related to your project.)
In grad school I proved the following:
A polynomial that is symmetric in the roots of two other polynomials is a polynomial in the coefficients of the other two polynomials.
As an example it is easy to construct a polynomial with integer coefficients that has sqrt(2) + sqrt(3) as a root. Just take the polynomial (x-sqrt(2)-sqrt(3))(x-sqrt(2)+sqrt(3))(x+sqrt(2)-sqrt(3))(x+sqrt(2)+sqrt(3)), multiply it out, and you'll get your answer. (In this case, x^4-2x+1.)
Kind of a cute result. So I took it to multiple mathematicians. None had heard of it until I talked to an old mathematician who told me that it sounded like a very old approach, and he encouraged me to look for an introductory algebra book from the 1800s.
I did, and lo and behold! It was once well-known, and a standard part of the undergraduate curriculum. Even better, two of the mathematicians who had not known the result worked in areas of math that STARTED with that observation! (One worked with the algebraic integers - that construction was how mathematicians first proved they formed a ring, and the other worked with symmetric polynomials - and that construction was the original reason why they were studied.)
This incident opened my eyes to the truth of how easily knowledge gets lost, and how little attention mathematicians pay to their own history.
Which has me puzzled. I can't think of anything offhand that I "discovered". I suppose you don't need to have been the first to discover it; it merely needs to be relatively unknown to most people? If that's the case, then nevermind: I'm full of obscure knowledge.
I've actually "invented" an impressive device, but I'm not sure if "something you invented" is what they're looking for.
Of course, I'm probably reading into this a little too much.
But it does help identify $80 nickels.
Now, that's a pretty big "if", but this isn't meant to be serious mathematics. It's just an interesting consequence of binary representation and the assumption that Pi might terminate, and is enough to make people do a bit of a double take when you phrase it along the lines of "the last bit of Pi [if it has one] is a 1".
... which Pi doesn't have, since it has no rightmost bits at all...
the last bit of Pi [if it has one] is a 1
This is true; it's also true that the last bit of Pi [if it has one] is a forty-two. A falsehood implies everything.
Or is this more like trying to answer "what is your greatest weakness" (terrible question!) with "I tend to work too hard"? :)