Here is a puzzle that no math person without knowing my research should be able to solve without using brute force:
What is next in the series?
1) 12 + 22 = 5
2) 32 + 42 = 25 = 52
3) 112 + 22 = 125 = 53
4) 72 + 242 = 625 = 54
5) 412 + 382 = 3125 = 55
6) 1172 + 442 = 15625 = 56
7) (?)2 + (?)2 = 78125 = 57
Of course just pulling from a prior post where I left off the last answer I derived, so the answer IS posted! So it's silly to solve the problem by brute force which is also cheating. Brute force involves programming a computer to just try different squares until you find some that fit. That only requires knowing how to code, and involves no mathematical intuition at all.
But the bigger question in my mind is: if some highly intelligent person is given that list can she or he figure out a pattern?
These type of problems are actually given in IQ tests I've seen or as mathematical challenge questions in other areas.
This particular one involves new math, so should stump any math expert in the world--unless they know of my research that is!
Or maybe not. Part of me wonders if some very high IQ person can figure it out. Or some person with exceptional math intuition. But I doubt it.
This particular result escaped Archimedes, Euler, Ramanujan and Gauss as well as every other major mathematical mind in human history, though before Archimedes I don't think it's fair to include anyone. So, let's say, every mathematical mind in the last two thousand years missed it. I doubt any human alive can just look at that series and figure it out. I couldn't have done it that way myself!
Putting it forward in this way can give a sense of what kind of an accomplishment it is, but of course news travels fast! So if you want to stump someone in your life you better hurry before this "think" is well-known.
Update January 5, 2015: One person answered the challenge on my Facebook page. So it can be done.
Update June 7, 2017: Removed link as I decided to delete my Some Math Facebook page some time ago.
This post generated a HUGE amount of interest in comparison to prior post and I consider it to be a great success. It has me thinking more about trying to find challenges from mathematical results as a better way to promote them.