Virtually the very first “math” problem I got interested in involved a 7th grade homework problem in 2005 that a colleague at work said her son had been given. I ended up commenting and helping on a number of further problems, which gave me some insight into the state of current public school teaching in mathematics. It was both encouraging and discouraging at the same time. I will join the math education commentary at a later date.
The problem was not that bad: What is the largest power of 2 that divides 800! without a remainder? (where “!” means “factorial”, for example, 5! = 5 x 4 x 3 x 2 x 1). I solved it in my usual pedestrian way. I showed it to a friend of mine (an algebraist!) and he of course had a nifty approach. He showed it to a colleague of his at NSF (a physicist) and he had the niftiest solution of all! (Most humbling.) See the Power of 2 Problem.
In 2011 I wrote about an amusing article in the 9 September 1978 Washington Post in which the reporter, Henry Allen, began thinking about the number of ancestors he had 10 generations ago. He figured that each generation had 2 parents, so the 10th generation would have 2^10 or over a thousand members. At 25 years per generation he started imagining the expanding number of direct ancestors he had going back through history, achieving astronomical numbers, which did not seem reasonable. As reporters do, he started interviewing people to get to the bottom of the problem. See the 
As my opening post, I begin with something I noticed back in 2011 when we purchased a new microwave that had a revolving carousel to distribute the heating. The glass plate was turning faster than the plastic support ring under it. I wondered if the speed depended on the size of the support ring and/or the size of its roller wheels. 