I spent part of the afternoon working out the properties of a cone without using references other than remembering the requisite formulas regarding circles and right triangles. So, if I ever need to know what shape to make a band to wrap around a conical surface, I’m set. (Yes, I do stuff like this for fun. #dontjudgeme)
Sadly, it’s easy to out-smart yourself. Not long out of high school I wondered how long it took for the hour and minute hands to overlap on a twelve-hour analog clock (assuming it ran at a rate of 1 second per second). I whipped out the calculus and other sundry mathy tools. Angular change and limits and…well, it was a pain and the answers never looked right–I blame arithmetic errors on my part.
Then I actually stopped and thought about the problem. From midnight to noon the minute hand goes around 12 times. In that time it overlaps the hour hand 11 times. Thus, the minute hand overlaps the hour hand every 12/11 hour (or every 65.454545… minutes; or every 1hr 5min 27.272727… sec). So easy and obvious.
The lesson here was that it was important to truly understand the problem–and ask the right question–before trying to attempt a solution. Otherwise you’re probably making a whole heck of a lot more work for yourself.