WARNING: if you have something better to do with your life, now would be a good time! The following post is one of those things I notice that would make everybody groan. If you're really curious, read on... but don't say I didn't warn you.
Today, as I rolled up the floor mat after table tennis, I found myself making odd mental calculations when I felt it getting harder to pull as I was pulling more of it – a perfectly natural thing.That is how the mat is rolled. It is essentially folded, but one end must be picked up and walked to the other end. This is done 3 or 4 times until it is not in the way anymore. As I was walking with the mat, I (naturally) noticed it getting heavier because I was pulling more of it.
When I went back to grab the now-doubled mat to fold it again, I realized that while it would start at zero again, it would increment at double the rate. Because the mat was at that time half the length of before, I made a mental calculation and realized that when the puller reaches the very end (and is about to drop) the force needed to pull (discounting for friction) is the same, no matter how many layers there are. I guess what I actually mean is that the amount of material being pulled is the same.
Not a particularly useful observation, and certainly not the simplest way to come to the same conclusion, but there's a bit of insight into my thought processes.
Also, speaking of folding, foldit!