One time when I was a kid I came across something about the Buffon needle problem - if you drop a needle randomly between two lines, the probability that it will cross one of the lines has a remarkably simple expression: [2*(length of the needle)]/[pi * distance between the lines]. This fact can be used to obtain an approximation for pi: just drop a lot of needles on a tiled floor, count how many intersect lines, and set the above expression equal to that; solve for pi. Without much ado, I went to the kitchen, emptied the contents of a few boxes of toothpics on the floor, crouched on my knees, and started counting.
Of course, there were a few hundred toothpics lying around and my parents came home long before I was finished. They were puzzled: why was I on the kitchen floor counting toothpics? I didn't want to give away my secret and said I had knocked them over accidentally. But why was I counting them? I shrugged. They looked at me, puzzled. Eventually I had to admit I was trying to approximate pi, much to my embarassement.
According to Wikipedia, the experiment was performed in 1901 by the Italian mathematician Mario Lazzarini, who obtained a value within 0.000001 of pi by tossing 3408 needles, though there is some debate over whether these results are genuine.