I just thought of this simple (at least I think it is simple) puzzle.
Consider a long loop of string and a number (at least 2) of round pegs of radius 1 inch each, parallel to each other. Drape the string around the pegs and pull the pegs so that the string is tight, as in the picture (which has only 3 pegs). Notice some sections of the string are straight and some are curved (shown in red in the picture).
Why is the total length of the curved sections equal to ?
This is related to a pulley puzzle of Harry Langman, published in 1949.