You've probably seen the type of puzzle which asks you to draw a certain pattern without lifting your pen from the page or going over any line twice; probably the most famous one is this:
 /  \
|\  /|
| \/ |
| /\ |
An interesting question is this: given a certain figure, how can one tell whether it's possible to draw it in one continuous stroke? The answer is quite simple; I think Euler may have been the mathematician to discover this, probably in relation to his study of the Bridges of Konisberg problem:

Look at all the vertices in the pattern. Count how many lines meet at each vertex, and note whether the number is even or odd. Count how many vertices have an odd number of lines going into them. If there are more than two of these "odd" vertices, then the figure cannot be drawn without lifting one's pen; otherwise, it can. For example, in the figure above, there are only two odd vertices (the bottom corners, which each have three lines going into them), so we know it's possible to draw in one stroke.

A couple more interesting facts: the number of odd vertices is always even (it can be zero, but zero's an even number as far as most mathematicians are concerned). If there are no odd vertices in the figure, then you may begin drawing at any point in the pattern. If there are two odd vertices, then you must begin drawing at one of them and end at the other (so for the picture above, every method of drawing it (and there are many) involves beginning at one of the bottom corners and ending at the other).