# Circle Puzzle Geometry

Two friends are playing a game. One friend stands in the middle of a circle radius 100m. His objective is to leave the circle. He may take one step at a time, distance 1m, in any direction. However, his friend may tell him to take that same step in the opposite direction instead. How does he leave the circle?

This is more of a maths puzzle than a classical maths question, but it stumped me nevertheless. I'm really curious to know the answer, after a friend posed this question to me.

• If you always take a step perpendicular to the line between you and the center of the circle, then you will get out in 10,000 steps.
– user856
Jun 14 '16 at 22:14
• 10,000 seems a little high, did u mean that in the casual sense of "it will take a while" or is that an exact upper bound, how did you derive it in the latter case? Jun 14 '16 at 22:16

A step of one meter perpendicular to the radius will increase the first friend's distance from the center, from $r$ to $\sqrt{r^2+1}$; and whether the step is clockwise or counterclockwise makes no difference. That is, it increases the square of the distance from the center by one square meter, and the second friend's instruction is irrelevant. It's easy to check that this strategy leads to an escape in exactly $10000$ steps.