1. The problem statement, all variables and given/known data This problem is from the 2010 AMC 12 High school math competition: A frog makes 3 jumps, each exactly 1 meter long. The directions of the jumps are chosen independently at random. What is the probability that the frog's final position is no more than 1 meter from its starting position? 2. Relevant equations I drew circles representing all of the points which can be reached after two jumps. This is just two circles with one of the circle's centers located on the circumference of the other circle. I believe the probability is 1/4 after two jumps. S=theta*r ;theta=pi/2 for the portion of the second circle's circumference contained in the first circle. 3. The attempt at a solution After the first two jumps the probability of being less than a meter from the starting point is 1/4. The problem is the third jump which can occur from any point on the second circle, each point with a different probability. If you could find the probability as a function of some angle theta around the second circle you could integrate it..yada yada...but this seems like the wrong way to go. Any suggestions? Thanks!