You are given a circle of radius 1 unit and two circles of radius a and b which touch each other and also touch the unit circle. Prove that you can always draw a 'flower' with six petals with the unit circle in the middle, and six circles around it having radii
a, b, b/a, 1/a, 1/b and a/b
such that each outer circle touches the unit circle and the two circles on either side of it.
http://nrich.maths.org/public/viewer.php?obj_id=2153

The Tower of Hanoi puzzle is an engaging and challenging problem for middle school students and an application of exponential functions. It also is the setting for an elegant proof by mathematical induction that the minimum number of moves M to move n disks from one disk to another is given by M = 2^n - 1.

Here is a Java implementation of the same puzzle.
http://nlvm.usu.edu/en/nav/frames_asid_118_g_3_t_2.html?open=instruction...