Koch's kinky curve is created by starting with a straight segment and replacing it with four segments, each 1/3 as long as the original segment. so, at the second stage the curve has three bends. at the next stage, each segments replaced by four segments, and so on. how many bends does this curve have at the third stage? the fourth stage? the nth stage?

Stage           # segments                     # bends

1st stage      1 segment                      1 - 1 = 0 bends

2nd stage     4*1 segments                 4 - 1 = 3 bends

3rd stage      4*4 = 16 segments        16 - 1 = 15 bends

The number of bends is equal to the number of segments less 1, becasuse two segments are required to make one bend.

4th stage      4*4*4 = 64 segments      64 - 1 = 63 bends.

nth stage      4^(n-1) segments             4^(n-1) - 1 bends