On a circular island we build n straight dams going from sea to sea so that every two intersect but no three go through the same point Use Euler's Formula to determine how many parts we get?

Answer 1

Consider following figure with n = 4.

Clearly, number of vertices (v) = 14, number of edges (e) = 24 and number of faces (f) = 12.

According to Euler's formula, v + f = e + 2

i.e. v + f = 14 + 12 = 26 = e + 2.

In general we have, v = 2n + nC2 and e = 2n + n2.

Hence, using Euler's formula,

f = e + 2 - v

= 2n + n2 + 2 - 2n + nC2

After simiplication we have,