One of my favorite little puzzles is “How many diagonals are there in a polygon?”

For solvers here I have three questions:

2. How many different routes of solution can we find? (different solutions here)
3. How can we modify this problem to make it easier? (easier versions, here)
One Comment leave one →
1. March 28, 2007 am31 7:58 am 7:58 am

Define
vertices: The points where two line segments come together (corners)
diagonal:a line segment connecting two non-adjacent vertices of a polygon: A closed plane figure made up of several line segments that are joined together.

Let n = number of vertices. Then the number of diagonals is:
n-3 + n – 3 + n – 4 + n – 5 + n – 6 + … until subtraction is n-1.
= (n-2)*n – 3 – sum(3:n-1)

So, for instance, you have 7 vertices in the picture above. That means n = 7. The number of diagonals is then 5*7 – 3 – 3-4-5-6 = 35-21 = 14.
A square would have 4 vertices. 2*4 – 3 – 3 = 2.
A pentagon would have 5 vertices: 3*5 – 3 – 3 – 4 = 5.

Unfortunately for me, it’s very late and I could be making lots of mistakes here.