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

For solvers here I have three questions:

1. What is the answer? (answers here)
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.