### Pinned Squares

The diagram shows a 5 by 5 geoboard with 25 pins set out in a square array. Squares are made by stretching rubber bands round specific pins. What is the total number of squares that can be made on a 5 by 5 board?

### AP Rectangles

An AP rectangle is one whose area is numerically equal to its perimeter. If you are given the length of a side can you always find an AP rectangle with one side the given length?

### Hot Pursuit

The sum of the first 'n' natural numbers is a 3 digit number in which all the digits are the same. How many numbers have been summed?

# Icosagram

##### Stage: 3 Challenge Level:

This problem was written by James from St Peter's College, Adelaide, Australia.

Draw a pentagon and draw in all the diagonals (lines joining the vertices not including the edges of the pentagon), how many diagonals is this?

This is called a pentagram. How many diagonals are there in a hexagram, heptagram, octagram, nonagram, decagram etc. up to a 20 sided shape.

Does any pattern occur when looking at the number of diagonals?

Why is this so?

Can you find a formula?