### 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:

Correct solutions with proofs came from James Page, Hethersett High School, Norfolk, Ruoyi Sun, the NLCS Puzzle Club and from girls in years 8, 9 and 10 at the Mount School York. Well done all of you. The diagram here is by Daniella Yule, Sophie Brook and Hollie Jefferson.

Thank you Nisha Doshi, Bella Heesom, Daniella Yule, Sophie Brook and Hollie Jefferson for this solution.

If we use $k$ to stand for the number of sides, there are also k vertices. Each vertex has $(k - 3)$ diagonals coming from it. This is $(k - 3) \times k$ diagonals. However, this method counts each of the diagonals twice, so the correct result is $\frac{1}{2}k(k - 3)$.