Once you have got to grips with Centred Shapes, have a look for relationships between different centred shapes, triangular numbers and square numbers.

If you think you have found a relationship, try to find a diagram to explain why it works.

Here are some relationships you could try to prove:

- 6 x N
^{th}Triangular Number + 1 = N^{th}Centred Hexagon - N² + (2N – 2)
^{th}Triangular Number = N^{th}Centred Hexagon - 3(N-1)
^{th}Triangular Number + N^{th}Centred Triangular Number = N^{th}Centred Hexagon - N(N+1) + N² = (2N)
^{th }Triangular Number - N
^{th }Centred Pentagon = N^{th}Triangular Number + (2N-2)^{th }Triangular Number - N
^{th }Centred Pentagon – N² + N = N^{th }Centred Triangle - N
^{th }Triangular Pyramid - N² = (N-2)^{th }Triangular Pyramid

For some further challenges, have a look at More Centred Shape Relationships.