Exploring Number Shapes

Have a look at this very short video which illustrates Centred Shapes:

Can you see how Centred Shapes are made?

Now, take a look at this video which shows, using pictures, how centred triangles are related to triangular numbers and square numbers:

Can you convince yourself that this will be the case for all Centred Triangles?

If you would like to read more, take a look at this Word document.

Now take a look at some of these relationships and see what you can discover. 

Can create some pictures to show that the relationships are true (a visual proof)?

a/ 3^rd Centred Hexagon Number + 1 = 2 of 3^rd Centred Triangle Number
(or N^th Centred Hexagon Number + 1 =  2(N^th Centred Triangle Number)

b/ 3^rd Centred Square Number – 2^nd Triangular Number = 3^rd Centred Triangle Number
(or N^th Centred Square Number – (N-1)^th Triangular Number = N^th Centred Triangle Number)

c/ 3² + 2² = 3^rd Centred Square Number
(or N² + (N-1)² = N^th Centred Square Number)

d/ 2^nd Triangular Pyramid Number + 3^rd Triangular Pyramid Number = 3^rd Square Based Pyramid Number
(or (N-1)^th Triangular Pyramid Number + N^th Triangular Pyramid Number = N^th Square Based Pyramid Number)

Once you've had a go at these, you may like to look at Centred Shapes - More to Explore.