Once you have explored some centred shapes, and taken a look at More to Explore, here are some more relationships you could try to prove if you're up for the challenge:
- 1½ of 3² - (1½ of 3) + 1 = 3rd Centred Triangle Number
{or One and a half of N² - (one and a half of N) + 1 = N th Centred Triangle Number } - 3rd Hexagonal Pyramid Number = 3³
{or N th Hexagonal Pyramid Number = N³} - 3rd Triangular Number + 2 of (2nd Triangular Number) + 1st Triangular Number = 3rd Centred Square Number
{or N th Triangular Number + two of (N-1) th Triangular Number + (N-2) th Triangular Number = N th Centred Square Number} - 6 of (3rd Square Based Pyramid Number) = Cuboid 3x4x7
{or 6 x N th Square Based Pyramid Number = N x (N+1) x (2N+1)} - 3rd Centred Triangle Number + 2nd T = 3rd Centred Square Number
{or N th Centred Triangle Number + (N-1) th T = N th Centred Square Number} - 3rd Centred Triangle Number – 3rd T = 2nd Square Number
{or N th Centred Triangle Number - N th T = (N-1) th Square Number}