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 = 3
^{rd}Centred Triangle Number

{or One and a half of N² - (one and a half of N) + 1 = N^{ th}Centred Triangle Number } - 3
^{rd}Hexagonal Pyramid Number = 3³

{or N^{ th}Hexagonal Pyramid Number = N³} - 3
^{rd}Triangular Number + 2 of (2^{nd}Triangular Number) + 1^{st}Triangular Number = 3^{rd}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 (3
^{rd}Square Based Pyramid Number) = Cuboid 3x4x7

{or 6 x N^{ th}Square Based Pyramid Number = N x (N+1) x (2N+1)} - 3
^{rd}Centred Triangle Number + 2^{nd}T = 3^{rd}Centred Square Number

{or N^{ th}Centred Triangle Number + (N-1)^{ th}T = N^{ th}Centred Square Number} - 3
^{rd}Centred Triangle Number – 3^{rd}T = 2^{nd}Square Number

{or N^{ th}Centred Triangle Number - N^{ th}T = (N-1)^{ th}Square Number}