More Centred Shape Relationships

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}