Fun with numbers - All about cubes 

The cubes have some amazing numerical characteristics. 

1³ = 1 (one odd )

2³ = 3 + 5 = 8 (two successive odds)

   = 1x2 + 1 + 2x3 -1

   = (2² -1)  + (2² = 1)

3³ = 7 + 9 + 11 = 27 (three successive odds)

   =( 2x3 + 1)  + ……+(3 x4 -1)

   = (3² – 2) + 3² + (3² +2)

4³ = 13+15+17+19  = 64 (Four succesivve odds)

   = (3x4+1) +……… +(4x5 -21)

  = (4² -3) +(4² – 1) +(4²+ 1) + (4² + 3) 

5³ = 21 + 23 + 25+ 27 + 29 = 125 (Five successiver odds)

   = (4x5 +1) +………….+ )5x6 – 1)

   = (5² -4) + (5² – 2) + 5² + (5²+ 2) + (5² + 4)

In general  

,n³ = ((n-1)n +1]+ ----------- + [n(n+1) -1]

     =(n² – n + 1)+ …………..+ (n² + n -1) 

Another property of cubes 

1³  = 6(0) + 1

2³ = 1³ + 6(1) + 1  = 6(1) + 2

3² = 2³ + 6(3) + 1 =  6(4) + 3

4³ = 3³  + 6(6) + 1 = 6(10) + 4

5³ = 4³  + 6(10)+ 1 = 6(20) + 5

6³ = 5³ + 6(15) + 1 = 6(35) + 6 = 6(36)+ 0 

In general 

,n³ = (n-1)³ +6 [n(n-1)/2} + 1 = 6 [ ₁ⁿ Σ n(n-1)/2] + n