Magical properties of cube numbers

Magical properties of cube numbers.

There are many hidden properties among cube numbers.Few are described below.

Different ways of expressing n³ 

4³ = 4² x 4 + 4x0²  = 16 x 4 +0

    = [4²  - 1x4](4+1) + 4 x 1² = 12 x 5 + 4 x 1

  =[ 4²- 2x4](4+2) + 4 x 2²  = 8 x 6 + 4 x 4

= [4²- 3 x4](4+3) + 4 x 3²  = 4 x 7 + 4 x 9

For 5³

5³ = 25 x 5 + 0

    = 20 x 6 + 5 x 1

= 15 x 7 + 5 x 4

= 10 x 8 + 5 x 9

= 5 x 9 + 5 x 16

1³ = 1

2³ = 1 +(1+6)= 1+7 = 8

3³= 1 +(1+6) +(1+6+12)= 1+7+19= 27

4³ = 1+(1+6)+(1+6+12)+(1+6+12+18)= 1+7+19+37 =64

5³= 1+(1+6)+(1+6+12)+(1+6+12+18)+(1+6+12+18+24)= 1+7+19+37+61= 125

1³ = 0²(6) +1

2³ = 1²(6)+2

3³= 2²(6) +3

4³ = (1² +3²) 6 + 4

5³ = (2² + 4² ) 6 + 5

6³ = (1² +3² +5²) 6 +6

7³ = (2²+4²+6²) 6 + 7

8³ = (1²+3²+5²+7²) 6 +8

9³ = (2²+4²+6²+ 8² )6 +9

10³ = (1² +3² +5² +7² + 9²) 6 + 10

In general  a³ = [ 1² +3² + 5² + ……. (a-l)²] 6 + a if a is even

                        = [ 2² + 4² + 6² +    (a-1)²] 6 + a if a is odd