For example choose the number 3:

Add up the cubes of all the numbers from 1 to 3:

1

^{3}+ 2

^{3}+ 3

^{3}= 1 + 8 + 27 = 36

Now, add up all the numbers from 1 to your number

1 + 2 + 3 = 6

and then square the result

6

^{2}= 36

The answers are the same.

In algebraic notation:

1

^{3}+ 2

^{3}+ 3

^{3}+ ... + n

^{3}= (1 + 2 + 3 + ... + n)

^{2}

We can show this with pictures:

It obviously works for 1:

1

^{3}= 1

^{2}

* = *

for 2:

(1 + 2)

^{2}=

* * * * * * * * *remove one to the side. This is the 1

^{3}. We need to show that the rest is 2

^{3}.

* * * * * * * * *The remaining has a 2

^{2}in it. Separate that out

* * * * * * * * *Move the two at the top to match the two at the side:

* * * * * * * * *Two lots of 2

^{2}makes 2

^{3}.

= 1

^{3}+ 2

^{3}

Now the case for 3:

(1 + 2 + 3)

^{2}=

* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *move 3

^{2}to one side. We already know that this is 1

^{3}+ 2

^{3}. So we need to show that the rest is 3

^{3}.

* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *The remaining has a 3

^{2}in it at the bottom left. Separate that out

* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *We have 3 lots of 3

^{2}which is 3

^{3}.

= 1

^{3}+ 2

^{3}+ 3

^{3}

We actually got a bit lucky with this one. The next case shows what we need to do for any size.

(1 + 2 + 3 + 4)

^{2}=

* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *move 6

^{2}to one side. We already know that this is 1

^{3}+ 2

^{3}+ 3

^{3}. So we need to show that the rest is 4

^{3}.

* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *The remaining has a 4

^{2}in it at the bottom left. Separate that out

* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *The parts above and beside the 4

^{2}have side length 1+2+3. Separate it out this way.

^{2}which is 4

^{3}

= 1

^{3}+ 2

^{3}+ 3

^{3}+ 4

^{3}

This procedure can be applied to the general case n.

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