Cube Sum is a Square

Pick a whole number. If you add up the cubes of all the whole numbers from 1 to your number, you will get the same as if you add up all the numbers from 1 to your number and then square the result.

For example choose the number 3:

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

1³ + 2³ + 3³ = 1 + 8 + 27 = 36

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

1 + 2 + 3 = 6

and then square the result

6² = 36

The answers are the same.

In algebraic notation:

1³ + 2³ + 3³ + ... + n³ = (1 + 2 + 3 + ... + n)²

We can show this with pictures:

It obviously works for 1:

1³ = 1²

* = *

for 2:

(1 + 2)² =

* * *
* * *
* * *

remove one to the side. This is the 1³. We need to show that the rest is 2³.

* *    *
* * *
* * *

The remaining has a 2² in it. Separate that out

* *    *

* *  *
* *  *

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

       *

* *  * *
* *  * *

Two lots of 2² makes 2³.

= 1³ + 2³

Now the case for 3:

(1 + 2 + 3)² =

* * * * * *
* * * * * *
* * * * * *
* * * * * *
* * * * * *
* * * * * *

move 3² to one side. We already know that this is 1³ + 2³. So we need to show that the rest is 3³.

* * *        * * *
* * *        * * *
* * *        * * *
* * * * * *
* * * * * *
* * * * * *

The remaining has a 3² in it at the bottom left. Separate that out

* * *        * * *
* * *        * * *
* * *        * * *

* * *   * * *
* * *   * * *
* * *   * * *

We have 3 lots of 3² which is 3³.

= 1³ + 2³ + 3³

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 6² to one side. We already know that this is 1³ + 2³ + 3³. So we need to show that the rest is 4³.

* * * *              * * * * * *
* * * *              * * * * * *
* * * *              * * * * * *
* * * *              * * * * * *
* * * *              * * * * * *
* * * *              * * * * * *
* * * * * * * * * *
* * * * * * * * * *
* * * * * * * * * *
* * * * * * * * * *

The remaining has a 4² in it at the bottom left. Separate that out

* * * *              * * * * * *
* * * *              * * * * * *
* * * *              * * * * * *
* * * *              * * * * * *
* * * *              * * * * * *
* * * *              * * * * * *

* * * *   * * * * * *
* * * *   * * * * * *
* * * *   * * * * * *
* * * *   * * * * * *

The parts above and beside the 4² have side length 1+2+3. Separate it out this way.

* * * *              * * * * * *
* * * *              * * * * * *
* * * *              * * * * * *
                     * * * * * *
* * * *              * * * * * *
* * * *              * * * * * *

* * * * 

* * * *   *  * *  * * *
* * * *   *  * *  * * *
* * * *   *  * *  * * *
* * * *   *  * *  * * *

Move those above the square to be with those beside the square. Matching the longest with the shortest and so on, 3 next to 1, 2 next to 2 and 1 next to 3.

                     * * * * * *
                     * * * * * *
                     * * * * * *
                     * * * * * *
                     * * * * * *
                     * * * * * *



* * * *   *  * * *    * *  * *    * * *  *
* * * *   *  * * *    * *  * *    * * *  *
* * * *   *  * * *    * *  * *    * * *  *
* * * *   *  * * *    * *  * *    * * *  *

Bunch together to make 4 squares

                     * * * * * *
                     * * * * * *
                     * * * * * *
                     * * * * * *
                     * * * * * *
                     * * * * * *



* * * *   * * * *    * * * *    * * * *
* * * *   * * * *    * * * *    * * * *
* * * *   * * * *    * * * *    * * * *
* * * *   * * * *    * * * *    * * * *

We now have 4 lots of 4² which is 4³

= 1³ + 2³ + 3³ + 4³

This procedure can be applied to the general case n.