Multiplication Sculptures and Sums of Cubes

A 15 by 15 multiplication sculpture built by Jane Kostick.

A 15 by 15 multiplication sculpture built by Jane Kostick.

Take a multiplication table and build a column of cubes over each entry. For each column, use as many cubes as the product it sits upon. The result is a “multiplication sculpture” or “multiplication tower.” The picture shows a 15 by 15 multiplication sculpture built by Jane Kostick in 2008. For more examples, check out Maria Droujkova’s photo collection at moebius noodles.

Last spring, club members at Girls’ Angle built a 4 by 4 version out of cubes that were 3 inches on a side. They thought about its properties, such as how many cubes make up an N by N multiplication sculpture.

We’re going to address this last question and end up with a nice, concrete proof of a famous algebraic identity.

Over the xy entry, there are xy cubes, so we have to add up all products xy where x and y range over the values from 1 to N, and this totals

(1+2+3+\dots+N)^2 = (\frac{N(N+1)}{2})^2.

Wait! Where else does that expression occur?

It is also the sum of the first N (positive) perfect cubes! In math notation,

1^3 + 2^3 + 3^3 + \dots + N^3 = (\frac{N(N+1)}{2})^2.

The implication is that the number of cubes sitting over the last row and column of the multiplication table must be N^3. If we can show this directly, we’d have a nice, concrete proof of the sum of cubes formula.

The columns of cubes over the last row of the multiplication table form a staircase with steps of height N. Saw these N columns off and get a flat, staircase-shaped plank, N cubes long and N^2 cubes high. Next, saw off the columns of cubes over the last column (of the multiplication table). You’ll get an almost identical staircase-shaped plank. The only difference is that the last step of total height N^2 is gone because it was removed when the columns over the last row (of the multiplication table) were sawed off.

Turn one of these planks over, and the two planks will fit together perfectly to form an N by N^2 rectangle, and an N by N^2 rectangle has N^3 cubes in it!

Thus, a secret key to the identity

1^3+2^3+3^3+\dots+N^3 = (\frac{N(N+1)}{2})^2

is hidden in plain sight in the multiplication table that many of us learned in elementary school!

To read about how Jane made this 15 by 15 multiplication sculpture, and see hints about more of its properties, check out the December, 2008 issue of the Girls’ Angle Bulletin, pages 12-14 and 25-28.

 

Advertisements

About girlsangle

We're a math club for girls.
This entry was posted in math and tagged , . Bookmark the permalink.

One Response to Multiplication Sculptures and Sums of Cubes

  1. Pingback: Virtual Reality! | Girls' Angle

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s