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 x, y 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
Wait! Where else does that expression occur?
It is also the sum of the first N (positive) perfect cubes! In math notation,
The implication is that the number of cubes sitting over the last row and column of the multiplication table must be . 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 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 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 by rectangle, and an by rectangle has cubes in it!
Thus, a secret key to the identity
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.