Completing the Cube
November 7, 1986
A review of "Completing the Cube" by Barbara Turner.
Copyright © 1998 Property of Deborah K. Fletcher. All rights reserved.
This article deals with the use of three-dimesional figures in the teaching of principles of advanced mathematics. The theory is that many properties of numbers may be understood more readily if there is a geometrical way to visualize them.
To introcuce the idea of geometrically demonstrating mathematical properties, the article gives a two-dimensional geometric proof of completing the square, using the formula:
The proof is in the form of the following figure:
This proof is for the case n=4, illustrating that
1 + 2 + 3 + 4 = 42 - ( 1 + 2 + 3 ),
noting that the ( 1 + 2 + 3 + 4 ) portion refers to the unshaded boxes, the 42 refers to the total number of boxes, and the ( 1 + 2 + 3 ) portion refers to the shaded boxes.
As a more general explanation, the figure would be an n x n square. The i-th column, containing i unshaded boxes, in completed by n-1 shaded boxes (one unshaded plus three shaded equals four, two unshaded plus two shaded equals four, etc.).
The article then progresses to the rather involved process of completing the cube. The formula for this is:
The following equation is derived from it:
The three-dimensional proof requires specially designed boxes called rods and squares (cuisenaire rods and squares are appropriate). The rods are of dimensions n x 1 x 1, and the squares are of dimensions n x n x 1. In each case, n is any positive integer. The volume of an n-rod is n; the volumbe of an n-square is n2. It is helpful to color-code the rods and squares according to size.
For this proof, we will take the case n=4. First, place a 3-square, a 2-square, and a 1-square, in that order, on top of a 4-square, flush to one corner, as shown:
Volume = 12 + 22 + 32 + 42.
Secondly, arrange a 1-rod, a 2-rod, and a 3-rod upright on the 2-square, 3-square, and 4-square, respectively, as shown:
Volume = 12 + 22 + 32 + 42 + 1 + 2 + 3.
Next, place two 1-squares on top of the 2-square, covering it completely, as shown:
Volume = 12 + 22 + 32 + 42 + 1 + 2 + 3 + 2 x 12.
The 3-square is now to be covered by placing two 2-squares upright on top of it, as shown:
Volume = 12 + 22 + 32 + 42 + 1 + 2 + 3 + 2 x 12 + 2 x 22.
The cube is completed by placing two 3-squares upright on the exposed surface of the 4-square, as shown:
Volume = 43.
Volume = 12 + 22 + 32 + 42 + 1 + 2 + 3 + 2 x 12 + 2 x 22 + 2 x 32.
Thus,
Hence,
The same will work for any integer n>1.
For further algebraic proofs, there are no geometric figures, but there is an equation for sums of kth powers of integers for k>2:
An equation for expressing
can be derived from the above equation, yielding the following:
This article is interesting and informative. It is an easy matter to follow the geometric proofs, and the algebra, although lengthy, is not overly complicated. I am glad that I read this article, because I feel that I now understand the concepts of completing the square and completing the cube.
Please View and Sign My Guestbook
Back to Debbie Fletcher