Friday, January 16, 2009

The Humble Multiplication Table, 1

A surprising relationship found in the multiplication table is that the sum of the entries in the main upwards diagonal and the diagonal above it is equal to the sum of the entries in the main downwards diagonal. What is also surprising is that this is but one among several observations about the multiplication table that can be expressed in terms of polygonal numbers.

This relationship involves three-dimensional triangular numbers (triangle-based pyramidal numbers, or tetrahedral numbers), and three-dimensional square numbers (square-based pyramidal numbers). Some values for these, and a few other polygonals, are shown below.

To see why this relationship holds, first note that the sum of the entries in the nth upward diagonal in the multiplication table is equal to the nth three-dimensional triangular number. 

Second, observe that he entries in the main down diagonal are square numbers (two-dimensional), so the sum of the main down diagonal is the nth three-dimensional square number. 

Finally, we use the fact that a square number (of any dimension) can be split into two triangular numbers (of the same dimension), which gives us the surprising result above.

The image below shows the relationship for a 4x4 multiplication table.