Neurobics

Product Squares

By Russell Dear

First thoughts suggested the problem required trial-and-error with number-crunching on a calculator. However, on further reflection an organised approach, to my delight, led directly to the solution -- no trial-and-error or calculator were required. You might like to accept the challenge. If at first you don't succeed check out the clues below.

You may have thought that the numbers 5, 15, 18 and 28 were placed to make the problem simpler. In fact they are there to give it a unique solution. There are other ways of placing the sixteen given numbers so that rows, columns and diagonals have the same product.

This raises the question: do these numbers give the least possible product for a four-by-four square with sixteen different whole numbers? I suspect that is the case.

So-called "magic squares", where numbers in rows, columns and diagonals have the same sum, have been popular for over a thousand years. According to Henry Dudeney, perhaps the leading mathematical puzzlist of all time, squares with rows, columns and diagonals having the same product were first mentioned towards the end of the eighteenth century.

Dudeney revived them in 1897 when he offered the challenge of placing nine different numbers in a three-by-three square so that the product of those in each row, column and diagonal was the same and least value. What is that least value?

Answer to 3 x 3 square:

The least product is 216.

Russell Dear is a Mathematician living in Invercargill