## Elegant Solution: An algebra problem from 1798

In John Cook’s excellent blog, The Endeavour, he recently posted a nice little problem taken from Benjamin Wardhaugh’s new book, “A Wealth of Numbers: An Anthology of 500 Years of Popular Mathematics Writing.” Mr. Wardhaugh took this problem from a magazine published between 1704 and 1841 in England called the Lady’s Diary.

The problem is as follows:

What two numbers are those whose product, difference of their squares, and the ratio or quotient of their cubes, are all equal to each other?

This is a great problem for a number of reasons. One of the main ones being that the solution is elegant.

If we write the problem algebraically we have:

Solving this equation graphically gives 4 solutions (which happen to be the vertices of a rhombus centre (0,0)). All of these solutions are based around the golden ratio. Can you see why?

However, this is only one quarter of the solution. I could have written the equation to start with in 4 different ways.

Each of these equations yield 4 seperate solutions. (See my solution sheet for an algebraic approach)