Lately I’ve had a slightly above-average obsession with books. My house is full of them: I go through spells of buying a load at once. And then spells of buying new bookcases to house them all. In an effort to both save money and to satisfy my curiosity, I’ve tried tracking down a few older second-hand books. The latest three are:

- Teaching as a Subversive Activity, Postman & Weingartner
- Learner Teacher, Otty
- Tomorrow’s Math, Ogilvy

The title of the post is a bit of a spoiler for which one I’m going to write about today.

## Back to the Future

The copy of Tomorrow’s Math that I’ve picked up is the second edition, published 1972. Inferring from the copyright information I think the book was first published in 1962. Subtitled “unsolved problems for the amateur”, the book’s intention is to demonstrate that not only are there a significant number of open problems in mathematics, but that many of the problem statements can be understood by laymen.

Before describing a couple of the problems, the preface is tantalising in a number of ways:

Which problems have been moved out? How does he source this ‘oversupply’ of accessible problems? Which problems have been solved?!

This book must provide an interesting snapshot of progress in mathematics: in the ten-year period since the first edition, a number of open problems have been tackled and solved. One of my longer term goals is to try and investigate what progress has been made on the other problems over the past forty or so years.

## Five Colours Suffice

Perhaps one of the most well-known problems in mathematics that remained unsolved for a long time is the question of how many colours are needed for a planar map so that no two regions sharing a common edge have the same colour. Here is how the situation is presented in *Tomorrow’s Math*:

At the time of his writing, a five colour theorem had been proven but the now infamous ‘Four colour theorem’ had yet to be demonstrated. I’m intrigued by the closing question in the second of the pages shown above: is it posed as a puzzle for the reader as a mathematically open question? These are the kind of things I hope to research when time and concentration permit!

## Deployment and Dispersal

The chapters of Tomorrow’s Math separate the problems into categories: applied, games, geometrical, arithmetical, topological, combinatorial and analysis. One of the applied problems that caught my interest was what Ogilvy refers to as *best deployment*.

He writes, “What is the best deployment of *n* stations (points) on a circular disk? More precisely, no point of the disk is at a distance greater than *k* from some one of the stations; what is the smallest possible value of *k* for various *n*? The answer is known for *n*≤5, but there are many values of *n*>5 for which the smallest *k* is not known, and a general solution seems remote at present.”

This is followed by the problem of dispersal: arranging a set of points on a sphere to maximise the least distance between them. This, he notes, has been solved for *n* = 2, 12, and 24. Adding that a recently discovered solution for *n*=11 gives no improvement over *n*=12.

I would personally have classified this as a geometric problem, but he includes the applications of monitoring, surveillance and satellites.

## Think, Pair, Share

- The nRich website has an excellent discussion about The Four Colour Theorem
- Robin Wilson published a book called Four Colours Suffice discussing the full history of the problem
- There are number of unsolved problems which are great for captivating students: Goldbach’s conjecture and the 3n+1 iterative process for example. Do you discuss open problems with your students?

I’ll let XKCD have the final word…