I must have been just nine or ten years old when I discovered the American magazine “Popular Mechanics” in our local library. It was the stuff of dreams. My real world, where my parents could not afford to buy me a bicycle, let alone own a car themselves, was replaced by a fantasy world where people (and, in my imagination, myself) built and flew model aeroplanes, owned workshops equipped with lathes and all manner of power tools, drove Cadillacs, Oldsmobiles and other exciting-sounding cars, and took to the water in speedboats. In my imagination I compared the merits of Johnson and Evinrude outboard motors, designed elaborate workshop layouts for my personal collection of power tools, and built radio sets and model cars.
Eventually I managed some of these things in reality (except for the speedboat bit), but I never suspected that the magazine that catalyzed my dreams still existed. But it does, and continues to catalyze the imagination, as I discovered when this intriguing heading popped up on my Twitter feed:
5 Simple Math Problems No One Can Solve
Like anyone with an interest in mathematics, I was well aware of many simple-sounding unproved conjectures – such as Goldbach’s conjecture, that every even number greater than 2 can be expressed as the sum of two primes (try it; it has been tested and found true for all even numbers up to 4×1018, but this does not mean that there may not be a higher exception!).
To prove (or disprove) Goldbach’s conjecture is likely to require new and deep insights, just as did Andrew Wiles’s famous proof of Fermat’s last theorem. The examples given in the Popular Mechanics article may well demand similar insights, but they have a teasing simplicity that makes them just right to draw young people into the entrancing world of mathematics. Even with a degree and life-long interest in pure mathematics, I hadn’t heard of most of them! Here they are:
The Collatz conjecture: Pick any number. If that number is even, divide it by 2. If it’s odd, multiply it by 3 and add 1. Now repeat the process with your new number. If you keep going, you’ll eventually end up at 1. Every time. But can you prove this (no one has so far)?
The moving sofa problem: What’s the shape of the largest sofa that you can fit around a right-angle corner joining two corridors?
The cuboid problem: Is there a box of “rectangular” shape where the diagonals and sides are all integers?
The happy ending problem: If you draw five random dots on a sheet of paper, you can always join four of them to form a convex quadrilateral. To guarantee a pentagon, you need nine dots. For a hexagon, it’s seventeen. But how many dots to guarantee a heptagon, or any larger convex figure?
That’s four of the problems. For further details, and for the fifth problem (which requires a diagram) you’ll have to see the original article at
Meanwhile, I’m off to share them with my teenage grandson Gabriel, who has just started to take accelerated maths course in high school. He is being taught about known maths, but if he is tempted to fiddle with these where the answers are not yet known, he’s likely to discover quite a bit for himself about what real maths is really like.