We’ve said it before and we’ll say it again, with its quirks of calculus and multiple infinities, maths is wonderfully weird at times.
Following on from their previous book about the weird world of mathematics, teenage maths whizz Agniijo Banerjee, and his tutor and science writer David Darling are back with even more exotic and unusual facts about maths in their new book Weirder Maths.
Here are a few of their more unusual maths facts.
Rubik’s Cube was invented in 1974 but it wasn’t until 2010 that mathematicians figured out the maximum number of moves needed to solve the puzzle from any starting position.
Known by Cube enthusiasts as God’s Number, it was finally calculated using by a team of researchers at Google, having burned through 35 CPU-years of computer time. God’s Number, it turns out, is just 20.
This surprisingly low number explains how top ‘speedcubers’ can solve the puzzle in under five seconds. The current world record is 3.47 seconds, from a random starting position, set by a Yusheng Du from China in 2018.
Strange but true: 0.999… = 1. At first this seems at odds with common sense because 0.9, 0.99, and so on are all less than 1, so it would seem that 0.999… (where the nines go on forever) should also be less than 1.
Yet it’s easy to show that 0.999… = 1. If x = 0.999…. Then 10x = 9.999… = x + 9. Subtracting x gives 9x = 9, so that x = 1.
We’ve proved in a few simple steps that 0.999… = 1 and, at the same time, that 1 – 0.999… isn’t some very tiny number but instead is exactly equal to 0.
We expect the number pi to turn up whenever circles are involved because its roots lie in this shape. But the wonder of pi is its habit of appearing even when there’s no circle in sight.
For instance, the series 1/12 + 1/22 + 1/32 + 1/42 + 1/52 … = 1 + 1/4 + 1/9 + 1/16 + 1/25… gets closer and closer to the value π2/6 = 1.645…, as we include more and more terms.
Turn this fraction upside-down and we get 6/π2, which is equal to the probability that two numbers, providing they’re big enough, are coprime – in other words that they have no common factors other than 1.
Pi, in fact, is intimately and, somewhat mysteriously, involved with how prime numbers (numbers that have no factors other than themselves and 1) are distributed.
One, two, many
In a remote Amazonian region of Brazil lives a tribe, the Piraha, whose couple of hundred members can’t count beyond two.
Their word for ‘one’ can also mean ‘a few’, while ‘two’ does double duty as ‘not many’. Anything else is simply ‘many’. They also have no way of saying ‘more’, ‘several’, or ‘all’.
Being hunter-gatherers, they have no need to count and so no need to practise doing it. American linguist Daniel Everett tried to teach the Pirahas some basic numeracy skills after they expressed concern that their lack of knowledge might make it easy for them to be cheated when trading with other tribes.
After eight months effort, however, not a single Piraha had learned how to count to 10 or even to add one and one. Both their culture and their previous experience left them totally unprepared to grasp even the rudiments of numbers.
Horn of plenty
Gabriel’s Horn is the surface formed by rotating the curve y = 1/x, a rectangular hyperbola, around the x-axis for values of x greater than one. The seventeenth Italian physicist and mathematician Evangelista Torricelli was astonished to discover that although the Horn has a finite volume, equal to π cubic units, it has an infinitely large surface area! This seems to imply that if the Horn were filled with paint there wouldn’t be enough of it even to coat the surface.
The way to avoid this problem is to realise that thickness of the coat can be made vanishingly small at a rate quickly enough to compensate for the ever- expanding area, enabling a strictly finite volume of paint to coat an infinitely large surface.
Torricelli lived just before calculus appeared on the scene. Otherwise, he would have understood that the apparent paradox of the Horn can be explained in terms of the infinitely small quantities known as infinitesimals.