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Guest post: Sequence Numbers

This is a guest post, sent in by David, who’s discovered an interesting property of numbers, and is looking for collaborators to take it further.

Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost. Rigour should be a signal to the historian that the maps have been made, and the real explorers have gone elsewhere.

– W. S. Anglin

A few years ago I saw a post on a website that showed that the inverse of 998,001 produces a decimal expansion that counts, using three digit strings, from 000 to 997 without error.

\[ \frac{1}{998,\!001} = 0.0000010020030040050060070080090100110120130\ldots \]

I immediately thought that this had to be a hoax. I decided to work it out to prove it was a hoax – after all some people put anything they want on the web whether it is true or not.

How many ways to shuffle a pack of cards?

This is an excerpt from friend of The Aperiodical, Matt Parker’s book, “Things to Make and Do in the Fourth Dimension”, which is out now in paperback.


There’s a lovely function in mathematics called the factorial function, which involves multiplying the input number by every number smaller than it. For example: $\operatorname{factorial}(5) = 5 \times 4 \times 3 \times 2 \times 1 = 120$. The values of factorials get alarmingly big so, conveniently, the function is written in shorthand as an exclamation mark. So when a mathematician writes things like $5! = 120$ and $13! = 6,\!227,\!020,\!800$ the exclamation mark represents both factorial and pure excitement. Factorials are mathematically interesting for several reasons, possibly the most common being that they represent the ways objects can be shuffled. If you have thirteen cards to shuffle, then there are thirteen possible cards you could put down first. You then have the remaining twelve cards as options for the second one, eleven for the next, and so on – giving just over 6 billion possibilities for arranging a mere thirteen cards.

The Other Half – Parable of the Polygons

Anna Haensch and Annie Rorem are the hosts of a new podcast, The Other Half. This is the second of two posts based on the first episode, about racism and segregation.

In the first part of episode one, we use the Racial Dot Map to get a sense of what race looks like in our country. And while it certainly gives us a picture of the stark racial lines segregating in our communities, it doesn’t necessarily help us understand how we got to be this way, and perhaps
more relevant, how we can fix this. In the second part of episode one we look at Parable of the Polygons, a playable blog post by Vi Hart and Nicky Case, to help us understand these slightly more nuanced questions.

parable of the polygons

The Racial Dot Map

Anna Haensch and Annie Rorem are the hosts of a new podcast, The Other Half. This post is based on the first episode, about racism and segregation.

In episode one of The Other Half, we look to mathematics as a potential tool for understanding racism and segregation in our society. To get a sense of the extent of segregation in the United States, we turn to a beautiful, startling tool to visualize it. Literally.

racial dot map

Apiological: mathematical speculations about bees (Part 3: Travelling Salesman)

This is part 3 of a three-part series of mathematical speculations about bees. Part 1 looked at honeycomb geometry, and part 2 looked at how bees estimate nest volumes.

The sight of bumblebees roaming around British gardens, foraging for nectar, is common and comforting. The movement of these fuzzy bees between flowers and plants can often seem deliberate yet erratic. Charles Darwin was intrigued by “humble-bee” routines, and observed them with the assistance of his six children, but always regretted not attaching strands of cotton wool to the bees so he could follow them more easily.

Within the last decade there has been renewed interest from a number of collaborating researchers into studying bumblebees’ movement between flowers and their foraging techniques. The prevailing journalistic spin on this research seems to be ‘Bees solve the Travelling Salesman Problem – a problem that mathematicians and computers cannot solve’. This is unfortunate, not least because it is gleefully misleading, confusing various meanings of ‘solve’, but also it obscures a lot of the fascinating underlying scientific investigations.

A geometrical approximation for π

If you were paying very close attention last week, you’ll have noticed my attempt to come up with an estimate of π, geometrically, as part of The Aperiodical’s π Day challenge (even if it’s not really π Day):