In geometric graph theory, the Hadwiger–Nelson problem poses the question: what is the minimum number of colours required to colour the 2-dimensional plane, so that no two points that are exactly one unit of distance apart are coloured the same colour? The answer to this question is called the **chromatic number** of the plane.

Up until now, we knew the answer was one of 4, 5, 6 or 7. The diagram shown here contains proofs of the upper and lower bounds: for the upper bound, you can assume the hexagons are each less than one unit in diameter, and so each point inside a hexagon is one unit away from a circle of points which use only the other six colours, as the colouring has no adjacent or adjacent-but-one hexagons the same colour.

The graph in the diagram also proves the minimum is four: known as the **Moser Spindle**, the graph consists of two pairs of equilateral triangles, and all the arcs are one unit long. It can be used to prove a three-colouring is not possible, as follows. The colour of the top left vertex forces the other two vertices of each of the two attached triangles to use the other two colours, but this in turn forces the two lower right vertices to be the same colour, and they are a unit apart. Contradiction!

Aubrey de Grey’s proof, uploaded to the arXiv earlier this month, uses a similar line of reasoning to prove it’s not possible to colour the plane with four colours in this way – only this graph has around 1600 vertices (see diagram, if you can), and the construction and checking of the graph is computer-assisted. The paper itself is still just on the arXiv for now as far as I can tell, but if we spot it being peer-reviewed and published anywhere we’ll make an update here.

In the wake of this announcement, a 16th Polymath project has been proposed to try to find a simpler graph, and improve on this result – some have conjectured it might not be possible to reduce the size of this graph further without redesigning it completely, as de Grey’s method already reduces it to as minimal as possible from the initial approach used (it involves taking smaller graphs and rotating them to align on top, much like the Moser Spindle).

Some blog posts on the topic, including one which makes attempts to verify the proof, and the paper itself, are listed below.

The chromatic number of the plane is at least 5, de Grey’s paper on the ArXiv

The chromatic number of the plane is at least 5, on Jordan Ellenberg’s blog

Amazing progress on longstanding open problems, on Scott Aaronson’s blog

The chromatic number of the plane is at least 5, on Dustin Mixon’s blog

Aubrey de Grey: The chromatic number of the plane is at least 5, on Gil Kalai’s blog

For a while now I’ve been fascinated by the story of Claude Shannon, the pioneer of information theory and the originator of many fundamental concepts now used in all modern manipulation and transmission of data. Being sent a copy of this biography to review was a great chance to find out more about his work and life.

**A Mind At Play: How Claude Shannon Invented the Information Age
**

The authors, who describe themselves as biographers and writers foremost, have taught themselves the mathematics they need to explain Shannon’s work, and weave in some excellent and succinct explanations of the concepts amongst a fascinating human story. From his early years as an enthusiastic maker and tinkerer, through his various university courses and his placement at Bell Labs, to his later years at MIT and retirement, Shannon’s life is chronicled in detail, with a spread of well-chosen photographs to accompany the story.

Claude Shannon is described as the father of information theory – his seminal 1948 paper outlined concepts including the fundamental nature of binary numbers (coining the word ‘bit’, a binary digit), information density, communication channels, and the theoretical Shannon Limit of how quickly digital information can theoretically be transmitted in a noisy channel. These ideas predated even simple computing machines, and Shannon’s work was perfectly timed to provide a foundation for those creating early computers.

The story gives a real sense of how Shannon was well placed to create the mathematics he did – with a sharp intellect that was torn between his love of abstract mathematical theory and his fondness for hands-on inventing and engineering, he had just the right mindset to see what communication theory would become and how it could be made rigorous in a mathematical framework.

It’s also fascinating to learn about Shannon’s other passions in life – nothing he did before or since comes close to the major impact his work on information theory had, but it was far from his only passion. Other areas of mathematics and engineering, as well as pastimes including juggling, stock market predictions, and building robots all fell to his mighty intellect and he brought huge joy to the people around him with his stories and ideas.

The book is well written and lovingly put together (and has a frankly beautiful cover in the hardback edition). It was enjoyable to read, and full of interesting facts and stories. I didn’t realise until reading this book that a wooden box I have at home, which has a switch on top that when flipped, engages a robotic arm that pops out and flips the switch back again, is a modern incarnation of an invention of Shannon’s – he called it ‘the ultimate machine’, one which switches itself off. Knowing this was his original creation, and the joy I find in it, gives me a real sense of connection to this brilliant mathematician whose work changed the world for all of us.

A Mind At Play: How Claude Shannon Invented the Information Age by Jimmy Soni and Rob Goodman is published by Simon and Schuster.

]]>The next issue of the Carnival of Mathematics, rounding up blog posts from the month of March, and compiled by Robin, is now online at Theorem of the Day.

The Carnival rounds up maths blog posts from all over the internet, including some from our own Aperiodical. See our Carnival of Mathematics page for more information.

]]>Here’s the final set of photos and video clips from the last week, and for the data fiends among you, a sneaky look at my spreadsheet of runs. With a graph, as requested by Hannah Fry.

Day 23 (aka @sportrelief day): stupid bloody GPS watch failed to log my distance, but I fixed it using the screen grab from my route planning map (which I won't share because it gives away where I live). A good pace! #pikmdotrun pic.twitter.com/6YWz03EeEl

— Katie Steckles (@stecks) March 23, 2018

Next up was this stroke of genius:

Day 25: had the genius idea of getting dropped of 3.14km away from the pub, where we're having a Sunday roast #winning #pikmdotrun pic.twitter.com/kndBgPY2gB

— Katie Steckles (@stecks) March 25, 2018

Augh! Just realised my tweet from yesterday didn't send! Day 27: ran out in the countryside near my nice hotel, missed the turning, had to just run 1.57km away and turn round. #pikmdotrun pic.twitter.com/62pzulsbUZ

— Katie Steckles (@stecks) March 28, 2018

Day 30: Final gym run (will be outdoors tomorrow). Accompanied by @aPaulTaylor & Waterbot. https://t.co/dtKU5Rj0qH pic.twitter.com/rm8AGaJqDQ

— Katie Steckles (@stecks) March 30, 2018

This final run video shows clips and photos from a bunch of the days, plus my triumphant final approach on Day 31:

There’s still time to chuck a final £3.14 on the pile at pikm.run, if you haven’t already. Thanks again to everyone! Now I’m going for a sit down.

]]>Extension and abstraction without apparent direction or purpose is fundamental to the discipline. Applicability is not the reason we work, and plenty that is not applicable contributes to the beauty and magnificence of our subject.

– Peter Rowlett, “The unplanned impact of mathematics”, Nature 475, 2011, pp. 166-169.Trying to solve real-world problems, researchers often discover that the tools they need were developed years, decades or even centuries earlier by mathematicians with no prospect of, or care for, applicability.

– Peter Rowlett, “The unplanned impact of mathematics”, Nature 475, 2011, pp. 166-169.There is no way to guarantee in advance what pure mathematics will later find application. We can only let the process of curiosity and abstraction take place, let mathematicians obsessively take results to their logical extremes, leaving relevance far behind, and wait to see which topics turn out to be extremely useful. If not, when the challenges of the future arrive, we won’t have the right piece of seemingly pointless mathematics to hand.

Peter Rowlett, “The unplanned impact of mathematics”, Nature 475, 2011, pp. 166-169.

Now, don’t get me wrong. I have every admiration for Peter and his work; his is a thoughtful voice of reason, and it’s not at all unreasonable for the Wikiquote page on mathematics to cite his writing.

It does seem unreasonable, though, that if a single paper by Peter merits three entries in the list, that the whole of womankind, from the whole of written history, only (currently) merits four (one from Hannah Arendt, one from Iris Murdoch, and two from Simone Weil under ‘quotes that mention mathematics’.)

Incidentally, I’m aware of the gender imbalance in Quotable Maths, which I curate at my blog. Of the last 110 quotes, only 16 are from women. While that’s not as bad as the Wikiquote ratio, it’s still not good enough; I’m working to improve that.

I would imagine, but haven’t checked, that there is a corresponding disparity in the number of quotes by mathematicians of colour, disabled mathematicians and LGBTQ+ mathematicians.

This isn’t ‘Nam, of course, there are rules – sadly, we can’t just jot down a selection of Katie’s carefully-crafted quips and spam the quotes page with them. According to the discussion page:

The quote ought to be understandable outside of its original context (or the context ought to be well enough known that most people (aquainted with the subject matter) will still understand it.

The quote ought to be one of the following:

- Known by many people.
- Uttered by a famous person.

NB: Both of the above need to be tempered by the context of the quote. I.e a quote need only be known by many people familiar with its subject matter.

The quote ought to be one of the following:- Interesting
- Funny
- Rude
- Of significance on its own
- Of significance given its source or some additional context

There must be dozens of those, right? What do you think about finding them and coordinating an effort to redress the balance?

I am not a Wikipedian, so you are at least as much of an expert in this as I am. Luckily, Katie and Peter have put together a Google Doc with the necessary details. We’ll be using this doc to coordinate real-life meet-ups and to document and discuss quotes that merit inclusion. We’ll also set up a Google Hangout for real-time coordination – please email Katie if you’d like to have the link sent to you; it will also be listed in the doc.

Editing wikis is straightforward. You don’t need to be a historian or a mathematician (although everyone is) to take part, you just have to be willing to roll up your sleeves and put some quotes in the right place. If you want to get started on the project ahead of time, it’d be helpful if you could find and add quotes to the Google doc ahead of time, so we have material to work with out of the gate.

So, I look forward to seeing you on **Saturday, May 12th from 10am** to help improve the Wikirepresentation of women in maths. No excuses! As Florence Nightingale herself said:

I attribute my success to this – I never gave or took any excuse.

WikiQuote: Mathematics

How to run an editathon

The Google Doc

Quotable maths at Flying Colours Maths

If I can make it to £1000 before the end of the month, I’ll be pretty pleased! Donate at pikm.run, or see below for my daily sweaty photos/videos/instagram posts.

Day 9: yep, I'm still doing this #pikmdotrun pic.twitter.com/MpFQvEImgr

— Katie Steckles (@stecks) March 9, 2018

On the 9th, I thought I’d make use of the mathematical properties of π to do a slightly silly one, and made a video:

Day 11: gym again. If you missed it yesterday, here's my video from Day 10: https://t.co/rdiPCJ534N pic.twitter.com/knyzH4L6f9

— Katie Steckles (@stecks) March 11, 2018

Day 13: still going. Thanks to everyone who's supported so far! https://t.co/dtKU5Rj0qH pic.twitter.com/p5qUUNncSD

— Katie Steckles (@stecks) March 13, 2018

I also managed to get in a few more joint runs with running companions:

Day 15: now officially kinda halfway! Buddy gym run again with @elsie_m_ #pikmdotrun pic.twitter.com/VhcVIsghll

— Katie Steckles (@stecks) March 15, 2018

Day 17: logistically complex. Managed to run πkm fully inside the B'ham NEC, as I've been working at @BigBangFair today. GPS watch gave up after 2.04km (no signal). #pikmdotrun pic.twitter.com/eebrhKjUgh

— Katie Steckles (@stecks) March 17, 2018

Day 19: staying in the gym due to the cold weather, on the world's shiniest treadmill #pikmdotrun pic.twitter.com/WO9xaPKaSV

— Katie Steckles (@stecks) March 19, 2018

I was also given an amazing gift by maths/knitting fan Linda Pollard, who came to see me perform at a show. She’s written up the knit of these magnificent π/sum gloves on a Ravelry page. I took the opportunity to test out their warmness on my next outdoor run:

This mild cry for help resulted in plenty of nice replies on Twitter, which has been a real boost – including Aperiodichum Colin Beveridge, who pointed out that my total is around $\pi^4$, a pleasing coincidence.

Day 21: Gym again (but went to a different gym for variety). Found it hard today. Encouragement please. #pikmdotrun pic.twitter.com/lIvuC7yBUZ

— Katie Steckles (@stecks) March 21, 2018

And of course, today’s effort:

Running continues. Watch this space for a final wrap-up and fundraising total at the end of the month.

Katie’s fundraising page at Sport Relief

More information about Sport Relief

You may remember back in September we posted about a mass-participation science experiment, aiming to model the spread of diseases in human populations using a smartphone app. The results of this experiment, presented by the contagiously loveable Hannah Fry, will be presented in a documentary this evening on BBC4. You can also see Hannah chatting about the experiment on this evening’s The One Show.

Contagion! The BBC4 Pandemic, on the BBC watch-o-tron

]]>]]>Robert P. Langlands wins the 2018 Abel Prize “for his visionary program connecting representation theory to number theory.” Congratulations! pic.twitter.com/HBiTJhChe0

— AbelPrize (@abel_prize) March 20, 2018

\begin{array}{l} \color{blue}13, \\ \color{blue}26, \\ \color{blue}39, \\ \color{blue}52 \end{array}

**What happened to $\color{blue}4$‽**

A while ago I was working through the $13$ times table for some boring reason, and I was in the kind of mood to find it really quite vexing that the first digits don’t go $1,2,3,4$. Furthermore, $400 \div 13 \approx 31$, so it takes a long time before you see a 4 at all, and that seemed *really* unfair.

I was being pretty unreasonable in my expectations of basic arithmetic, but I wasn’t completely brain-dead: I smelled an integer sequence! How about

$a(n)$ = least $k$ such that $k \times n$ starts with a $4$.

That’s not particularly interesting, and someone who comes across this sequence in the OEIS might think “why $4$?” So, I did a bit more thinking and came up with this:

$a(n)$ = least $k$ such that $\{ \text{first digit of } j \times n, \, 0 \leq j \leq k \} = \{ 0,1,2, \dots 9 \}$

I wrote a bit of Python, and in a few minutes I had some numbers:

$n$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$a(n)$ | 9 | 45 | 27 | 23 | 18 | 15 | 13 | 12 | 9 | 9 | 9 | 42 | 62 |

And hey, $13$ is a record-setter. I’m really beginning to dislike this number. Anyway, I searched the OEIS for my sequence and it wasn’t there, so I submitted it and it was duly accepted as A249067.

Along the way, OEIS editor Charles R Greathouse IV added this intriguing conjecture:

Conjecture:$a(n) \leq N$ for all $n$. Perhaps $N$ can be taken as $81$.

Why $81$? Maybe look at the graph produced automatically by the OEIS:

The record of $81$ is reached at $a(112)$. And at $a(1112)$. And $a(11112)$. That’s because they’re very slightly bigger than $\frac{1}{9} \times 10^m$, so nine times $1 \dots 12$ is *just* bigger than $9 \dots 9$, i.e. a number starting with a $1$, so it takes nine times nine steps down the times table before you see a number with $9$ as its first digit.

This pattern repeats at every power of $10$, and in fact every pattern in this sequence repeats (more or less) at every power of 10: this animated plot of the sequence with different horizontal scales shows that it’s self-similar:

(The fuzziness in the bigger plots is because each plot just takes a sample of points, and interpolates between them)

So the conjecture *looks* true, and this is my sequence, so I should prove it.

It isn’t surprising that this thing repeats when you multiply by $10$: we’re only looking at the first digit, and obviously the first digit of $n$ is the same as the first digit of $10n$. That doesn’t suffice as a proof of Charles Greathouse’s conjecture though: numbers which don’t end in a $0$ might do something unhelpful.

Fortunately, the day after I thought this sequence up was MathsJam night. I decided I’d set the Charles Grey pub’s brightest minds on the problem. I had a few ideas but I’m not particularly quick at putting thoughts together.

Ji proposed an application of the pigeonhole principle: if you look at the first *two* digits of the numbers you see in $n$’s times table, you can write out everything you might see in a $9 \times 10$ grid:

\begin{array}{}

10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 \\

20 & 21 & 22 & 23 & 24 & 25 & 26 & 27 & 28 & 29 \\

30 & 31 & 32 & 33 & 34 & 35 & 36 & 37 & 38 & 39 \\

40 & 41 & 42 & 43 & 44 & 45 & 46 & 47 & 48 & 49 \\

50 & 51 & 52 & 53 & 54 & 55 & 56 & 57 & 58 & 59 \\

60 & 61 & 62 & 63 & 64 & 65 & 66 & 67 & 68 & 69 \\

70 & 71 & 72 & 73 & 74 & 75 & 76 & 77 & 78 & 79 \\

80 & 81 & 82 & 83 & 84 & 85 & 86 & 87 & 88 & 89 \\

90 & 91 & 92 & 93 & 94 & 95 & 96 & 97 & 98 & 99

\end{array}

The $n$ times table will dance around this grid until all nine rows have been visited. The longest it can do that is by visiting all 80 cells not in the last line. If the process doesn’t visit the same place twice before it hits every row, that means that the latest you can put off visiting the last row is the 81^{st} iteration. So we need to show that you can’t visit the same spot twice before visiting each row once.

Unfortunately, that’s not true. The $12$ times table visits the ’12’ cell at $12 \times 1 = 12$ and again at $12 \times 10 = 120$, before all possible first digits have been seen.

So, we need another explanation.

Katie Steckles and the Manchester MathsJam crowd came up with an alternate explanation: if you can prove $\left\lceil \frac{1}{9} \cdot 10^m \right\rceil$ (that is, $112$, $1112$, $\ldots$) takes 81 steps for all $m \geq 3$, then that’s the maximum, as any $m$-digit number bigger than that will reach $9 \times 10^m$ in at most as many steps, and will definitely have seen all the other initial digits before then, and any $m$-digit number smaller than $\left\lceil \frac{1}{9} \cdot 10^m \right\rceil$ will visit every first digit in the first 9 multiples.

There’s some evidence for this: the $m+3$-digit numbers that take 81 steps seem to be the ones between $11\ldots112$ and $112499\ldots99$.

I don’t know if there’s a clever way of showing that $\left\lceil \frac{1}{9} \cdot 10^m \right\rceil$ takes 81 steps, but would it convince you if I said that $112$ takes that long, and adding more $1$s in the middle can’t make it any worse? Anyway, that’s good enough for me.

I think I can now answer my question: *exactly how bad is the $13$ times table?* Let’s compute the record-setters for A24097: the numbers that take longer than any smaller number to see every possible leading digit:

$n$ | 1 | 2 | 13 | 112 |
---|---|---|---|---|

$a(n)$ | 9 | 45 | 62 | 81 |

$13$ is a record-setter in the sequence, which means it’s pretty bad, but it’s not the worst: we’ve shown above that $112$ takes the longest possible number of steps to see every digit. And the number $2$ comes under scrutiny for taking way longer than its neighbours. So really, $13$ is just unlucky to find itself in such company.

If you’re interested in the working-out I did for this post, I’ve put my Jupyter notebook online.

]]>The Online Encyclopedia of Integer Sequences just keeps on growing: at the end of last month it added its 300,000^{th} entry.

Especially round entry numbers are set aside for particularly nice sequences to mark the passing of major milestones in the encyclopedia’s size; this time, we have four nice sequences starting at A300000. These were sequences that were originally submitted with indexes in the high 200,000s but were bumped up to get the attention associated with passing this milestone.

Here they are:

1, 10, 99, 999, 9990, 99900, 999000, 9990000, 99900000, 999000000, 9990000000, 99899999991, 998999999919, 9989999999190, 99899999991900, 998999999918991, 9989999999189910, 99899999991899109, 998999999918991090, 9989999999189910900, 99899999991899108991, 998999999918991089910, 998999999918991089910

The number formed by concatenating the first three digits in the sequence is $110 = 1 + 10 + 99$. This has a Golomb sequence vibe about it, though it’s a bit more straightforward to generate.

This sequence was submitted by Eric Angelini, a Belgian TV producer who has added countless sequences to the OEIS, usually generated like this by picking a constraint and working out what the sequence would need to look like in order to obey it.

1, 0, 0, 2, 0, 3, 4, 4, 3, 4, 5, 6, 4, 5, 6, 4, 5, 6, 5, 6, 6, 5, 7, 6, 5, 7, 6, 6, 7, 6, 7, 7, 6, 7, 7, 6, 7, 7, 8, 7, 7, 8, 7, 8, 8, 7, 8, 8, 7, 8, 9, 8, 8, 9, 8, 8, 9, 8, 9, 9, 8, 9, 9, 8, 9, 9, 9, 10, 9, 9, 10, 9, 9, 10, 9, 10, 10, 9, 10, 10, 9, 10, 10, 10, 10, 10, 11, 10, 10, 11, 10, 10, 11, 10, 11, 11, 10, 11, 11, 10

I’m amazed this one wasn’t already in! Seems like exactly the kind of thing that would appear in something like *Dudeney’s Amusements*. There’s an associated paper on the arXiv, by Ales Drapal and Carlo Hamalainen, which notes that some of the earliest work on triangle dissections was done by Bill Tutte, of Bletchley Park fame.

The entry page contains some fab plaintext-art drawings of solutions for a few different $n$.

1, 2, 4, 3, 6, 5, 9, 16, 14, 20, 7, 15, 8, 12, 18, 31, 26, 27, 40, 30, 49, 38, 19, 10, 23, 53, 11

The definition of this one is a bit opaque if you’re not in the right frame of mind, but it’s really neat. If you plot the sequence, as the OEIS can automatically do for you, you get this:

Or, if you want to do this in your head, think of the set of points $(n, a(n))$.

Now, if you pick any polynomial of degree $k$, there’s no subset of $k+2$ of the points on the scatter plot that lie on that polynomial. It’s a ‘duck-and-dive’ sequence – it always picks the smallest number that won’t be on any of the $2^{n-1}$ polynomials defined by the sequence leading up to $a(n)$.

The OEIS entry contains a conjecture that this sequence is a permutation of the natural numbers. It’s easily shown that it contains no duplicates – otherwise, if the number $m$ is repeated, there’d be two elements lying on the line $y=m$, a degree-0 polynomial. What’s not obvious is that every number will eventually turn up. It’d be pretty wild if some numbers never did – and that’d form a new sequence, too!

1, 1, 1, 1, 1, 3, 2, 1, 6, 15, 2, 1, 10, 52, 55, 2, 1, 15, 129, 389, 184, 2, 1, 21, 266, 1563, 2539, 648, 2, 1, 28, 487, 4642, 16445, 16604, 2111, 2, 1, 36, 820, 11407, 69863, 169034, 105365, 6352, 2, 1, 45, 1297, 24600, 228613, 1016341, 1686534, 654030, 17337, 2

I don’t like “triangle read by rows” entries, purely because the OEIS’s web interface doesn’t make them easy to read. It’s debatable whether sequences generated by two parameters are even ‘sequences’, but that’s not a fight worth having, because there are some truly fab bits of maths hiding in the OEIS’s triangles.

This one looks at what you can do by starting with the list of numbers $1,2, \ldots, n$, and repeatedly picking a block of adjacent numbers and reversing their order. It’s like a generalised version of the Oval Track puzzle.

]]>