Enrico Fermi apparently had a knack for making rough estimates with very little data. Fermi problems are problems which ask for estimations for which very little data is available. Some standard Fermi problems:

- How many piano tuners are there in New York City?
- How many hairs are there on a bear?
- How many miles does a person walk in a lifetime?
- How many people in the world are talking on their mobile phones right now?

Hopefully you get the idea. These are problems for which little data is available, but for which intelligent guesses can be made. I have used problems of this type with students as an exercise in estimation and making assumptions. Inspired by a tweet from Alison Kiddle, I have set these up as a comparison of which is bigger from two unknowable things. Are there more cats in Sheffield or train carriages passing through Sheffield station every day? That sort of thing.

The point of these is not to look up information or make wild guesses, but instead to come up with a back-of-the-envelope, ‘wrong, but useful‘, orders of magnitude estimate. Some ‘rules’, if you want to play with these the way I would:

- don’t look up information;
- don’t make precise calculations using calculator or computer;
- be imprecise — there are 400 days in a year, people are 2m tall, etc.;
- round numbers where possible and calculate in your head.

One approach is to estimate by bounding – come up with numbers that are definitely too small and too large, and then use an estimate that is an average of these. But which average?

Say I think some quantity is bigger than 2 but smaller than 400. The arithmetic mean would be $\mathrm{AM}(2,400)=\frac{2+400}{2}=201$. The geometric mean would be $ \mathrm{GM}(2,400)=\sqrt{2\times 400} = 28.28\!\ldots$.

Which is a better estimate? The arithmetic mean is half the upper bound, but 100 times the lower bound. On this basis, for an ‘order of magnitude’-type estimate, you might agree that the geometric mean is a better average to use here. Following my Maths Jam talk, Rob Low said that the geometric mean makes more sense for an order of magnitude estimate, since it corresponds to the arithmetic mean of logs. To see this, consider \[

\begin{align*}

\log(\mathrm{GM}(A, B)) &= \log(\sqrt{AB}) \\

&= \log((AB)^{\frac{1}{2}}) \\

&= \frac{1}{2}\log(AB) \\

&= \frac{1}{2}(\log(A) + \log(B)) = \mathrm{AM}(\log(A), \log(B)) \text{.}

\end{align*}

\]

So, geometric mean it is. However, taking a square root is not usually easy in your head, and we want to avoid making precise calculations by calculator or computer. Enter the approximate geometric mean.

For the approximate geometric mean, take $2=2 \times 10^0$ and $400=4 \times 10^2$, then the AGM of $2$ and $400$ is: \[ \begin{align*}

\frac{2+4}{2} \times 10^{\frac{0+2}{2}} &= 3 \times 10^1\\

&= 30 \approx 28.28\!\ldots = \sqrt{2\times 400} = \mathrm{GM}(2,400) \text{.}

\end{align*} \]

Why does this work? Let $A=a \times 10^x$ and $B=b \times 10^y$. Then \[

\begin{align*}

\mathrm{GM}(A,B)=\sqrt{AB}&=\sqrt{ab \times 10^{x+y}}\\

&=\sqrt{ab} \times 10^{\frac{x+y}{2}} \text{,}

\end{align*}

\]

and \[\mathrm{AGM}(A,B) = \frac{a+b}{2} \times 10^{\frac{x+y}{2}}\text{.}\]

Setting aside the $10^{\frac{x+y}{2}}$ term, which appears in both averages, is it obvious that, for single digit numbers $>0$, \[\mathrm{GM}(a,b)=\sqrt{ab} \approx \frac{a+b}{2}=\mathrm{AM}(a,b) \text{?} \]

There is a standard result that says \[ \begin{align*}

0 \le (x-y)^2 &= x^2 – 2xy + y^2\\

&= x^2 + 2xy + y^2 – 4xy\\

&= (x+y)^2 – 4xy \text{.}

\end{align*} \]

Hence \[ \begin{align*}

4xy &\le (x+y)^2\\

\sqrt{xy} &\le \frac{x+y}{2} \text{,}

\end{align*} \]

with equality iff $x=y$. So $\mathrm{GM}(a,b)\le\mathrm{AM}(a,b)$, but are they necessarily close?

By exhaustion, it is straightforward to show (for single-digit integers, given the rule to round numbers where possible) that the largest error occurs when $a=1$ and $b=9$. Then \[ \sqrt{1 \times 9} = 3 \ne 5 = \frac{1+9}{2} \] and the error is $2$ which, relative to the biggest number $9$ might be seen as quite significant.

I’d say you are not likely to use this method if the numbers are of the same order of magnitude, because the idea is to come up with fairly wild approximations and if they were quite close it might be sensible to think of them as not really different. Then the error is going to be at least one order of magnitude smaller than the upper bound, i.e. $10^\frac{x+y}{2} \ll 10^y$. For example, if your numbers were $1$ and $900$ (as a pretty bad case), then: \[ \mathrm{GM}(1,900)=\sqrt{900}=30 \ne 50=\mathrm{AGM}(1,900) \] and a difference of $20$ on a top value of $900$ is not as significant as a difference of $2$ was on a top value of $9$.

So I suppose I would argue that this makes the error relatively insignificant. However, this thinking left me somewhat unsatisfied. I felt there ought to be a nicer way to demonstrate why the approximate geometric mean works as an approximation for the geometric mean. Following my talk at Maths Jam, Philipp Reinhard has been thinking about this, and he will share his thoughts in a post here in a few days.

I didn’t have time to fit into my talk what I would recommend if the two numbers differed by an odd number of orders of magnitude. For example, $\mathrm{AGM}(1,1000)$ generates another square root in $1 \times 10^{\frac{3}{2}}$ – precisely what we were trying to avoid! What I have recommended to students is to simply rewrite one of the numbers so that the difference in exponents is even. For example, writing $1=1 \times 10^0$ and $1000 = 10 \times 10^2$ gives \[\mathrm{AGM}(1,1000)=5.5 \times 10^{1} \text{.}\]

Following Maths Jam, the esteemed Colin Beveridge made the sensible suggestion of just treating $10^{\frac{1}{2}}$ as $3$, making \[

\begin{align*}

&\mathrm{AGM}(1,1000)\\

&= 1 \times 10^{\frac{3}{2}}\\

&\approx 1 \times 3^3 = 27\text{.}

\end{align*}

\]

This increases our problems, though, because we have the potential to deal with larger differences (hence larger errors) than when dealing with single-digit numbers. Actually, it was wondering why this increased error happens that got me thinking seriously on this topic in the first place. I’ll stop now to let Philipp share what he has been thinking on this.

]]>I just noticed that last Wednesday was ten years since that lecture. It was basic maths for forensic science students. I was given a booklet of notes and told to either use it or write my own (I used it), had a short chat about how the module might work with another lecturer, and there I was in front of the students. That was spring in the academic year 2007/8 and this is the 21st teaching semester since then. This one is the 15th semester during which I have taught — the last 12 in a row, during which I got a full-time contract and ended ten years of part-time working.

I have this awful feeling this might lead people to imagine I’m one of the people who knows what they are doing.

P.S. The other thing that I started when I started working for the IMA was blogging – yesterday marks ten years since my first post. So this post represents the start of my second ten years of blogging.

]]>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.

]]>If you pay attention to United States politics you have probably noticed that mathematics is currently enjoying a rare moment of relevance. You probably also know this is not happening because all of a sudden politicians have decided that mathematics is clearly the coolest thing in the world, even though it clearly is, but instead because *gerrymandering* has become one of the major issues du jour.

For those of you lucky enough not to know what gerrymandering is, let me give you a quick précis. Named after Elbridge Gerry – it should be pronounced like *Gary* and not *Jerry* – and a congressional district which slightly resembled a salamander he signed into law as the governor of Massachusetts, gerrymandering has come to be the blanket term for the redrawing of political districts in the United States in a way that provides political gain for the party conducting the redrawing. This is primarily done through either *packing*, drawing a district so all of your opponents’ votes are concentrated in a small number of districts and therefore can not meaningfully affect others, or *cracking*, splitting up the opponents’ votes among many different districts so they have less influence on any of them. This has generally been considered to be totally legitimate, and smart, political maneuvering in the US and upheld as legal in the courts, unless it can be proven the gerrymandering was done based on partisanship and not on race.

The reason gerrymandering is such a hot topic is because the courts might just be changing their views regarding partisan gerrymandering, and a big factor behind this is mathematics. There was an argument in front of the supreme court, in the case of *Gill v. Whitford* late last year about partisan gerrymandering in my home state of Wisconsin, which had mathematics as a central pillar in the arguments against the current district lines. Even more recently the Pennsylvania Supreme Court threw out their current districts and demanded they be redrawn, and while I am not sure if mathematics played a large role in them getting thrown out it certainly will when they are redrawn.

As gerrymandering is enjoying its moment in the sun, it is only fair the mathematician playing the biggest role in changing how it is all being thought about is called Moon. Moon Duchin is an Associate Professor at Tufts University and the creator of the Metric Geometry and Gerrymandering Group which, through a series of conferences, is applying cutting-edge mathematics to the redistricting problem, training mathematicians to be expert witnesses on gerrymandering for court proceedings, and providing teachers with lesson plans and guidance on how to implement them (there is one more conference in California coming up in March and a big workshop happening in August if you want to get involved).

The really big news though is, as of January 26 Duchin is working as a consultant for Governor Wolf in Pennsylvania, with the job of helping to make sure their redrawn congressional district map is fair. I have had the joy of talking to Moon for my podcast Relatively Prime about her work with the MGGG and watched her give a talk about gerrymandering at the 2018 Joint Mathematics Meeting. I do not think I have ever seen a more enraptured audience at a mathematics conference, there were a lot of people in the room and each and every one of them was paying attention. I can not think of a better person from a mathematical ability perspective, as well as a public engagement one, to be the face of this for mathematics.

It is too bad it has taken something so awful as gerrymandering to get mathematics a seat at the table in US political discourse, and even though I have spent a huge amount of my life trying to convince people mathematics is something we should all care about I would happily not have people talk about it if it meant we had no gerrymandering. That said, I am glad we have mathematicians like Moon Duchin who are willing to take this battle on in front of not only the mathematical community but an ever increasing portion of the politically engaged public, not to mention Governor Wolf and lawyers like those in *Gill v. Whitford* who are willing to reach past their comfort zone and let mathematics play a central role in their work. There is not going to be a clean, perfect solution to all of this, but hopefully with mathematicians like Moon involved in this it will end up a lot better than where it is now.

Writing about maths, especially deep technical maths, so that a reader can follow along with it is *hard* – the Venn diagram of the set of people who can write clearly and the set of people who understand the maths, two relatively small sets, has a yet smaller intersection.

Vicky Neale sits squarely inside it, and *Closing The Gap* has gone straight into my top ten “books to give to interested students”.

Here’s a clever way to structure a maths book (I have taken copious notes): follow the development of a difficult idea or discovery chronologically, but intersperse the action with background that puts the discovery in context. That’s not a new structure – but it’s tricky to pull off: you have to keep the difficult idea from getting too difficult, and keep the background at a level where an interested reader can follow along and either say “yes, that’s plausible” or better “wait, let me get a pen!”. This is where Closing The Gap excels.

Neale takes as the difficult idea the Twin Primes Conjecture, and specifically the work that followed from Yitang Zhang’s lightning-bolt discovery in 2013 that infinitely many pairs of primes are separated by at most 70,000,000 (which sounds like a lot… but is very small compared to “no upper limit”) – especially the Polymath projects and the work of James Maynard in reducing the bound to either 600 (unconditionally) or 12 (if the Elliott-Halberstam conjecture is true – a bound later reduced to 6 by Polymath8b).

The Elliott-Halberstam conjecture? What’s that? Neale takes the time to explain, by way of a mathematical pencil, the flavour of the conjecture, without getting bogged down in the technical details; she tells us enough that the story makes sense, and enough that we could go and find out more if we wanted.

Because of Neale’s position in the Venn diagram, she can pull off this kind of thing, making maths accessible without losing accuracy – she’s meticulous about saying “there’s more to this” when there’s more to something.

This attention to detail is possibly overdone in places – I found myself rolling my eyes from time to time about in-text reminders that I met Terry Tao in a previous chapter, or that we’d hear more about such-and-such in a future one, which I suppose is an upshot of deciding to do without footnotes. This is literally my only mild criticism of the book; I’m even in thrall to the quality of the paper it’s printed on.

Closing The Gap communicates the excitement, frustration and interconnectedness of top-tier mathematical research, including the relatively new approaches pioneered by Tim Gowers (among others) with the Polymath project. The book’s introduction starts with an extended analogy comparing mathematics to climbing (we know a MathsJam talk about that!) – how something impossible gradually becomes possible, then difficult, then accessible to novices with the help of a guide. Neale sets herself up as this guide, and succeeds brilliantly.

]]>The odds of being crushed by a meteor are considerably lower (i.e. more likely) than those of winning the jackpot on the National Lottery.

— Quite Interesting (@qikipedia) January 11, 2018

In the account’s usual citationless factoid style, the Elves state that you’re more likely to be crushed by a meteor than to win the jackpot on the lottery.

The replies to this tweet were mainly along the lines of this one from my internet acquaintance Chris Mingay:

Should we not be getting almost weekly stories of people being crushed by a meteor then ?

— Chris Mingay (@GhostMutt) January 11, 2018

Yeah, why don’t we hear about people being squished by interplanetary rocks all the time? I’d tune in to that!

A couple of other helpful sorts have provided some extra data as context for this fact:

I asked on Twitter if any turbonerds keep a record of every jackpot ever, and of course they do: Peter Rowlett and Tim Stirrup both provided me with a link to Richard K. Lloyd’s comprehensive table, which reckons there have been 4749 winners, of which 3220 became millionaires.

4750 people have ever won the lottery (for a definition of ‘won’ that might not be the one we want, but it gives us an order of magnitude)

According to their website,4750 people have become millionaires since 1994 from UK lotto wins, so how many have been crushed?

— ste-b (@worldwarste) January 11, 2018

And only one person ever has been crushed by a meteor:

How can this possibly be true when only one person has ever been hit by a meteor in recorded history?

— Dan (@dev_meltus) January 11, 2018

I immediately hit the

(Hey, QI like to do it to their guests, so why can’t I?)

The statement sounds wildly incorrect on first inspection, so I reckon we’re not talking about the same kinds of odds.

It must be the case that:

- someone has worked out the odds of being killed by a meteor,
- someone has worked out the odds of winning the lottery, and
- someone has compared those two numbers.

I assume at least the first two someones were not QI Elves, and I reckon the third one probably wasn’t either. So, where did QI get their fact?

A search for “meteor lottery odds” got me this story on independent.co.uk published five days before QI’s tweet, so that’s probably their source. That links to “Review Journal”, a generically authoritative-sounding title, which turns out to be the Las Vegas Review Journal, who in 2015 published an article by someone affiliated with gobankingrates.com titled “20 things more likely to happen to you than winning the lottery”. That cheery listicle cites a 2008 article by Phil Plait on his Bad Astronomy blog where he cites Alan Harris’ answer to the Fermi question of working out your odds of being killed by a meteor, directly or indirectly. The “crushed” phrasing, which is a stronger statement than the one Harris looked at, seems to originate with the Las Vegas Review Journal. Maddeningly, Plait doesn’t give a citation for Alan Harris’s calculation and I can’t find a better source on Google, so the search stops here.

After all of that chasing, I’ve got a kind-of reputable source for the “1 in 700,000” odds of being killed by a meteor presented in the Independent article. That’s much better odds than the 1 in 45,057,474 chance of winning the lottery claimed by operators Camelot. We hear about people winning the lottery fairly often, so why isn’t “meteor squish” a journalistic cliché like “bus plunge”?

Well, the meteor figure is your *lifetime* odds of being killed, and the lottery figure is your odds of winning *each time you play*. That’s it – they’re measured in different units, effectively. Plait’s *Bad Astronomy* piece contained a good explanation of what the odds meant, but that got lost when the headline figure was spread in factoid form.

So we can’t compare the two numbers as stated – that’d be like me saying I’m taller than you are old. What can we do to get numbers for meteors and lotteries that we *can* compare?

One option is to assume both take place once – a meteor hits Earth, and you play the lottery. We know the odds of winning the lottery in one attempt, and one of Harris’s assumptions in his model was that an asteroid impact would kill everyone – so your probability of being killed is 1. No contest – you’re way more likely to be killed by an asteroid that hits Earth than for the lottery ticket you just bought to be a winner.

A more reasonable approach might be to look at your odds within a certain period of time. We’ve already got a figure of around 1 in 700,000 for being killed by a meteor in a 70-year lifespan, so we just need to get the corresponding figure – what are your odds of winning the National Lottery at least once in your lifetime?

Clearly, it depends on how often you play. My personal odds are zero – I’ve never so much as bought a scratchcard. Conversely, if you buy enough tickets, you can guarantee you win, a tactic executed to great success by Voltaire and later on some MIT students. Those strategies both relied on oversights in the rules of their respective lotteries to make them profitable, but if you’ve got a fortune to spare you could buy a National Lottery ticket corresponding to each combination of six balls and guarantee that exactly one of them will win.

For the sake of getting a reasonable number, let’s say you buy one ticket for each draw. There are two draws each week, so 104 draws each year. So your odds of winning the lottery at least once in 70 years is

At this point I wanted to use the fact that you can only play the lottery once you’re 16, and the life expectancy in the UK is 81.2 years, but I’ll stick with 70 years of playing so we can compare with the meteor number.

\[ 1 – \left( 1 – \frac{1}{45057474} \right)^{(104 \times 70)} \approx 1 \text{ in } 6190 \]

That’s a way, way lower number than the meteor number. So you’re vastly more likely to win the lottery in your lifetime than you are to be killed by a world-ending meteor – over 100 times more likely, in fact.

And if a meteor did kill everyone, you’d be unlikely to read about it in the news the next day.

]]>The results of the Royal Statistical Society’s Statistic of the Year competition, which we covered here when they announced a call for entries, have been announced. The winners include a UK Statistic of the Year (on the density of building in the UK – it’s lower than you think), an International Statistic of the Year (comparing the risk of a US citizen being killed by a terrorist or a lawnmower -guess which one is 34.5 times more dangerous), and five ‘highly commended’ entries.

This meant that stats from various different topics could be recognised, ranging from environmental issues, housing and teen pregnancy, to technological advances and the fundamentals of scientific research. Happily, the story was picked up by a couple of the papers, hopefully raising awareness of some of this (actually quite interesting and important) actual maths.

The full list of results, on the RSS website

Best statistics of 2017 are 69 and 0.1, at The Financial Times

The next London Mathematical Society Mary Cartwright Lecture, presented by Carola-Bibiane Schönlieb (University of Cambridge), will be held on Friday 2 March 2018 at De Morgan House in London. The lecture will be followed by a wine reception and dinner.

Organised annually by the LMS’s Women in Mathematics committee, the lecture is named for the British mathematician and chaos theorist. The event features two speakers – a warm-up from Andrea Bertozzi (UCLA) on Geometric graph-based methods for high dimensional data, followed by the Mary Cartrwight Lecture itself, given by Carola Bibiane-Schönlieb (University of Cambridge) on Model-based learning in imaging.

More information and how to register

We (Katie, Peter and Christian, and briefly Paul) were guests on one episode of Samuel Hansen’s Relatively Prime podcast just before Christmas – on one of our favourite hobby-horses, ridiculous made-up formulae in the press. We explained how the UK press does it better than anyone to Samuel’s US-based audience, then went on to tear a few of them apart. If you won’t be made sad by being reminded how it’s not Christmas any more, you can listen to the podcast through the RelPrime Website or on iTunes.

Formulaic Perfection, on RelPrime.com

]]>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.

]]>We now know 50 Mersenne primes! The latest indivisible mammoth, $2^{77,232,917}-1$, was discovered by Great Internet Mersenne Prime Search user Jonathan Pace on the 26th of December 2017. As well as being the biggest Mersenne prime ever known, it’s also the biggest prime of any sort discovered to date.

GIMPS works by distributing the job of checking candidate numbers for primality to computers running the software around the world. It took over six days of computing to prove that this number is prime, which has since been verified on four other systems.

Pace, a 51-year old Electrical Engineer from Tennessee, has been running the GIMPS software to look for primes for over 14 years, and has been rewarded with a \$3,000 prize. When a prime with over 100 million digits is found, the discoverer will earn a \$50,000 prize. That probably won’t be for quite a while: this new prime has $23{,}249{,}425$ decimal digits, just under a million more than the previous biggest prime, discovered in 2016.

If you’re really interested, the entire decimal representation of the number can be found in a 10MB ZIP file hosted at mersenne.org. Spoiler: it begins with a 4.

**More information: **press release at mersenne.org, home of the Great Internet Mersenne Prime Search.

This recording came about when I was working the *Codebreakers and Groundbreakers* exhibition for The Fitzwilliam Museum, Cambridge. The exhibition focuses on two codebreakers, the first being Michael Ventris, an architect and linguist who in 1952 deciphered the ancient Greek script known as Linear B. This breakthrough added about 500 years to our knowledge of ancient Greek history and is considered one of the great advances in classical scholarships.

The second codebreaker featured in the Fitzwilliam exhibition is Alan Turing, one of the leading British codebreakers at Bletchley Park who broke the German code machine Enigma during World War II. This gang of codebreakers was a mix of mathematicians, linguists, and puzzle-solvers. One of our aims for the exhibition is to show how this collaboration and sharing of skills made breaking Enigma possible.

The exhibition will run until the beginning of February 2018, and features unique Linear B clay tablets from the palace at Knossos, Crete; a rare U-boat Enigma machine and British TypeX machine both on loan from GCHQ; as well as various archival material from Michael Ventris and Alan Turing.

The exhibition also contains audio extracts from a BBC broadcast from Michael Ventris. So it was disappointing that we could not feature an equivalent audio extract from Alan Turing. However, the Turing archives at King’s College, Cambridge do contain Turing’s script for *Can Computer’s Think?* This script is available online via the Turing Digital Archive.

I was surprised how good Turing’s script was. Instead of the dense, academic language I feared, the script was a clear and simple explanation of the future of computing written for the layman – with perhaps the exception of the long desert island analogy, which I did not get on with at all. The second thing that surprised me was, for a script written in 1951, how current it all seemed. Turing is often described as ahead of his time, and the evidence is right here.

Turing talks about the computer’s ability to imitate any kind of calculating machine, a property known as universality, and considers if a machine will ever be able to imitate a brain. Turing then goes on to discuss whether a computer is capable of originality, or indeed free-will.

Turing then makes one firm prediction, that by the end of the 20th century computers would be able to answer questions in a manner indistinguishable from a human being – this is the famous Turing test. Turing’s prediction may have been a couple of decades early, but with the rise of digital assistants I would have to say he was completely right.

I was fascinated by Turing’s prediction of how unsettling it would be to design machines to look like people, an effect we now call the Uncanny Valley. Turing also saw no limits to what a computer would be able to achieve, and saw the future of programming to be closer to that of teaching, which is what we see today in the areas of Deep Learning and Neural Networks.

Throughout the lecture, Turing’s language is friendly and inclusive. He is also charmingly humble, admitting there are many other opinions and that these were just his own. I was also pleased to see Turing acknowledge the legacy of computing with a quotation from Ada Lovelace speaking about Charles Babbage’s Analytical Engine. I found the script so pleasing that I decided it would be nice to rerecord it so that new audiences would be able to hear Turing’s words as he intended.

Allan Jones has written more on this series of broadcasts by the BBC.

The *Codebreakers and Groundbreakers* exhibition is open at the Fitzwilliam Museum, Cambridge until Sunday the 4th of February 2018.

A transcript of the broadcast (PDF) has been submitted by Lewis Baxter.

]]>