Maths on the Back of an Envelope: Clever Ways to (Roughly) Calculate Anything

by Rob Eastaway

  Often, if you’re looking to forecast what’s going to happen in the near future, a straight-line extrapolation from the past is a good starting point. Here, for example, is a chart showing the proportion of all grocery spending that was done online from 2013 (2.1%) to 2017 (4.8%). Care to guess what happened in 2018?10

  The online market was clearly growing each year, though not by a fixed amount. The annual growth was just 0.4% in 2014, and for the next three years it was in the range 0.6% to 0.8%. A guess of 0.7% growth seems sensible for 2017–18. Of course it might be as high as 1% or as low as 0.3%, or it might do something cataclysmic, we can’t be sure, but like steering an oil tanker, statistics that are heading steadily in one direction can take a lot of effort to shift to a different path. So a prediction of 5.5% (ish) for 2018 is a relatively safe bet – and as it happens, 5.5% is exactly what the growth was that year. But I’ve been a bit lucky here – informed guesses aren’t always as accurate as this.

  The further into the future you are trying to forecast, the more risky it is to extrapolate a trend from the data you have. And be careful: while the grocery purchasing habits of the USA are clearly revealing a changing pattern of consumer behaviour, it’s possible for an ‘upward trend’ to happen purely at random. If I repeatedly toss 10 coins and get four heads the first time, five heads the second time and six heads the third, it might look like an upward trend, but the stats say that the most likely outcome next time is going to be five.

  Finally, buried within a long-term trend, there might well be short periods where the data moves in the opposite direction. Some climate-change deniers liked to cherry-pick the period 2001–13 to ‘prove’ that global warming had stopped. Over that time-frame, the average global temperatures moved up and down with no obvious trend. Zoom out to look at the data over a century, however, and the evidence of an upward trend is compelling. That in itself is not proof, of course. But most scientists prefer to look at the long-term statistics, not short-term blips.

  PREMIER LEAGUE GOALS: CERTAINTY WITHIN UNCERTAINTY

  Here’s a prediction. Next season, 1,000 goals will be scored in the Premier League.

  OK, it will probably be a handful more than that, but that figure is probably right to within 5%, which is astonishingly accurate by back-of-envelope standards.

  How come this prediction is such a conveniently round number? Mainly it’s a coincidence, but our confidence in it comes from history. If you look at all the seasons since 1995/6 (when the league settled at 20 teams), the highest number of goals scored was 1,066 in 2011/12, and the lowest was 931 in 2006/7. In six of the nine seasons between 2009 and 2018, the total number of goals was in the range 1,052 to 1,066.

  In total there are 380 Premier League matches played in a season, with an average of about 2.6 goals per match. This means that were the league for some reason to change the number of teams up or down, we could still make a decent estimate of how many goals there would be. Increase the league from 20 to 22 teams, and we’d now have 462 matches. That’s roughly a 20% increase in the number of games, so we’d expect about 1,200 goals.

  If we increase a league to 24 teams, there are 552 matches; that’s almost a 50% increase on the Premier League, so we might expect, say 1,450 goals. As it happens, the second division, known as The Championship, does have 24 teams. And sure enough, the average number of goals per season is roughly 1,450.

  It might seem remarkable that a game that is popular for its drama and unpredictability can be so surprisingly predictable when you look at the big picture – but that’s not unusual in statistics.

  4

  FIGURING WITH FERMI

  THE FERMI APPROACH

  The man often credited as the guru of back-of-envelope calculations was Enrico Fermi. Fermi was a physicist who was involved in the creation of the very first nuclear reactor. Most famously, he was present at the detonation of the first nuclear bomb, the so-called Trinity Test in New Mexico, USA, in July 1945. At the time, scientists still weren’t sure how big the explosion would be. Some even feared it might be large enough to set off a chain reaction that would destroy the planet.

  The story goes that Fermi and others were sheltering from the explosion in a bunker about six miles from ground zero. When the bomb went off, Fermi waited until the wind from the explosion reached the bunker. He stood up and released some confetti from his hand, and when it had landed, he paced out how far the confetti had travelled. He then used that information to make an estimate of the strength of the explosion. We don't know for certain how Fermi did this, but it probably involved him estimating the wind speed and working out how much energy was required to push out a ‘hemisphere’ of air from the centre of the explosion.

  Fermi’s estimate of the bomb’s strength was 10 kilotons. Later, more rigorous calculations revealed that the real strength had been nearer to 18 kilotons, in other words Fermi’s answer was out by a factor of nearly two. Anyone submitting an answer that far out in a maths exam would probably get no marks, yet Fermi got huge credit for the accuracy of his back-of-envelope answer. The important thing was that his answer was in the right order of magnitude, and gave scientists a much better understanding of the potential impact of the weapon they were now dealing with.

  What was so impressive about Fermi’s calculation was that he did it based on such crude data. His results also show that being in the right ballpark can sometimes mean being quite a long way from the exact right answer, and that an ‘inaccurate’ answer can still be useful.

  Calculations that are done without access to much proper data have become known as ‘Fermi problems’. These calculations are typically done as an intellectual exercise: working things out for the sake of it.

  There is, however, a practical benefit to honing your skills at solving Fermi problems, because this type of problem is notoriously used in interviews. I can still remember one of the questions I was asked at my university entrance interview: ‘What does an Egyptian pyramid weigh?’

  One way to figure this out would be:

  (1) estimate the dimensions of the pyramid and hence its volume;

  (2) estimate the density of stone (in kilograms per cubic metre) and work out its volume;

  (3) multiply the volume by the density to get a figure for the weight.

  I doubt my interviewers were interested in the answer, not least because nobody has ever weighed an Egyptian pyramid. What the interviewers were looking for was the thinking process. I don’t remember what answer I gave, but it can’t have been too bad because I was offered a place.

  For many employers it’s the same: companies like Google and Microsoft are famed for the (Fermi) questions they have posed to would-be employees, to see how they think on their feet.

  Whether you’re practising for interviews, or are just exercising your brain for the fun of it, Fermi problems are an excellent work-out. In this section I have come up with a selection of Fermi problems that captured my imagination. In each case, I’ve shown the approach I would take when tackling them. You might well take a different approach. The only thing you and I can guarantee is that we are very unlikely to come up with the same answer, though with any luck we will at least both end up in the same ballpark.

  COUNTING

  When embarking on Fermi-style estimations, a good place to start is counting. Answering the question ‘How many …?’ is the most primitive mathematical challenge – though it often turns out to require a surprising level of skill, and is sometimes highly contentious too.

  Who can forget Donald Trump’s anger when his own view that he had ‘probably the biggest crowds ever’ at his inauguration was contradicted by several sources who suggested the crowd was a lot smaller than his predecessor’s?

  We’ve already seen examples of where even meticulous attempts to count exactly the right number often miss the mark (see vote-counting in elections here, for example). Fortunately, there are many situations where only a decent estimate is required. Here are some examples.

&n
bsp; HOW MANY WORDS ARE THERE IN YOUR FAVOURITE BOOK?

  Publishers like to do word counts – sometimes they even pay a rate per word. So how many words are there in a manuscript, or a book you pluck off the shelf?

  There is, of course, in most word-processing software, a button that can be clicked to find the precise answer to this question: but that’s only for the electronic version of the book. For the paper version, you can either laboriously count each word or – more practically – make an estimate.

  Word counting lends itself to the technique of taking a sample and then extrapolating that to estimate the whole book: word length and density tends to be pretty consistent through most books, so if you turn to a random page of full text somewhere in the middle of the book, that’s likely to be a good representation of every page.

  How many words are there in, say, Pride and Prejudice?

  The ultimate cavalier approach is to count the words in one line and base the entire estimate on that. Twelve words in the line, 38 lines on the page, 345 pages, using Zequals that’s

  10 × 40 × 300 = 120,000 h 100,000 words.

  But it’s not much effort to take a bigger sample and come up with a more accurate estimate. Even counting three lines of text to get an average will improve the estimate considerably. If the word count over three lines is 34, that’s an average of about 11 words per line. And with the indents, sentences that end in the middle of the page, and half-pages at the end of the forty-odd chapters, we might say there’s the equivalent of about 36 full lines of text per page, and about 320 pages.

  The approximate answer hasn’t changed: 11 × 36 × 320 h 10 × 40 × 300 = 120,000 h 100,000. But these better estimates of words per page and number of pages justify a more precise estimate: 11 × 36 ∼ 400, and 400 × 320 = 128,000.

  Impressively, but perhaps fortuitously, it turns out that this estimate is remarkably close to the official word count of 122,000.

  Working out the word count in the book you are reading now is a bit trickier – tables, numbers, diagrams and in-fill boxes make things a bit more complicated. But you can still make reasonable estimates to come up with a figure.

  HOW MANY HAIRS ARE THERE ON AN ADULT HUMAN’S HEAD?

  The number of hairs on a person’s head will, of course, vary hugely from person to person. Let’s try to figure out a number for somebody with a full head of hair, say. And let’s do it just using the imagination, and without looking at a scalp.

  Picture (if you can) a square centimetre of scalp on a human head.

  How far apart are the follicles? We can start by putting some upper and lower bounds on it. If hairs were 2 mm apart then the hair wouldn’t be dense at all, and you’d think we’d be able to see the scalp shining through. But if hairs are as little as 0.5 mm apart, the scalp would be so densely packed, it would be more like fur. So let’s go for 1 mm as a reasonable compromise.

  That means that within each square centimetre of scalp, we’re looking at 10 × 10 = 100 hair follicles.

  What’s the surface area of a scalp? Put your hands around your head, fingers at the top of your forehead, thumbs on your neck at the base of the scalp – call that a circle maybe up to 25 cm (10 inches) across.

  The area of a circle with a diameter of 25 cm is: π × radius2, and the radius in this case is 12.5 cm. But this is all so rough and ready that Zequals is called for:

  3.14 × 12.52 h 3 × 102 = 300 cm2.

  The surface area of the scalp isn’t a circle, it’s more like a hemisphere. For the same radius there will be more surface area on a hemisphere than on a circle, so let’s double it to 600 cm2.

  The total hairs on the typical head is therefore, according to this estimate, 600 × 100 = 60,000.

  That will vary a lot – the number probably going up to 100,000 at most, and down to 30,000 at least (if a receding hairline or thinning hasn’t set in).

  The knowledge that a person has, at most, 100,000 hairs on their head means you can state with absolute certainty that there are at least two people in (say) Huddersfield1 who have exactly the same number of hairs on their head as each other.

  The proof is found by first supposing that everyone in Huddersfield has a different number of hairs on their head. We know that the most hairs anyone has is going to be around 100,000. So let’s suppose that everyone else has fewer hairs, and that they all have a different number of hairs on their heads. Imagine now lining them up, starting with the person with no hair, then the one with one hair, two hairs etc.

  We have 100,000 people with different numbers of hairs on their head. But what do we do with person 100,001? Or indeed with the other 50,000 or more Huddersfield residents? Inevitably, they are going to match with somebody. Hence we have proved our assertion that at least two people in Huddersfield must have the same number of hairs as each other, and indeed that there will be tens of thousands of people whose hair numbers are not unique. This form of proof is known as the pigeonhole principle, and is often used by mathematicans in problems rather more abstract than counting hairs.

  DO MORE PEOPLE GO TO FOOTBALL MATCHES AT THE WEEKEND THAN GO TO CHURCH?

  Which is Britain’s biggest religion, Christianity or football? It’s a question that crops up periodically, especially when new statistics are published showing the decline in church attendance. It’s a hard comparison to make, because quantifying church attendance requires a lot of sweeping assumptions and definitions, including what counts as a ‘church’, and what counts as ‘going to church’.

  Do weddings and funerals count, for example? And how many church weddings and funerals are there?

  Wedding and funeral attendances are an interesting thing to estimate in their own right. There are around 250,000 weddings per year (see where this estimate comes from here). I also know that these days most weddings don’t happen in church. Let’s suppose one-third of them do. That means maybe 80,000 church weddings per year. Let’s say there’s an average of 50 people at a church wedding. That suggests maybe:

  80,000 church weddings per year × 50 per wedding ÷ 52 weeks

  h 80,000 people go to a church wedding each week.

  That’s not much higher than the number attending a Manchester United game at Old Trafford.

  Funerals are more significant: if the UK population is roughly stable and is evenly spread across all age groups, then we’d expect a cohort of 900,000 to die each year. However, most deaths are still among those in the pre-Baby Boomer generations when the population was much smaller, which is one reason why the annual death toll in recent years has been nearer 500,000. But everyone who dies has a funeral, and church funerals are still the most common. So that suggests at least 250,000 church funerals per year, or 5,000 per week, with an average attendance of, I don’t know, maybe 50 people? That would mean a substantial 250,000 funeral attenders in church each week.

  All in all, weddings and funerals account for perhaps 350,000 church attendances per week.

  Whether we ignore weddings and funerals or not, any official statement of church attendance has to be taken with a pinch of salt. In 2018, the Church of England claimed that an average of 750,000 people attended a church service each week, but unlike in a football match, there is no turnstile or pre-paid seating from which a headcount can be determined. There is a tradition in some churches of doing an annual census known as the ‘October count’. But sources tell me that these figures include the occasional vicar scanning the congregation, putting her finger in the air, adding an optimistic 20% and saying ‘I reckon we had 40 in this week, verger.’

  Football crowds are much easier to count, as are the number of matches.2 Over a weekend, 10 Premier League games (35,000 per game?), 12 Championship matches (15,000?), and 24 in Leagues 1 and 2 (10,000?) gives us about 750,000. Of course there are hundreds of other matches too, and rather than count them, we can probably apply something like the 80:20 rule here3 and say that the rest of the matches are going to have a headcount of no more than 200,000.

  So that’s arou
nd a million people attending a football match in England in a typical week – that’s more than the number of people going to a Church of England service. But by the time we add in Catholics, Methodists, Baptists, Pentecostalists and other denominations, that number of Christian church attendees can probably be doubled. So the Church undoubtedly has the edge over football. For the time being, at least.

  HOW MANY TENNIS BALLS ARE USED AT WIMBLEDON?

  Some employers have a reputation for asking curve-ball questions during job interviews, to see how well candidates can think on their feet. The Wimbledon tennis-ball question is attributed to the international consultancy firm Accenture, though it is not clear if this has ever actually been posed in an interview or is just an urban myth. Would an employer really ask a question that requires a reasonable knowledge of tennis? Well, maybe. And in any case, it’s an interesting Fermi question.

  Let’s assume we’re talking purely about Wimbledon fortnight – the men’s and women’s singles, men’s and women’s doubles, and the mixed doubles.

  First of all, how many matches are there?

  Wimbledon is a knockout tournament, and once the main tournament starts, there are no byes. As with all such knockout tournaments, this means that the number of contestants must be a power of 2: there are 2 players in the final, 4 in the semi-final, 8 in the quarters, and then 16, 32, 64, 128 as you go back through to the first round.

  There’s a short cut for working out how many matches there will be in a knockout tournament. Since each match knocks out one player, and the tournament finishes with only one player who hasn’t been knocked out, the number of matches must always be one less than the total number who enter the tournament.