A Year in Numbers E-Book

Kyle D. Evans

• Herausgeber: Allen & Unwin
• Kategorie: Wissenschaft und neue Technologien
• Serie: Kyle D. Evans - maths gift books
• Sprache: Englisch

Seitenzahl: 357

Veröffentlichungsjahr: 2023

Beschreibung

***A Strong Words 2023 Book of the Year*** 'A highly enjoyable and serendipitous read!' - Rob Eastaway, bestselling co-author of Headscratchers: The New Scientist Puzzle Book Did you know: -Only around 100 people have ever lived beyond a million hours (that's about 114 years) -Around 7% of everyone who has ever lived is currently alive -The '12 days of Christmas' song, when sung in full, results in 364 gifts being given - one for every day... except Christmas Broken down into 12 chapters that correspond roughly to months of the year - from going 'back to school' with arithmetic and times tables through prime numbers and all the way to the 12 Days of Chris-maths, this book features a collection of 365 fascinating numerical 'nuggets', accompanied by clear, bite-size explanations of the mathematics that underpin them.

Leseprobe

For Edwin & Juno

First published in the United Kingdom in 2023 by Allen & Unwin, an imprint of Atlantic Books Ltd.

This paperback edition published in 2025 by Allen & Unwin.

Copyright © Kyle D. Evans, 2023

The moral right of Kyle D. Evans to be identified as the author of this work has been asserted by him in accordance with the Copyright, Designs and Patents Act of 1988.

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of both the copyright owner and the above publisher of this book.

No part of this book may be used in any manner in the learning, training or development of generative artificial intelligence technologies (including but not limited to machine learning models and large language models (LLMs)), whether by data scraping, data mining or use in any way to create or form a part of data sets or in any other way.

Every effort has been made to trace or contact all copyright holders.

The publishers will be pleased to make good any omissions or rectify any mistakes brought to their attention at the earliest opportunity.

10 9 8 7 6 5 4 3 2 1

A CIP catalogue record for this book is available from the British Library.

Paperback ISBN: 978 1 83895 893 0

E-book ISBN: 978 1 83895 892 3

Illustrations on pages 13, 28, 55, 56, 64, 133, 162, 168, 197, 199, 200, 224-25, 240, 241, 248, 281, 284, 305 by Hana Ayoob.

The picture acknowledgements on p. 341 constitute an extension of this copyright page.

Printed in Great Britain

Allen & Unwin

An imprint of Atlantic Books Ltd

Ormond House

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www.atlantic-books.co.uk

Product safety EU representative: Authorised Rep Compliance Ltd., Ground Floor, 71 Lower Baggot Street, Dublin, D02 P593, Ireland. www.arccompliance.com

CONTENTS

Cover

Title

Copyright

Contents

Introduction

January: Building Blocks

February: About Time

March: Power Up

April: Harder, Better, Faster, Stronger 82

May: Measure for Measure

June: Going for Gold

July: Prime Time

August: ‘e’asy as Pi

September: When Will I Use This in the Real World?

October: That’s Entertainment!

November: The Chapter that Goes On and On . . .

December: The 31 Days of Christmaths

Afterword

Further Notes

Glossary

Picture Credits

Acknowledgements

Landmarks

Cover

Contents

Start

INTRODUCTION

Welcome, dear reader. I’m sure you don’t need instructions on how to read a book, but if I may make a few recommendations before you begin.

In my day-to-day life as a maths teacher I am constantly barraged by what I call ‘nuggets’: fascinating little numerical titbits that catch the attention and stick in my mind when I get home and my six-year-old asks me what I learnt at school today. These nuggets may come from students, my fellow teachers or a conversation with a colleague that jogs a memory of something I once knew but was buried in the back of my brain. I have attempted to collect a year’s worth of these little gems in loosely connected themed monthly chapters.

This book is intended to be read a little bit per day, perhaps while brushing your teeth or completing another short morning routine. I appreciate, however, that you may find the book just too compelling to read in such tiny chunks and want to speed on ahead. That’s fine: there’s no real reason it couldn’t be read like any other book.

Sometimes the factoids are so exciting and intriguing that I have had to spin them out over several days. The upshot of this is that if you seek out your birthday / anniversary / the day that you met Lee Latchford-Evans from Steps at Fleet Services, there’s a small chance that you’ll be dumped in the middle of something that’s hard to understand in isolation. My apologies, but I’m sure you’ll work it all out in the long run (actually, the kind of person who’d skip ahead to a certain day would almost certainly not read the introduction anyway).

Some of the maths is a little complicated, but I promise it’s never for the sake of it. In some places where you may be interested in longer explanations, I have expanded in the Further Notes at the back of the book. There’s also a glossary in the back in case you forget any word definitions along the way – glossary words are underlined the first time they appear.

I hope you enjoy this numerical trip around the sun as much as I enjoyed putting it together.

JANUARY

Building Blocks

1 January

Happy New Year! May this year leave you healthier, wealthier and wiser than the last. Speaking of wealth, I’m afraid to report that Amazon founder Jeff Bezos has already earned more than you will this year. Even if you’re actually reading this on 1 January. Actually, even if you started reading this entry at midnight on New Year’s Day (and honestly, why wouldn’t you?) then by the time you’ve reached the end of this sentence Bezos will have earned more than you will this year.

Bezos’ earnings reportedly come in at around 9 billion dollars per month, which equates to about $3500 per second. Some estimates have the figure as low as a measly $2000 per second, but unless I have any billionaires reading my books (give us a tenner?) it’s safe to say that Bezos has already earned more than you will this year. In an entirely unrelated point, around 10% of the world’s population live in extreme poverty – that is, on less than $2 per day.

2 January

In this chapter we’ll lay the foundations for everything that follows, so what better place to start than with 0 and 1? Think of them as the Lennon and McCartney of numbers: in many ways polar opposites, but unarguably the building blocks of everything else that follows. It may therefore be surprising that, much in the same way that the can opener was invented fifty years after canned food, the numeral for 0 was not seen until thousands of years after that for 1.

Cultures in Egypt, Mesopotamia and China, among others, all converged upon the familiar single stroke numeral for the number 1 between 3500 and 1500 bc.

Table showing Egyptian numerals, from Kurt Sethe, A History of Mathematical Notations Vol. I (1928). It includes hieroglyphic, hieratic and demotic numeral symbols. Note the agreement over the numeral for 1 and the shocking lack of zeros anywhere.

Though there were various symbols for the absence of a number across the years – the Sumerians first opting for a slanted double wedge around 5000 years ago – we would not see an empty circle to indicate zero until around the eighth century ad in China. Britain and Europe used Roman numerals up to around ad 900 – there is no zero whatsoever in this numbering system. That must have made people absolutely 51 6 500.*

3 January

In Vegas, I got into a long argument with the man at the roulette wheel over what I considered to be an odd number.

Steven Wright

There should be nothing clearer in mathematics than the designation of odd and even numbers: even numbers are those that can be shared into two equal piles; odd numbers are all the other ones. But what about zero? Can something that is essentially the absence of a number have the property of oddness or evenness? Well, yes – a pile of no sweets can be shared fairly between two people: none each. Also, zero being even preserves the even, odd, even, odd sequence that all other numbers follow. All of this is essentially a preamble to clear a path for the following revelation: every single odd number has an ‘e’ in it. When spelt in English, at least. Don’t believe me? Check a few…

4 January

Many cultures have their own superstitions, including lucky and unlucky numbers. In China, the pronunciation of the number four sounds similar to ‘death’, meaning that four is considered an unlucky number.* This means that parking spaces or the floors of buildings will often go straight from 3 to 5, but it has also had a more unexpected practical downside for the superstitious.

For some time, Beijing has implemented a traffic calming measure that means that only certain cars can use inner city streets on certain days. Cars are split into five equal groups by means of the last digit in their car registration (tail) number, as shown in the following table (correct as of 2019):

Can you see the potential issue? Chinese drivers actively avoid having a 4 in their tail number, to an extent that is quite marked:

Credit: From Anderson et al., ‘Superstitions, Street Traffic and Subjective Well-being’, Journal of Public Economics, vol. 45 (2016), pp. 1–10.

Fewer drivers choosing tail numbers featuring a 4 means that days when 4-plates are allowed to drive have been measured as significantly less busy in terms of traffic. It’s rare to get actual, concrete evidence of how not being superstitious can improve your daily life, but there it is: pick a 4-plate and get to work quicker.

5 January

Here’s a neat and entirely non-superstitious feature of the number 4: if you write out every number in English, ‘four’ is the only number that has as many letters as the size of the number itself.

This leads to a quick and fun word game: pick any number and spell it. Count the letters in the spelling and that is your new number. Spell this number, and keep repeating the process until you eventually end up at 4. Try to form the longest chain you can:

Fifteen (seven letters) → Seven (five letters) → Five (four letters) → Four (four letters)

Seventy-seven (twelve letters) → Twelve (six letters) → Six (three letters) → Three (five letters) → Five (four letters) → Four (four letters)

If English isn’t your language of choice you can play this game just as well in other languages, but you might end up at more than one final destination, or in an endless loop! (This happens in French, where there isn’t a number in which the number of letters in the spelling is the same as the number itself.)

6 January

There are definitely two people in London with exactly the same number of hairs on their heads. Around 9 million people live in London, but there are only about 100,000 hairs on a human head, so if you tried to match all the Londoners to a list of hair numbers you would exhaust the list pretty quickly. In fact, any town or city with a population over 100,000 (Gillingham, Woking, St Helens) has two people with exactly the same number of hairs on their head.

You might think this is obvious – all you have to do is find two bald people in London; perhaps Right Said Fred are playing tonight? So how about this instead:there are definitely two people in London with exactly the same number of hairs on their entire body. The average body has about 5 million hair follicles, so still less than London’s population. This idea is called the pigeonhole principle, which essentially says that if you have more pigeons than holes, you either have to put two pigeons in the same hole or a pigeon is going to go homeless.

7 January

Speaking of baldness, how many hairs would you say you’d have to remove from your head to become bald? 10,000 hairs? 50,000? 100,000? Do you think a person could ever become bald by removing a single hair?

Most people’s answer to this would be: of course not. Hairs fall out all the time; every time you comb or wash your hair you lose a couple. You’d never come out of the shower and find that your spouse no longer recognizes you due to your sudden baldness.

But there lies a paradox within this. Imagine removing a single hair from your head, and then another, and then another. Of course it logically follows that you will eventually not have a single hair on your head; not even a Rab C. Nesbitt or a Homer Simpson wisp to comb over. So we have to accept the contradiction that removing a single hair could never make a person bald, but by repeatedly removing a single hair a person can absolutely go from a full head of hair to completely bald. This is sometimes known as the bald man paradox, and the nub of it lies in our rather woolly definition of what ‘bald’ actually means.

8 January

The letter ‘e’ is by far the most commonly used in the English language, accounting for over 12% of all letter usage.

Graph showing the frequency of use of letters in the English language.

‘e’ is also the most commonly used letter in Spanish, French, German, Italian, Dutch, Swedish, Norwegian, Finnish and Hungarian, but it’s pipped by ‘a’ in Turkish, Icelandic and Slovak and by ‘i’ in Polish.

9 January

One of the most ubiquitous paradoxes in mathematics – thanks largely to Mark Haddon’s excellent novel The Curious Incident of the Dog in the Night-Time – is the Monty Hall problem, in which the contestant on a game show is offered a choice of three doors, behind which lies a prize they will take home: two of the doors have a goat behind them and one hides a car. Contestants pick a door, but rather than being shown what is behind their chosen door they are shown the goat that lies behind one of the doors they didn’t pick. The contestant now faces two closed doors: one of which they have picked, the other they haven’t; one hides a goat, the other a car. The question is: should the contestant stick to the door they chose, or switch to the other door?

Most people will not think that it matters whether you stick or twist, since the odds seem 50/50: two doors, one good outcome. For this reason most people tend to stick to their original choice, not wanting to switch and then lose – a bit like the worry of changing your lottery numbers on the week that the jackpot comes in on your old numbers. But you should always switch! Yes, it’s counter-intuitive and frustrating, but it’s correct.

The best way to visualize this is probably to imagine six people who play the game; we’ll label them players A–F. Players A and B choose door 1, C and D choose door 2, E and F choose door 3. We’ll say that the car is behind door 1, though of course it makes no difference to the rest of our calculations. So, as things stand, players A and B are in a winning position and everyone else is taking a goat home.*

Now, at the next step, half of the players will keep their door and half will change. So A stays at door 1 and wins a car, B switches to another door and gets goat; C and E stick with their original door and get goat; D and F switch from goat to car. Pulling this together into a table:

The crucial thing now is to look at the outcomes for the stickers or switchers: the stickers get the goat two times out of three, but the switchers get the car two times out of three. It is always better to switch.

The final choice of goat or car is indeed 50/50, but you’re more likely to already have chosen a goat! So on average you’re better to switch, as two times out of three you’ll be improving your lot.

You’re still not convinced, are you?

10 January

All of the numbers used in the book so far are confined to the world of positive whole numbers (or positive integers), but of course negative numbers are equally valid numbers and their existence helps us to incorporate concepts such as debt into mathematical calculations. The best way to think about negative numbers is to visualize a thermometer with hot and cold temperatures: –8 is less than –3 because –8 °C is colder than –3 °C, for example. This is a concept that most children are taught at the age of 11 or 12, but just because they’re taught it doesn’t necessarily mean they learn it.

In 2007 in the UK, a National Lottery scratchcard based on negative numbers and temperatures had to be withdrawn from sale due to a large number of players not understanding whether or not they had won. The ‘Cool Cash’ card required players to scratch off a number – representing a temperature – and if it was lower (or colder, if you will) than the number printed on the card then the player was a winner. Since the card had a winter theme the temperatures involved were usually negative, leading to much confusion. The following marvellous quote was given willingly to the Manchester Evening News at the time, though I will protect the identity of the person who gave it:

I phoned Camelot and they fobbed me off with some story that –6 is higher, not lower, than –8, but I’m not having it. I think Camelot are giving people the wrong impression – the card doesn’t say to look for a colder or warmer temperature, it says to look for a higher or lower number. Six is a lower number than 8. Imagine how many people have been misled.

11 January

On the subject of chilly temperatures, the only temperature that’s the same in both the Fahrenheit and Celsius/Centigrade scales* is –40 degrees. The formula for converting from Centigrade to Fahrenheit is to multiply by 1.8 (or ) and then add 32 (or to use my Dad’s shorthand, ‘double it and add 30’, which is much easier mentally). This means a little equation can be formed and solved:

So if ever it’s –40 degrees, in either scale, and someone asks you what the temperature is, you can proudly respond ‘forty under’ without having to worry about if they’re from the United States, the Bahamas, the Cayman Islands, Liberia, Palau, the Federated States of Micronesia, the Marshall Islands or any other country that primarily uses Fahrenheit. Oh, hang on, that’s an exhaustive list. Maybe those countries could just use the scale the rest of the world uses instead?

12 January

Many people’s strongest recollection of mathematics lessons in school will be learning or even reciting multiplication tables. They may have stuck for life, or you may have some grey areas around the trickier products, but everyone did them. I personally have worked with several mathematicians who have a secure understanding of the most abstract and esoteric mathematical concepts, but who are unsure whether seven 8s are 52 or 54.*

Speaking of which, the answer to 7 × 8 seems to have taken on a certain folklore status in the British press and media, largely due to its regular use as a potential banana skin for politicians under questioning. In 1998 the Labour MP Stephen Byers gave the answer of ‘54’ when asked this question in a radio interview on school standards. At the time the British prime minister, Tony Blair, responded that he ‘applaud[ed] anything which gets up in lights the issues we are seeking to promote’ and that ‘it is one of those character-forming events’, all of which seems perfectly reasonable. George Osborne had clearly been paying attention though, as by the time the then chancellor was asked the same question by seven-year-old Samuel Raddings on a Sky News youth panel event in 2014, he refused to answer, stating that he ‘made it a rule in life not to answer a whole load of maths questions’, a response indicative of the determination of a certain type of modern politician to put not getting caught out ahead of all else.

13 January

Some times tables are clearly easier than others, but I think we’d all agree that 2, 5, 10 and 11 are fairly easy numbers to multiply by. Oh, and 526,315,789,473,684,210. I’m serious! This very special number is incredibly easy to multiply by numbers up to 18. From this point on I’m going to refer to 526,315,789,473,684,210 as ‘the beast’, just to save on typing.

Notice that the beast has two of every digit from 1 to 8, as well as a single 0 and 9. If you want to multiply it by any number from 2 to 9, simply ‘cut’ the beast after the number you’re multiplying by, as long as the neighbour on the other side of the cut is smaller than it. This is easier displayed than explained, so bear with me…

Say you wanted to multiply the beast by 7. There are two 7s in the beast; one followed by an 8 and the other followed by a 3. So we cut between the 7 and the 3. From our cut, we read through the beast, returning to the start when we get to the end, and adding a zero to the end as a cherry on top:

Similarly, to multiply the beast by 5, the options are to cut between 5 and 2 or 5 and 7. We choose the lower option, which is to cut between 5 and 2, and then read off the rest of the number as before, adding the final zero:

14 January

If, like many people, you’ve often struggled with seven 8s, you may find it useful to realize that the first eight numerals form a coincidental and useful pattern:

15 January

The 9 times table is without doubt the most enjoyable times table, because of the very satisfying embedded rules:

The two digits in the solution always add up to 9, and the first digit increases by 1 as the second digit decreases by 1. Also, the first digit of the answer is 1 less than the number being multiplied by, so if you want to know seven 9s, you know the first digit of your answer must be 1 less than 7, i.e. 6, and the other digit must be whatever makes the sum to 9, that is, 3. So you have your answer: 63.

Unfortunately schoolchildren don’t go as far as their 1089 times tables, but if they did they would discover some even more impressive symmetry:

Not only do the digits always add up to 18, but the first two digits rise by 1 every time, while the other two drop by 1 every time. And, like the 9 times table, each answer’s mirror image can be found elsewhere in the list.

16 January

The fact that the numerals ‘6’ and ‘9’ have rotational symmetry (one is the other upside down) is a narrative device that has often been exploited: it provides a pivotal twist in two award-winning BBC shows – Line of Duty and Numberblocks – when a fallen ‘9’ on a front door leads to the house number being misread and a consequent case of mistaken identity.*

Without this rotational property it would be impossible to construct a ‘cubic calendar’. You’ll almost certainly have encountered these – perhaps you even own one?

This is a calendar where each face displays a number, and the order of the two cubes can be swapped when required. There are twelve faces available to us and only ten different digits that need to be displayed, so it should be easy, right? Except each block needs a 1 and a 2, or we couldn’t display the 11th or 22nd of the month. We also need a 0 on each block: if there were only a 0 on one block then we could only make six of the nine required dates that involve a 0.

This creates a problem: we require space for thirteen digits with only twelve faces available to us. But since 6 and 9 are never needed at the same time, we can have a 6 on one cube with no need for a 9 on the other. Here’s a possible arrangement:

17 January

Anyone in the Western world who is familiar with meaningless superstition will be aware of the supposedly unlucky nature of Friday 13th, and if you regularly frequent pub quizzes (or read the footnote on page 12) you may even be aware that the fear of the number 13 is known as triskaidekaphobia.

The origin for the superstition over the number 13 may originate from a Norse myth about Loki being the thirteenth guest at a dinner party, or from the thirteen diners present at Jesus’s Last Supper. Personally, the thought of thirteen people eating in my house fills me with terror, but for the very rational and non-superstitious reason of having to maintain all that chit-chat and do the washing-up afterwards. Particular fear of Friday 13th may originate from T. W. Lawson’s 1907 novel Friday the Thirteenth, but it’s hard to find any solid evidence of why Fridays in particular should be unlucky.

Italians would find Friday 17th to be an unluckier day than Friday 13th, since 17 is deemed to be a bad omen. This is because the Roman numerals for 17, XVII, can be rearranged as VIXI, meaning ‘I have lived.’

18 January

The footballer Paddy Kenny felt the brunt of the Italian fear of the number 17 in 2014 when he was unceremoniously sent home from Leeds United’s pre-season tour – and later sacked by the club – for having the audacity to be born on 17 May. Leeds’s notoriously hotheaded chairman at the time, Massimo Cellino, was terrified of the number 17 and believed that having any player born on that date would be a bad omen for the club.

Cellino left Leeds United in 2017 (of course). They were promoted to the Premier League three years later under new ownership.

19 January

Every integer (whole number) can be written as a sum of powers of 2 in one unique way. We’ll cover powers more in March, but for now you just need to know that the powers of 2 are simply a doubling sequence starting from 1, so: 1, 2, 4, 8, 16, 32, 64, and so on.

In fact, when you represent a number in this form you are actually converting it to binary, the counting system of 0s and 1s that computers use.

So the binary representation of 18 is 10010 (no need to include the zeros to the left of the first 1), and similarly 31 and 100 in binary are 11111 and 1100100 respectively.

20 January

Next time you’re in St Gallen, Switzerland, do pop along and check out the huge binary clock at the train station:

A reminder from yesterday: the rightmost column represents 1, then the column one to the left represents 2, the next column to the left 4, and so on. If there is a symbol in that column then that value is ‘on’; if there isn’t, it’s ‘off’. Add up all the ‘on’ values to convert from binary to decimal.

So the time displayed here in hours, minutes and seconds is (in binary) 1001:11001:101110, and:

So the time pictured is 09:25:46, but by the time you’ve worked that out you’ve probably missed your train.

21 January

Utilizing binary equivalents in a clever way reveals that we can multiply two numbers together by simply halving one and doubling the other. If your times tables are as weak as a British MP’s then this might be the method for you. For example, let’s say we wanted to multiply 18 by 35. Put them alongside each other, and halve the smaller number until you get to 1. (If halving means you hit a decimal number, round it down; so, instead of 6.5, write 6.) Every time you halve the smaller number, double the larger number. So we get something like this:

Then cross out any rows with an even number in the halving column, and add up the remaining numbers in the doubling column:

22 January

How on earth does Russian peasant multiplication, which we saw yesterday, work? Consider the halving sequence that takes you from 18 down to 1, and at each step consider whether there is a remainder when dividing by 2:

If you now read the bold remainders from bottom to top, you will find the binary equivalent of 18: 10010. This means that 18 can be written as 16 + 2, as we saw earlier. Therefore

which is exactly the calculation that we were led to with the ‘Russian peasant’ method.

23 January

This 23rd day of the year feels an appropriate time to announce that any room with 23 people in it has a 50/50 chance of two people sharing a birthday.* This legitimately mind-bending idea is often referred to as the birthday problem and I highly recommend you test it next time you find yourself in a room with 23 people in it (though you may draw strange glances in the greengrocer’s queue).

The most commonly guessed (wrong) answer to the question of how many people would be needed before there was a 50/50 chance of two of them sharing a birthday is 182ish, in other words approximately one person for every other day of the year. But with over 180 people in a room the chances that they all have a unique birthday is absolutely minuscule, because there are so many potential pairings between all of the people in the room.

The key to understanding this problem is not to think of the number of people in the room, but rather the number of potential ‘meetings’ (shared birthdays) that could occur. Imagine one person alone in a room, who is then joined by one more. They shake hands and check whether they have the same birthday; a tiny 1 in 365 chance (ish… damn that pesky leap year). Then another person joins, and crucially there are now two people with whom they can check birthdays. When the fourth person joins that’s three more potential checks, and for every person that joins we are adding more potential shared birthdays.

These complete graphs for one through to five people show the potential numbers of meetings: 0, 1, 3, 6, 10. The number of potential meetings is growing quickly, and this is why it takes fewer people than you might expect to reach a common birthday.

To calculate the precise solution to this problem we should consider the probability of not finding anyone with the same birthday. When the second person enters the room there is a chance that they have a different birthday to the person already in the room (in other words a probability very close to 1). When the next person enters, there are now only 364 birthdays they could have that wouldn’t clash with the others already in the room. So the probability of reaching this point with no birthday clashes is : still very close to 1. Repeat this process, however, and you’ll find that adding the 23rd person tips the probability past 50%.

24 January

A room with 23 people will give you a 50% chance that two people share a birthday, but if you don’t fancy those odds, perhaps wait for a roomful of 40 people: this would garner nearly a 90% chance of a joint birthday. Fifty people takes you all the way to a 97% chance of success (but by the time you’ve checked all 50 people, many of them will have lost interest).

Graph to show the probability of finding a birthday pair for different numbers of people in a room; 23 people gives a 50% chance.

And once you’re up to 88 people, you have a better than 50% chance that three people are all born on the same day.

25 January

Since a group of 23 people have a 50% chance of sharing a birthday with someone in the group, every other football match should feature a common birthday somewhere on the field (11 players on each team, plus the referee).

As an example, let us consider England’s two greatest footballing successes: winning the 1966 men’s World Cup and the 2021 women’s European Championships. In the former game, nobody who started the match shared a birthday, but in the latter there was a birthday pair: England strikers Beth Mead and Ellen White were both born on 9 May.

One could reasonably argue, though, that a football match (or indeed any sporting event) is not a fair situation in which to check the birthday problem, since sportspeople are not equally likely to be born on any day of the year. This is due to what the author Malcolm Gladwell calls the ‘Relative Age Effect’, which says that people born towards the start of the academic year in their country of birth are more likely to excel at sport. The thinking goes as follows: children born earlier in the academic year will be the largest and strongest in their early years at school, when proportionally the age difference – and hence difference in size, coordination and balance – between the oldest and youngest in the year is greatest. These children are more likely to excel at a young age due to the size/strength imbalance, which makes them more likely to reach elite teams and receive top-level training, thus exacerbating the talent gap in a continuing cycle. The theory is borne out in the English football leagues: a 2021 survey by casino.co.uk found that 58% of players in the top two divisions of English football were born in the first half of the academic year, with just 18% born in the fourth quarter of the year. Manchester United have a staggering five out of every six players born in the first half of the academic year, though whether this is evidence of the Relative Age Effect producing quality players is up for debate.*

26 January

Here are the powers of 2, written in binary: 1, 10, 100, 1000, and so on. Add these together and you have a long list of 1s: 1111…

Adding another 1 to this number would change all of the 1s to 0s, and require an additional 1 to be added to the far left: 10000…, which is the next power of 2.

So the sum of a list of powers of 2 will always be 1 less than the next power of 2.

27 January

Happy Birthday Charles Lutwidge Dodgson, aka Lewis Carroll! Both a mathematician and writer, Dodgson penned dozens of books, including A Syllabus of Plane Algebraic Geometry, Symbolic Logic Parts I & II, The Fifth Book of Euclid Treated Algebraically, Alice’s Adventures in Wonderland and, of course, An Elementary Treatise on Determinants, With Their Application to Simultaneous Linear Equations and Algebraic Equations.

Legend has it that it was the last of these books that Queen Victoria was sent when she appealed to the author for a copy of some of his other work, having been so charmed by Alice. Dodgson always strenuously denied this story, though, so we had better not use it for today’s fact. Instead, let’s have this: an avid puzzler, Dodgson also created the ‘word ladder’ puzzle, still going strong in newspapers to this day. He called them ‘doublets’ and here’s one that was published in Vanity Fair in 1897 – can you get from ‘head’ to ‘tail’ in five steps? You can change only one letter per line, and every word you form along the way must be a real word in the English language. The answer is in the Further Notes at the back of the book.

28 January

Here are some more unlucky numbers from other cultures:

• 7 in Japan, because the word for seven sounds like ‘torture’

• 26 in India, because many natural disasters and tragedies have occurred on this date (with a large pinch of superstition and numerology thrown in too…)

• 39 in Afghanistan, because it sounds like ‘morda-gow’, meaning dead cow

• 666 in some Christian cultures, as it’s mentioned in the bible as the number of the beast. Ridiculously long names for phobias are always good fun, so here’s another: fear of the number 666 is called Hexakosioihexekontahexaphobia.

29 January

Here is a beautiful ‘complete graph’ showing ten vertices or nodes, which might represent the number of potential meetings between ten people. But how many lines, or edges, are required to join all the vertices? Please don’t try to count them. The answer is given after the diagram.

30 January

These four symbols are synonymous with the musician Ed Sheeran and, to a lesser extent, mathematics. They can be found on the cover of maths textbooks around the world, but one of these symbols is more controversial than the others.

The division symbol, something like this (÷), can often lead to confusion that you wouldn’t necessarily encounter with the other symbols. For example, the following question can give an answer of either 16 or 1, depending on which way the wind is blowing:

16 ÷ 4(2 + 2)

Actually I believe the answer is 16, but more importantly it’s clearly an ambiguous question, largely due to the lack of certainty over how to apply the division operation. If we were to use a horizontal fraction bar, it becomes much clearer what we are trying to ask:

What many people don’t realize is that the division symbol is literally a fraction bar with a blob to signify some unknown value above and below, ‘blob over blob’: ÷

For many this is a life-changing realization to rank alongside first noticing the little arrow on your petrol meter that shows you which side of the car the petrol cap is on, or the first time you’re told that the word ‘helicopter’ is not made up of the parts ‘heli’ and ‘copter’, but rather ‘helico’ meaning to spin, as in ‘helix’, and ‘pter’ meaning wing, as in the start of ‘pterodactyl’.

31 January

Every two-digit number that ends in 9 is equal to the product of its digits added to the sum of its digits. In other words, multiply the digits and add on both digits. Behold!

It always works! The trick here is to realize that the original number, which looks like x9, where x is the digit in the tens column, would more correctly be represented as 10x + 9. This leads to a digits product of 9x and a digits sum of (x + 9). Put this all together and you get:

* That’s LI VI D. But you knew that, right?

* Conversely, the number 5 in Thai sounds like ‘ha’, so 555 is a sort of shorthand for laughing, the equivalent to the detestable Western ‘lol’.

* My students often like the idea of winning the goat, but I fear they would change their minds when they actually had to take it home and put it somewhere.

* Celsius and Centigrade are essentially the same thing now, though at one time Celsius ran the other way, so water froze at 100 degrees and boiled at 0 degrees.

* Just teasing, I know it’s 58… I mean 56!

* It’s worth noting that this comes from a British school, where it is standard that students learn their multiplication tables up to 12 × 12. Many countries stop at 10 × 10; some go on to 15 × 15 or even 20 × 20. I have heard from a Sri Lankan friend that they learnt tables 1 to 12 and 14 to 16 – good news for any 13-fearing triskaidekaphobics. The reason for stopping at 12s seems to stem from pre-decimal British currency in which there were 12 pence in a shilling, so mental multiplication with 12s was useful. Given the bizarre current push for the re-introduction of imperial measures in the UK, this could yet become useful once again…

* Minor apologies for the spoiler, but it happens literally within the first five minutes of Line of Duty, and Numberblocks is a five-minute CBBC show for young schoolchildren.

* I enjoy the thought that someone is certainly reading this entry on the birthday problem on their actual birthday. Happy Birthday to you, if so! And I mean that sincerely.

* I’m writing this rather obvious gag in Spring 2023, so if Manchester United have enjoyed a glorious period of success since then, I will have egg on my face. They haven’t though, have they?