DK HOW TO BE A MATH GENIUS

Complex character Newton was famously
absentminded. He was
Newton was a genius. Not only did once found boiling his
he establish many laws of physics—
he also invented a new branch of math watch with an egg
called calculus (math that studies in his hand!
changing quantities). However,
Newton wasted a lot of time on
alchemy—the search for a recipe
that would turn base metals, such
as lead, into gold. He could also be
an unforgiving man. He had a
lifelong falling-out with British
scientist Robert Hooke about
optics, and a bitter dispute with
German mathematician Leibniz
over who had invented calculus.

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The law of gravity The vast mass of the Sun The gravity Sir Isaac Newton
holds the planets of the of Earth holds us on
Newton studied the work of Italian scientist Solar System in orbit its surface and the In 1696, Newton was appointed warden of
around it Moon in its orbit the Royal Mint, where Britain’s money is
Galileo and German astronomer Johannes made. At that time, coins were made of gold
The force of Moon and silver. People would snip valuable metal
Kepler. Putting their ideas together, Newton gravity decreases from the edges of coins, or make fake coins
out of cheap metal. Newton found ways to
realized that there is a force of attraction, with distance make both of these practices more difficult.
He also tracked down counterfeiters—even
gravity, that extends throughout the disguising himself as one to do so. In 1705,
Queen Anne made Newton a knight in
Universe. The greater the mass of an gratitude for his work at the mint. When the
great scientist died in 1727, he was buried in
object (m), the stronger its gravity (F), Westminster Abbey, London, among
England’s kings and queens.
but the force of gravity decreases with
99
distance (r). He discovered a law for

calculating the force between two objects

of mass (m1 and m2) using G to represent

the constant strength of gravity: F=G mm
12

r²

PROBABILITY

Probability is the branch of math that deals with the What are the chances?
chance that something will happen. Mathematicians
express probability using a number from zero to one. Working out chances is quite simple. First, you
A probability of zero means that something definitely need to count the number of possible outcomes.
won’t happen, while a probability of one means that it The chance of throwing a die and getting a four
definitely will. Anything in-between is something that is one in six (1⁄6), because there are six ways for
may happen and can be calculated as the die to fall, just one of which is the four. The
a fraction or percentage of one. chances of throwing an odd number (1, 3, or 5)
is one in two (3⁄6 = ½) or 50 percent.

How chance adds up

The chance of a tossed coin being a
head is ½ (one in two). The chances of
a head then a tail is ½ x ½ = ¼. The
chances of a head then another head
(which can be written HH to save space)
is also ¼. The chances of three tails
in a row (TTT) is ½ x ½ x ½ = 1⁄8.

1st toss 2nd toss 3rd toss
½H

½H ½T
½ H ½T ½H

½ T ½H ½T
½T ½H

½T
½H

½T

chance: ½ chance: ¼ chance: 1⁄8 Chaos

But don’t risk it! Some things, such as where a
pinball will bounce, are almost
It’s tempting to think that if you have impossible to predict. Each ball you
tossed four heads in a row, the next fire takes a slightly different route.
toss is more likely to be a head. But it’s
equally likely to be a tail: The chance Even the tiniest differences in the
of HHHHT is ½ x ½ x ½ x ½ x ½ = 1⁄32, ball’s starting position and how
and the chance of HHHHH is much you press the flipper or pull
exactly the same.
the spring become magnified into
major changes in direction as the
ball bounces around the table.
This unpredictable behavior is
called “chaotic.”

100

The house always wins ACTIVITY

Ever wondered how casinos make money? They What are the odds?
make sure the chances of winning are stacked in
their favor. Casino games give the “house” (the Sometimes our brain misleads us. We can
casino itself) a statistical edge that means it wins be influenced by things that aren’t really true.
more often than it loses. For example, if you bet on For example, books and blockbuster films lead
a number in a game of roulette, you have a 1 in 36 us to believe that sharks are very dangerous to
chance of winning. But a roulette wheel also has a humans. In reality, however, more people are killed
37th space for zero. This ultimately gives the casino by hippos than sharks. Try putting these causes
an advantage. It will win more games than it loses, of death in order of probability:
since it doesn’t pay anyone if the ball lands on the
zero. It’s this zero that gives the house its “edge.” Computer game exhaustion

The chance of a shuffled pack of Snake bite
cards being in the right order is
less than one in a trillion trillion Hippo attack

trillion trillion trillion. Walking into a lamppost

Predictions Falling down a manhole

Using probability, you can try to predict or forecast things that Playing soccer
are going to happen. For example, imagine you have a bag
containing five red balls, six blue balls, and seven yellow balls. Hit by a falling coconut
What color ball are you most likely to pull out—red, blue, or
yellow? The answer is yellow because there are more yellow Struck by lightning
balls in the bag, so the probability is higher for this color.
Predictions aren’t always correct. You could pull out a red Hit by a meteorite
or blue ball—it’s just less likely to happen.
Shark attack

101

BRAIN GAMES

DISPLAYING A crime tally

DATA It can be tough for a superhero to decide
When you want to know what’s going on which villain to tackle first. A simple tally
in the world, you need the facts—or data. of their evil deeds is a great way to see at
This will often be in the form of a lot of a glance who’s the worst threat to the city.
numbers that don’t tell you much at first, Numero
but present them in the right way and
you’ll get the picture. Here’s the latest Pi Man
data on superhero activity...

Graphic picture 100 There were a record
number of crimes
Using a line graph to plot data
over time—like the number of 80 during a prison break
crimes in your city—it’s easy to
spot those times when villains 60
are in town. And if a superhero
can find a pattern, it makes the 40

job a lot easier! 20

0
Jan Feb Mar Apr May Jun Jul Aug Sept Oct Nov Dec

100 60 supervillains Bar none
were born with

their powers One way to fight tomorrow’s evildoers

80 is to find out where supervillains get

60 Alien powers are the their powers from and use this knowledge
most common kind
Government to beat them. A bar chart like this, where
experiments
Magical item
Gamma radiation
Genetics
Alien
40 the heights of the bars show the number

of supervillains with powers of a certain

20 source, compares them at a glance.

0

102

Name Secret identity Sidekicks Hero or villain Arch-nemesis What’s on the table?
Hero Numero
Math Man Yes Yes Hero None Knowing all the facts about your fellow
Hero None heroes and villains can come in very
Calcutron No No Villain Math Man handy. A simple table of information is
Villain None helpful because it presents multiple
The Human Shape No Yes facts in a clear and effective manner.

Numero Yes No

Pi Man Yes Yes

Arrival of Depletion of Pie in the sky Strength
supervillain powers 40% 17%
Even superheroes can fail
44% sometimes. What went wrong? Speed
Handy pie charts like this one, 8%
Cape in which the area of each slice
malfunction is a fraction of the whole, show
the major areas for concern.
12%

Other 4%

Math Man Calcutron Check it out Intelligence
40%
Organic lifeform Flight Synthetic lifeform If you’re looking to recruit new
superheroes into your legion, and Agility
Super strength Super Immortal you need to know if they’ve got 12%
intelligence X-ray vision what it takes, a simple checklist
of their powers can help you Powers
Wears a cape Doesn’t wear make an informed decision. 23%
a cape
Flight Profile pictogram
Who does what? Super strength
Invisibility When drawing up the profile of
When you put a team of superheroes Telekinesis the perfect superhero, a pictogram
together, you need a good range of Super intelligence is a great way to show the balance
skills. Venn diagrams are an ideal Psychic powers of qualities you’re looking for. It
method of comparing characteristics combines a range of information
and showing which are shared—and within one fact-packed picture.
which aren’t. Here, the depths of the coloured
areas reflect the mix of qualitites.

103

BRAIN GAMES

LOGIC PUZZLES

AND PARADOXES

To solve these puzzles, you need to think
carefully. This area of math is called logic—you
get the answer by working through the problem,
step by step. But watch out, one puzzle here is a
paradox, a statement that seems to be absurd or
to contradict itself.

Logical square Black or white?

Each of the colored squares below contains a Amy, Beth, and Claire are wearing hats, which they know
different hidden number from 1–8. Using the are either black or white. They also know that not all
clues, can you work out which number goes where? three are white. Amy can see Beth and Claire’s hats, Beth
can see Amy and Claire’s, and Claire is blindfolded. Each
is asked in turn if they know the color of their own hat.
The answers are: Amy—no, Beth—no, and Claire—yes.
What color is Claire’s hat, and how does she know?

0 4

9 ?? ?

5 A barber’s dilemma Four digits

7 A village barber cuts the hair of What is the four-digit
6 everybody who doesn’t cut their number in which the
own. But who cuts his hair? first digit is one-third of
• The numbers in the dark blue and dark green the second, the third is
squares add up to 3. • If he does, then he is one the sum of the first and
• The number in the red square is even. of those people who cuts their second, and the last is
• The number in the red square and the number own hair. three times the second?
below it add up to 10. • But he doesn’t cut the hair of
• The number in the light green square is twice the people who cut their own hair. ?? ??
number in the dark green square. So he doesn’t cut his own hair.
• The sum of the numbers in the last column is 11 • But he is the man who cuts the
and their difference is 1. hair of everyone who doesn’t cut
• The number in the orange square is odd. their own hair.
• The numbers in the yellow and light green squares • So he does cut his hair… which
add up to one of the numbers in the bottom row. takes us back to the start again.

28 31

104

People with pets Cat Fish Dog Parrot

Four friends each have a pet. There’s a cat, a fish, I’m allergic to fur,
a dog, and a parrot. The pets’ names are Nibbles, so my pet doesn’t
Buttons, Snappy, and Goldy. From what the friends have any, and my
are saying below, can you figure out who has which pet has the second
pet, and the names of each animal? shortest name of

My pet isn’t a I don’t have the four.
goldfish or a dog, a dog...
but it is named

Nibbles.

My pet is named ...and I know Cecilia
Buttons and likes Goldy is a cat.

Anna swimming. Dave

Bob If you’re having trouble, try
drawing a grid with the

people’s names in the first
column, and then filling in any

clues you’ve worked out.

Lost at sea

It’s a foggy gray day at sea
and, viewed from the air, you
can only make out some
empty blue water and parts
of ships. Can you find out
where the rest of the fleet is
located? Every ship is
surrounded on all sides by
squares of empty water.

Fleet:

6 Dinghies:

4 Yachts:

2 Cruisers:

105

106 Charles Babbage

Agnes Meyer Driscoll The English mathematician Babbage
(1791–1871) not only invented the first
Driscoll (1889–1971) was one of the best true computer—he was also a
code breakers of the 20th century. Working champion code breaker. In 1854, he
for the U.S. Navy, she broke some of the
most difficult codes of the time, including broke a famous military cipher that used
many of those used in the world wars. 26 alphabets to encipher each message.
Sometimes known as “Madame X,” Babbage’s discovery was used to decipher
Driscoll also helped develop Russian messages during the Crimean War.
code-breaking machines and
teach other code breakers. Thomas Jefferson

Sir Francis Walsingham A decade before he became president, Jefferson
(1743–1826) invented a revolutionary coding
During the reign of Elizabeth I of England, spying
was widespread. Walsingham (c. 1532–90) was a machine called a “wheel cypher.” He went on to
master spy. He discovered an assassination plot oversee and develop several other ciphers,

by Elizabeth’s cousin, Mary, Queen of Scots, by too. They were used to send messages
intercepting Mary’s messages. His code-breaking to Europe and to keep in touch with
secret missions. The U.S. military
expertise led to Mary’s execution. adopted Jefferson’s wheel ciphers,
using them from 1922 until 1942.

ACTIVITY

Codes everywhere

There are codes all around us, and many of
them are designed to be read by machines.
If you have a smartphone, there will be at least
one “bar code scanner” app for it. You can use
this to read the bar codes of all kinds of products
in stores or even just items on your kitchen
shelves. See what information comes up about
them. Try using it to scan library books, too.

BREAKING Hacker

CODES A hacker is someone who breaks into computer
If you have a secret message to read, call in the systems, either just for fun or to steal valuable
experts! Both codes and ciphers make readable
messages unreadable, and both can be cracked information. Hacking into a computer system
using math. Codes change each word to a code often involves decrypting (decoding) computer code
word, symbol, or number. Ciphers jumble the
letters or replace them with different symbols. or messages. Sometimes, hackers are employed
by computer companies to test their systems
and make them more secure. These hackers

are sometimes nicknamed “white-hat hackers.”

To test security, an IBM
employee named Scott Lunsford
tried to hack into the computer
system of a nuclear power plant.

It took him just a day.

Encryption Public key encryption

Financial transactions are almost always sent by One big math breakthrough of the early 1970s was public key
computer, and they need to be kept secret so that encryption. A key is the name for the information needed to make
no one can steal information about the sender’s or or decode a cipher. Used for all e-mails and texts, this system
receiver’s bank accounts. The transactions travel through means that only the intended recipient can read the message. The
the Internet, along wires, and through space as radio way it works is that the recipient’s computer system invents a pair
signals. Since these messages are easy to tap, they are of keys: one to encrypt and one to decrypt. The sender uses the
turned into ciphers, or encrypted, for their journeys. encrypt key to encrypt a message to the recipient. This message
can be read only by the recipient because only he or she has the
Frequency analysis decrypt key to unlock its contents.

Simple ciphers can be broken by frequency analysis
(counting how often each symbol occurs). Each symbol
stands for a letter in the original text (the plaintext), so
the most common symbols should represent the most
common letters. In English, the most common letters
are E and T, in German they’re T and A, and in Spanish
they’re E and A. By substituting these letters for their
encrypted versions, the plaintext can be worked out.

107

108 Make a cipher wheel

Caesar cipher To write and decode a substitution cipher easily, you need BRAIN GAMES
a cipher wheel. Both you and the person you are sending
The Caesar cipher is named after the Roman General the message to need one, and you both need to know the
Julius Caesar and is a code based on substitution— key to the letter substitution.
replacing one letter of the alphabet with another. For
example, you could replace each letter with the one that You will need:
follows it, so “b” becomes “c”, “c” becomes “d”, and so
on. In trickier versions, the letters might be two or three • Paper • Pencil
steps ahead, so “a” would be “d”, and “b” would be “e”.
Can you can decipher the following message: • Scissors • Ruler

ZHOO GRQH WKLV LV D KDUG FRGH • Paper fastener

To figure out what the letter-gap is, think Step 1
about what the one-letter word might be.
Copy the two wheels shown here
Substitution cipher Look for Y in the bottom and cut them out. Put the smaller
row, and then look above one on top, divide them both into
In a Caesar cipher, the coded alphabet runs in order: it to see which letter it 26 equal parts as marked, and
It’s just the position that changes. In a substitution really is—C fasten them together in the middle.
cipher, however, the coded alphabet is not in order.
Using the cipher below, what does this message say? YTRJWYLKCJPFK Step 2

ABCDEFGHIJKLMNOPQRSTUVWXYZ In the outer circle write the
LCYRJPDOAVZHBKTXGSWUFEMINQ alphabet in the correct order,

and in the inner circle write
the cipher alphabet (you

can use the one at left, or
make your own). Decide on a
key letter, for example X = P.

Caesar’s cipher B C D E
might have been A F
simple, but it was ZL Y Step 3
effective because at
the time people WY CODES AND HI Give the wheel to a
weren’t used to the MX G OA friend and provide them
CIPHERS with the key letter as a
idea of codes. starting point. When they
align the inner X to the
outer P, the rest of the
cipher will be revealed.

UV JK Here are some codes and ciphers for B
T FE VZ L you two make and break. There are
also instructions for making a cipher
U H wheel that will make it easy for you
Polybius cipher and your friends to turn messages
109 S Q NO P M
Polybius (c. 200–118 BCE) was a into top secret text.
Greek historian who worked for R G KT
the Romans and devised a new =X
kind of cipher. An English =X Shape code
version would look like the one =X
below. To use it, you take the =X Each of the 11 colored shapes stands
pair of numbers that represents =X for a number between 0 and 12. Can
each letter in the message. H =X you work out the value of each shape
is in row 2, column 3, so its using math and logic?
ciphered number is 23.
=X
12 345
=X
1A BCDE
=X
2F GHIJ
=X Start by thinking
3K LMNO =X about which
numbers could
4P QRST make this
equation work
5 U V W X YZ
= XX
Step 1

Using the cipher above, decode the following message.
The letters all run together, making it trickier.

45 23 24 44 24 44 11 52 15

43 55 35 32 14 13 35 14 15

Step 2

Give a copy of the cipher to a friend and send each
other hidden messages. You can also create your
own cipher by mixing up the order of the letters—
just be sure that everybody has the same cipher!

British prime minister Winston Churchill
once said that Turing’s work shortened

World War II by two years.

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Early life

Turing TuringwasborninLondonon
June 23, 1912. His father worked

as a civil servant in India, and not

It was Alan Turing’s brilliant mathematical mind that long afterward his parents returned
helped the Allies win World War II by developing new there, leaving Turing and his older
brother in the care of family friends

types of code-breaking machines. He then went on to in England. As a boy, Turing excelled
build some of the world’s first computers, and was at math and science. At the age of
16, he came across the work of the

a pioneer in the development of intelligent machines, great scientist Albert Einstein and

the science we now call artificial intelligence. became fascinated by his big ideas.

The Turing machine

In 1931, Turing went to King’s College, Cambridge, to
study mathematics. It was here that he published a
paper in 1936 about an imaginary device that carried
out mathematical operations by reading and writing
on a long strip of paper. Later known as a “Turing
machine,” his idea described how a computer could
work long before the technology existed to build
one. Later the same year, Turing went to the United
States to study at Princeton University.

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Turing was a world-class Turing returned to England in 1938, where
marathon runner. He the British government asked him to work on
came in fifth in the deciphering German codes. When World War II
qualifying heats for the broke out, Turing moved to Bletchley Park,
1949 Olympic Games. the secret headquarters of the Code and
Cipher School. With his colleague Gordon
Welchman, Turing developed the “Bombe,”
a machine that could decipher German
messages encrypted (coded) on a
typewriter-like device called the
Enigma machine (right).

110

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various

The first computers

After the war, Turing moved to the
Britain’s National Physical Laboratory,
where he designed a computer called the
Automatic Computing Engine (ACE) that
was able to store program instructions in
an electronic memory. It was never built,
but it led to the development of the Pilot
ACE, one of the first general-purpose
computers. In 1948, Turing moved to
Manchester University to work on
computer programming there. Some of
these early computers were vast, filling
whole rooms and weighing many tons.

Turing was given
an award for service to

his country during
World War II.

Turing’s test Tragic suicide

Turing wanted to know whether a Turing was gay at a time when
machine could be considered capable homosexuality was illegal in
of thinking. In 1950, he devised an Britain. Because of this, he faced
experiment to see whether a computer persecution and the threat of
could convince someone asking it imprisonment. In 1954, Turing took
questions that it was, in fact, human. his own life. This statue of him is in
Turing’s “imitation game”, now known Bletchley Park, today a museum
as the Turing test, is still used to about the secret code-breaking
determine a machine’s ability to activities of World War II.
show humanlike intelligence.

111

ALG3BRA

There’s an important area of math Find the formula
called algebra that replaces numbers
with symbols (often letters of the Algebra uses formulas to solve problems. A formula
alphabet) in order to solve a problem. is kind of like a recipe—it gives you the ingredients
In addition to mathematicians, and tells you what to do with them. Scientists use
scientists use algebra to find out formulas for all kinds of things. For example, having
things about the world. received a radio signal from a spacecraft, scientists
can work out how far away the spacecraft is using
Simple algebra the formula below. It uses the metric system, which
is used internationally in science.
The difference between arithmetic and algebra can be
seen by writing the same calculation two ways: Distance = Time x Velocity (speed) of radio waves

In arithmetic: 4 + 5 = 5 + 4 Put in the information you know to get the answer.
In algebra: x + y = y + x
Time—in this case the radio waves reached Earth
You can tell that, in this case, x = 4 and y = 5. in 10 seconds

The first is a simple equation. The algebra, however, Velocity—radio waves travel at 3 million km/second
gives you the rule for any numbers you want to use as
x and y. You can see this in the following example: So the distance = 10 seconds x 3 million km/second

x+y=z Distance = 30 million km

If you are given the values of x and y, then you can work
out what z is. So if x = 3 and y = 5:

3+5=z

z=8

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total mass = +roecqkueitp+mceanpts+ulcere+wfuel
112

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comes from the name of
a book by the ancient Arab
mathematician Al Khwarizimi.

Finding patterns

When there is a pattern to numbers, you
can use it to figure out other information.
For example, scientists on the planet Zog
want to build a rocket 110 urgs (u) long
and need to know how many vons of krool
to use. They have this data about other
rockets they have constructed:

Length Vons of krool

30 u 140
60 u 200
80 u 240
100 u 280

The pattern that fits all the numbers
above leads to the following equation:

Vons of krool = length x 2 + 80

So their new rocket would need
110 x 2 + 80 = 300 vons of krool.

ACTIVITY

Lunar lightness

Try this problem for yourself. The table below shows the
weights of various objects on the Earth and on the Moon.

Object Weight on Earth Weight on Moon

Apple 6 oz 1 oz
Robot 300 lb 50 lb
Moon lander 18 tons 3 tons

Can you find the equation that relates weight on the
Earth to weight on the Moon? How much would you
weigh on the Moon?

113

BRAIN GAMES

BRAINTEASERS

You use algebra to solve problems all the time—
you just don’t notice it. As you think through the
puzzles on these pages, you’ll be using algebra,
but when it’s disguised in everyday situations
or a fun brainteaser, it’s not that scary!

In algebra, “x” means Cake bake
an unknown number.
Jim has been asked to bake a cake for a friend’s
That’s why an birthday, and is given the following recipe:
unknown quality in a • 16 tbsp butter
person is called “the • 2 cups sugar
• 4 eggs
x-factor.” • 4 cups flour
At the last minute, Jim realizes that he does not have
AItmnnwhueuehmalanotctiubpuhtetlhsirmfleeidiodbnpewiatenpehtetrterehtbarmeenlslsioodiashwdf,mal,aevetne.hdCwpebafiaeennyneutndytmooatauhbmdeeldfiaregasksdnueosrantewnhoedeur t 3 enough eggs. The stores are closed, so he decides to
adapt the recipe to work with three eggs. What are the
to the third flower? new quantities of butter, sugar, and flour he should use?

1 6

2 56 7 7 96 2 4?9

5 83

114

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There is an apple tree and a beech tree in 18 golf balls
a park, each with some birds on it. If one
bird from the apple tree were to fly to the
beech tree, then both trees would have
the same number of birds. What is the
difference between the numbers of birds?

15 golf balls

A fruity challenge You’re on your own

Each type of fruit in these grids is
worth a different number. Can you work
out what the numbers are? When you
have this information, figure out what
the missing sum values at the end of
every row and column are.

With a little help 71

54 115

48

Once you know what 60 70 80
a pineapple is worth,
you can then find out Start here to work
the value for oranges out how much a
pineapple is worth

SECRETS OF THE

UNIVERSE
In the hands of scientists, math has the power
to explain the Universe. Science is all about Plant breeding project
proving theories—and to do that, scientists
need to use math to make predictions from All the flowers One-quarter of the
the theory. If the predictions turn out to be are pink, so flowers are white,
correct, then the theory probably is, too. pink must be the so the pink flowers
dominant color must have been
A world of math carrying white
Three-quarters flower genes
In the 16th century, the great scientist and inventor of the flowers
Galileo Galilei (1564–1642) discovered that many are pink
things that happen in the world can be described
by simple mathematics, from falling objects and
the strength of bridges to musical notes. After
Galileo, nearly all scientists tried to find the

mathematical laws to describe
exactly how things work.

The math of life

Gregor Mendel (1822–1884), an
Austrian scientist and monk, found
that simple math applies to some of the
characteristics of living things. Features
such as flower color or eye color, for
example, are always passed on from
parents to offspring according to
probabilities (see pages 100–101).
His work was the beginning of

the science of genetics.

Galileo built powerful
telescopes to study
astronomy, and
sold others to the
military for spotting
enemy ships.

116

The simple truth

Until the early 20th century, math was used
to add detail to scientific theories, to prove
them, and to use them. Since then, however,
math has often been used to suggest theories. When
there are several theories to choose from, the one that
is mathematically simplest is usually right. Genius

physicist Albert Einstein (1879–
1955) found the correct
equations to explain gravity
by choosing the simplest.

Mighty machines

Mathematicians are more likely
to spend their time looking for

patterns, coming up with ideas, or
trying to prove new theorems than

doing calculations. Why bother,
when there are computers to do
the work for us? The most powerful
supercomputers can work billions of
times faster than any human being.

Their extraordinary number-
crunching abilities mean scientists

can test their theories more
thoroughly than ever before.

Nothing’s perfect Professor Stan Gudder said,
“the essence of mathematics is
In 1931, Austrian-born mathematician
Kurt Gödel (1906–1978) published a not to make simple things
revolutionary theorem. He showed that complicated, but to make
it is impossible for any complicated complicated things simple.”
mathematical theory to be complete—
there will always be gaps, and there will
always be statements in the theory that
can’t be proven. Math was never
the same again!

it ibnAevinoOlptlsoiUtvnkthrtfhwheoeereneeasvuoitovUoetvmhteferdmrhnenyretii.mhsnlvanstIeeegktittadnrsresabstisi.nhearetueTeogeryspohiomsstotsiahsftbottachthiejfonttaeemaihmmylccotlesastetrutsosoleidhtslerilytesniyvsshcreebtpcaattreseihoaatli.cnenrmvenagtixiobuacgpnrsdllaeleeaytsien

117

BRAIN GAMES 2 How long is 3.1 hours? 3 What is ó + ѿ? Learn some
useful facts
1 Is midnight… A 3 hours and 10 minutes A ²/7 by heart, like
multiplication
A 12:00am B 3 hours and 6 minutes B ²/12
B 12:00pm tables.
C Neither C 3 hours and 1 minute C 7/12

4 What is ¼ x ¼? THE BIG

A ¹/2 QUIZ
B ¹/16
C ¹¹/16 6 Which of these is true?

5 Which is smallest? A 2.1% of 43,000 is bigger than 0.21% of 4,300

A 8.35 B 2.1% of 43,000 is equal to 0.21% of 4,300
B 8Ҁ
C 8.53 C 2.1% of 43,000 is smaller than 0.21% of 4,300

7 How many cuts do you need 8 If you add 10% to 100 you For a difficult or long
to make through a loaf of get 110. If you subtract 10% calculation, estimate
bread to make yourself from 110, what do you get? your solution before
4 large sandwiches if you working out the exact
don’t like crust? A 90 answer, especially when
A7 using a calculator.
B8 B 99
C9
C 100

Try different ways to learn facts, 9 What is -1 + -2?
figures, and formulas: Say the
A -3
words out loud, make up a rhyme,
or even draw a helpful picture. B -1

C3

118

10 What is 0.1 x 0.1? 11 What is 2.3 x 10? 12 Using a piece of string, which
of these triangles will enclose
A 0.01 A 2.30 the largest area?

B 0.11 B 20.3 A Right

C 0.1 C 23 B Equilateral

C Scalene

Are there certain math questions that 13 Which of these shapes
always make you stop and think... then has the smallest number
scratch your head and think again? If so, of faces?
you’re not alone. There are math traps it is
easy to fall into, until you know what to look A Cube
out for. Here are some of the most common
confusions, and some other tricky questions B Square-based pyramid
to add to the fun.
C Tetrahedron

14 If you start baking a cake 20 When you get an
minutes after a quarter to 6:00, incorrect answer, make
and it takes 65 minutes to cook, sure you understand
what time will it be finished?
what went wrong.
A 6:50

B 7:10 17 Using a piece of string,
which of these shapes
C 6:45 will enclose the
largest area?

A Circle

15 Which of these shapes is 16 What is 3 ÷ ¼? B Square
the odd one out? C Triangle
A Rectangle A 0.75
B Cube
C Triangle B ¹/12

C 12

Make sure you 18 What is 1 ÷ 0?
really understand a question,
A1
especially if it is asked out
loud. Write the question down, or B0

ask the teacher to repeat it! C Infinity

119

GLOSSARY

algebra consecutive estimate

The use of letters or symbols in place of Numbers that follow one after the other. To work out a rough answer.
numbers to study patterns in math.
cube even number
angle
Either a solid shape with six faces, or an A number that can be divided
A measure of how far a line needs to instruction to multiply a number by itself exactly by two.
rotate to meet another. An angle is usually three times, for example 3 x 3 x 3 = 27.
measured in degrees, for example 45°. This can be written 33. faces

area data The surfaces of a 3-D shape.

The amount of space inside a 2-D shape. Factual information, such factors
Area is measured in units squared, as measurements.
for example in2 (cm2). The numbers that can be multiplied
decimal together to give a third number. For
arithmetic example, 2 and 4 are factors of 8.
A number system based on 10, using the
Calculations that involve addition, digits 0–9. Also a number that contains a formula
subtraction, multiplication, or division. decimal place.
A mathematical rule, usually written
axis (plural axes) decimal place in symbols.

The line on a graph. The distances The position of a digit after the fraction
of points are measured from it. The decimal point.
horizontal axis is called the x-axis, The result of dividing one number
and the vertical axis is called the y-axis. decimal point by another.

bar graph The dot separating the whole part frequency
of a number and the fractions of it,
A type of graph that uses the heights of for example 2.5. How often something happens within
bars to show quantities. The higher the a fixed period of time.
bar, the greater the quantity. degrees
geometry
billion The unit of measurement of an angle,
represented by the symbol °. The area of math that explores shapes.
A thousand million, or 1,000,000,000.
diameter graph
chart
The greatest distance across a shape. A chart that shows how two sets of
A picture that makes mathematical information are related, for example the
information easy to understand, such digit speed and position of a moving object.
as a graph, table, or map.
A single-character number, such as 1 or 9. hexagon
cipher
encrypt A flat shape with six straight sides.
A code that replaces each letter with
another letter, or the key to that code. To turn a message into code to keep horizontal
the information secret.
circumference Parallel to the horizon. A horizontal line
equation runs between left and right, at right angles
The distance around the edge of a circle. to the vertical. Also describes a surface
A mathematical statement that two that is flat, straight, and level.
code things are equal.
isosceles triangle
A system of letters, numbers, or symbols equilateral triangle
used to replace the letters of a text to A triangle with at least two sides of
hide its meaning. A triangle that has three angles of 60°, equal length and two equal angles.
and sides of equal length.

120

line of symmetry pyramid tetrahedron

If a shape has a line of symmetry, you A 3-D shape with a square base A triangular-based pyramid.
can place a mirror along the line and the and triangular faces that meet in
reflection will give an exact copy of half a point at the top. theorem
the original shape.
quadrilateral A math idea or rule that has been,
measurement or can be, proved to be true.
A 2-D shape with four straight sides and
A number that gives the amount or size four angles. Trapeziums and rectangles theory
of something, written in units such as are both examples of quadrilaterals.
seconds or feet. A detailed, tested explanation
radius of something.
octagon
The distance from the center of 3-D (three-
A flat shape with eight straight sides. a circle to its edge. dimensional)

odd number range The term used to describe objects that
have height, width, and depth.
A number that gives a fraction with The difference between the smallest
0.5 at the end when divided by two. and largest numbers in a collection triangle
of numbers.
parallel A 2-D shape with three straight sides.
ratio
Two straight lines are parallel if they are 2-D (two-dimensional)
always the same distance apart. The relationship between two numbers,
expressed as the number of times one is A flat object that has only length and width.
pentagon bigger or smaller than another.
velocity
A flat shape with five straight sides. right angle
The speed in a direction.
percentage/percent An angle that is exactly 90°.
Venn diagram
The number of parts out of a hundred. scalene triangle
Percentage is shown by the symbol %. A method of using overlapping circles to
A triangle with three different angles and compare two or more sets of data.
pi sides that are three different lengths.
vertex (plural
The circumference of any circle divided sequence vertices)
by its diameter gives the answer pi. It is
represented by the Greek symbol ∏. A list of numbers generated according The corner or point at which surfaces
to a rule, for example 2, 4, 6, 8, 10. or lines meet within shapes.
polygon
square vertical
A 2-D shape with three or more
straight sides. A 2-D shape with four straight equal sides A vertical line runs up and down, at right
and four right angles. angles to the horizon.
polyhedron
squared number whole number
A 3-D shape with faces that are
all flat polygons. A number multiplied by itself, for example A number that is not a decimal
4 x 4 = 16. This can also be written 42. or a fraction.
positive
sum
A number that is greater than zero.
The total, or result, when numbers are
prime factors added together.

Prime numbers that are multiplied symmetry
to give a third number. For example,
3 and 5 are the prime factors of 15. A shape or object has symmetry (or is
described as symmetrical) if it looks
prime number unchanged after it has been partially
rotated, reflected, or translated.
A number greater than one that can only
be divided exactly by itself and one. table

probability A list of organized information, usually
made up of rows and columns.
The likelihood that something will happen.
tessellation
product
A pattern of geometric shapes that covers
The answer when two or more numbers a surface without leaving any gaps.
are multiplied together.

121

Answers Panel puzzle 34-35 Thinking
The extra piece is B outside the box
6-7 A world of math Profit margin
Bumper cars: 60 percent of 12 is 7.2 1 Changing places
FC Number of sessions: 4 x 8 = 32 Second place.
Fares per session: 32 x $2 = $64 2 Pop!
D A $64 x 7.2 = $460.80 Use a balloon that’s not inflated.
E Cost to run: $460.80 - $144 = $316.8 3 What are the odds?
Profit: $316.80 per day 1 in 2.
12-13 Math skills 4 Sister act
A game of chance They’re two of a set of triplets.
Spot the shape There is a 1 in 9 chance of winning: 5 In the money
1D 90 (customers) x 3 (throws) = 270 Both are worth the same amount.
2C 270 ÷ 30 (coconuts) = 9 6 How many?
3C You would need 10 children.
4C 22–23 Seeing the 7 Left or right?
solution Turn the glove inside out.
18-19 Problems 8 The lonely man
with numbers What do you see? The man lived in a lighthouse.
1 Toothbrush, apple, lamp 9 A cut above
A useful survey? 2 Bicycle, pen, swan Because it would be more profitable.
1 The survey may be biased because it 3 Guitar, fish, boat 10 Half full
was carried out by the Association for 4 Chess piece, scissors, shoe Pour the contents of the second
More Skyscrapers. cup into the fifth.
2 They only surveyed three of the Thinking in 2-D 11 At a loss
30 parks (1 in 10). This is too small The very rich man started off as a
a sample to be able to arrive at a Visual sequencing billionaire and made a loss.
conclusion about all the parks. Tile 3 12 Whodunnit?
3 We don’t know how many visitors The carpenter, truck driver, and mechanic
went to the third park. Seeing is understanding are all women. Note that the question
4 The fact that the other two parks had The snake is 30 ft (9 m) long. says fireman, not firefighter.
fewer than 25 visitors all day suggests the 13 Frozen!
survey took place over one day, too short a 3-D vision The match!
time frame to draw useful conclusions. Cube 2 14 Crash!
Nowhere—you don’t bury survivors.
The bigger picture 30-31 Big zero 15 Leave it to them
Because tin helmets were effective at One pile.
saving lives, more soldiers survived head Roman homework 16 Home
injuries, rather than dying from them. So This question was designed to show why The house is at the North Pole so the
the number of head injuries increased, place value makes math so much easier. bear must be a white polar bear.
but the number of deaths decreased. The quickest way to solve the problem is
to convert the numbers and the answer: 36-37 Number patterns
CCCIX (309) + DCCCV (805) = 1,114 (MCXIV).
Prison break
As you work through the puzzle, you
should begin to see a pattern in the
numbers on the doors left open—they are
all square numbers. So the answer is 7:
1, 4, 9, 16, 25, 36, and 49.
 
Shaking hands
3 people = 3 handshakes
4 people = 6 handshakes
5 people = 10 handshakes
The answers are all triangle numbers.

A perfect solution?
The next perfect number is 28. All perfect
numbers end in either 6 or 8.

122

44-45 How big? How far? 52-53 Pascal’s 56-57 Missing numbers
triangle
Measure the Earth Sudoku starter
360° ÷ 7.2° = 50 Braille challenge
50 x 500 miles (800 km) = 25,000 miles Look at row 6 of Pascal’s triangle 1 76 489 32 5
(40,000 km). and add up the numbers to get 64. 5 89 723 146
This means there are 64 different 4236 51 897
50-51 Seeing sequences ways to arrange the dots. For a 392847561
four-point pattern, go to row 4 of the 814536 279
What’s the pattern? triangle, which adds up to 16, showing 65791 248 3
A 1, 100, 10,000, 1,000,000 that there are 16 possible ways to 945 368 712
B 3, 7, 11, 15, 19, 23 arrange the dots. 7 31 2 94 65 8
C 64, 32, 16, 8 2 6817 5934
D 1, 4, 9, 16, 25, 36, 49
E 11, 9, 12, 8, 13, 7, 14 Slightly harder
F 1, 2, 4, 7, 11, 16, 22
G 1, 3, 6, 10, 15, 21 785 693124
H 2, 6, 12, 20, 30, 42 9 64 5 2 13 8 7
21 3847659
54-55 Magic squares Your own 356 789 241
magic square 849 216 573
Making magic 127 35 489 6
571 96 8 43 2
276 7 4 9 14 24 18 32 3 11 23 11 24 7 20 3 498132 765
2 25 4 27 22 31 17 5 13 21 9 632475918
5 11 2 16 34 9 1 10 36 21
9 5 1 10 6 15 3 23 6 19 2 15 Sujiko
4 3 8 12 13 8 1
6 26 30 28 5 16

33 14 29 8 20 7 4 12 25 8 16

12 19 15 35 17 13 10 18 1 14 22

66-67 Puzzling primes Prime cubes 796

Sifting for primes 28 3 28 3 28 9 21 18
643
1 2 3 4 5 6 7 8 9 10 469 649 57 1 32 1
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30 75 1 57 1 28 3 14 15
31 32 33 34 35 36 37 38 39 40 469
41 42 43 44 45 46 47 48 49 50 15 7 548
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70 28 9 28 3 28 1 Kakuro
71 72 73 74 75 76 77 78 79 80 463 649 647
81 82 83 84 85 86 87 88 89 90 75 1 51 7 95 3
91 92 93 94 95 96 97 98 99 100
28 9 28 9 28 7 17 20 3 8
643 463 461 16 4
51 7 15 7 59 3 7 3
9 1 5
28 7
641 22 10
35 9
13 5 15 2 2

8 8

12

21

849

28 1 28 7 9 8
467 641
59 3 95 3 1 53

7 16 7 3

17 1

19

13 7 12 8

6 4

28 7 28 1 28 1
461 647 467
53 9 35 9 53 9

123

70-71 Triangles 74-75 Shape shifting Shapes within shapes

Measuring areas Triangle tally
The areas of the triangles are: There are 26 triangles in total.
3 x 7 = 21 21 ÷ 2 = 10.5
3 x 5 = 15 15 ÷ 2 = 7.5 Tantalizing tangrams
4 x 4 = 16 16 ÷ 2 = 8
4 x 8 = 32 32 ÷ 2 = 16 Arrow

Add them together: Matchstick mayhem
10.5 + 7.5 + 8 + 16 = 42 square units
Fox Candle
80-81 3-D Shape
puzzles Dare to be square 4 x 4 grid
3 x 3 grid You can
Constructing cubes You can
A+D draw this draw this
H+I with just using just
E+G 4 squares. 6 squares.
B+C
F is the odd one out.

Boxing up
Net D will not make a cube.

Face recognition
There are many ways to do this.
Here’s just one:

123

456 7 86-87 Amazing mazes

How many more can you find? Experiment Simple mazes Complex mazes Weave mazes
with different starting shapes. Also, try
making a line of shapes that form a circle.

Trace a trail
You can trace a trail around the octahedron,
but not the tetrahedron or cube. This is
because the journey is impossible if more
than two corners of a shape have an odd
number of connections to other corners.

Building blocks 96-97 Mapping 100-101 Probability
A 10 cm3
B 19 cm3

On the map What are the odds? 6 Struck by lightning
Church = 44,01 The order of likelihood is: 7 Hit by a falling coconut
Campsite = 42,03 1 Playing soccer 8 Shark attack
2 Snake bite 9 Walking into
3 Falling down a manhole a lamppost
4 Computer game 10 Hit by a meteorite
exhaustion
5 Hippo attack

124

104-105 Logic puzzles and paradoxes

Logical square Black or white? People with pets
The hat is black. Amy could only know Anna: Nibbles (parrot).
85 her hat color if both Beth and Claire were Bob: Buttons (dog).
34 wearing white (since she knows that not Cecilia: Snappy (fish).
all three hats are white), but Amy answers Dave: Goldy (cat).
26 “No.” That means there must be a black
17 hat on at least one of the others. Beth Lost at sea
realizes this and looks at Claire to see if
her hat is white, which would mean Beth’s
was the black one. But it isn’t, so Beth
answers “No.” So Claire must have the
black hat, and she knows this because
she heard the other sisters’ answers.

The barber’s dilemma
This story is a paradox.

Four digits
The answer is 1,349.

108-109 Codes 114-115 Brainteasers
and ciphers
A number of petals In the balance
Caesar cipher The answer is 117. The pattern is adding You need 12 golf balls.
The message reads: “Well done this up the three smallest numbers, and
is a hard code.” multiplying the total by the largest number. A fruity challenge Banana = 20
So (3 + 4 + 6) x 9 = 117. Pineapple = 12 Strawberry = 15
Substitution cipher Orange = 18 Grapes = 16
This message reads: “Codes can be fun.” Cake bake Apple = 6
Jim will need 12 tbsp butter, 1.5 cups 70
Polybius cipher sugar, and 3 cups flour. 54
The cipher reads: “This is a very old code.” 71
In a flap 42
Shape code The difference is two. If there were seven 71
birds in the apple tree, for example, and one 48
=0 =6 left to even out the numbers, there would 66
=1 =8 need to be five in the beech tree. 48
=2 =9 70 62 80 66
= 3 = 10 54 36 42 60
= 4 = 12
=5 118-119 Big quiz 12 B—To find the area of a triangle work
out ½ x height x base. An equilateral
112-113 Algebra 1 A—Midnight is 12:00 am . triangle will give the highest and widest
2 B—3 hours and 6 minutes . dimensions and therefore the largest area.
Lunar lightness 3 C—7/12 13 C—A tetrahedron has just four faces.
Object are six times lighter on the Moon 4 B—1/16 14 B—7:10
than on the Earth. So, to find out how 5 A—8.35 15 B—A cube is a 3-D shape, while
much you would weigh on the Moon, 6 A—2.1% of 43,000 is bigger the others are 2-D.
divide your weight by six. 16 A—The question is not asking what is a
than 0.21% of 4,300. quarter of 3 but how many quarters in 3.
7 C—You need to make 9 cuts . 17 A—A circle’s edge is always the
8 B—99 maximum distance from the centre point.
9 A—You are adding together two negative 18 A—This is a trick question, you can’t
numbers, so (-1) + (-2) = -3. divide numbers by zero. Try it on a
10 A—0.01 calculator and it will register an “error”.
11 C—23

125

INDEX

A neurons 11, 16 codes and code breakers 67, electronic networks 87
and 3-D vision 79 106–111 ellipses 77
absolute zero 31 visualization 12, 13, 19, encryption 67, 107
Academy of Science, 22–23 coins Enigma machine 110
brain games counterfeit 99 equations 112–113
St. Petersburg 84, 85 big quiz 118–119 tossing 52, 100 equilateral triangle 70
algebra 112–115 brainteasers 114–115 Eratosthenes 45, 66
angles 43 calculation tips 38–39 colors 12, 89 Escher, Maurits Cornelis 61
codes and ciphers comets 98 eternity 60
right angle 70 108–109 computer languages 21 Euclid 26
animals 14–15, 73 displaying data 102–103 computers 17, 50 Euler, Leonhard 84–85
Archimedean screw 40 lateral thinking 34–35 eyes 11, 79, 90
Archimedes 40–41 logic puzzles 104–105 binary system 29
areas, measuring 71 magic squares 54–57 encryption 67, 107 see also visualization
arithmetic sequences 50 magic tricks 64–65 hackers 107
art 11, 61 math skills 12–13 history of 21, 53, 110–111 F
artificial intelligence 17, 110 mazes 86–87 supercomputers 117
asteroids 46 measurement 44–45 contour lines 96 factors 37
astronomy 43, 46, 58, 98, 99 optical illusions 23, counting 12, 26–27, 28 prime 67
atomic clocks 95 88–91 crime, measuring 102–103
avalanches 47 shape shifting 74–75 crystals 78 Faraday, Michael 18
3-D shapes 80–83 cubes 33, 78, 80, 81, 83, 84 Fermat, Pierre de 16
B visualization 22–23 cubic numbers 37 Fibonacci (Leonardo of Pisa) 27
curves 59, 77 Fibonacci sequence 51, 53
Babbage, Charles 21, 106 C fingerprints 42
babies 14 D Fizz-Buzz 27
Babylonians, Ancient Caesar cipher 108 flatfish 73
calculation 6, 27, 44–45 data, displaying 102–103 food
and base-60 system decagon 72
27, 28, 63 tricks and shortcuts decibels (dB) 46 measurements 47
and degrees 43 38–39 decimal system 28, 31 supply 50
number system 28–29 calculus 41, 99 degrees 43 footprints 43
place-value system 27 Cantor, Georg 61 diameter 76, 77 forensic science 42–43
and zero 30 casinos 101 diatoms 73 formulas 112
Bacon, Roger 19 cerebrum 10–11 digits 26, 28, 39 fractions 26, 33
balance 115 Ceres 58 distance, measuring 44, 71, 95 frequency analysis 107
bar charts 102 chance 7, 100, 101 division 39 Fujita scale 47
bar codes 106 chaos 100 dodecagon 72
barn (measurement) 47 checklists 103 dodecahedron 33, 84 G
base-10 see decimal chess 55 domes 82
system children 14–15 doughnuts, shape of 81 Galileo Galilei 99, 116
base-60 system 27, 28, 63 chili peppers 47 Driscoll, Agnes Meyer 106 Gauss, Karl 58–59
beard growth 46 China 62, 63, 74 dyscalculia 18 genetics 116
bell curve 59 number system 28–29 geometric sequences 50
binary system 29 Christianity 62, 63 E geometry see shapes
Bletchley Park 110, 111 cicadas 67 Gödel, Kurt 117
body see human body cipher wheel 108 Earth, measuring 45 golden ratio 51
Bombe 110 ciphers 106–109 eggs 79, 82 golden rectangle 51
Brahmagupta 30 circles 76–77 Egyptians, Ancient 27, 40, 44 golden spiral 51
Braille 53 circumference 76 GPS (Global Positioning
brain 10–11, 12–13, 14, 73 clocks 95 and fractions 26
vs. machine 16–17 number system 28–29 System) 97
memory 13, 15 time system 94 graph theory 85
Einstein, Albert 117 graphs 59, 102

126

gravity 79, 85, 99, 113, 117 L patterns of 36–37, prisms 81, 98
Great Internet Mersenne 50–53, 113 probability 52, 100–101
labyrinth, Cretan 86, 87 perfect 37 prodigies 16
Prime Search (GIMPS) 67 lateral symmetry 73 prime 66–67 profit margin 7
Greeks, Ancient 26–27, 36, 37, lateral thinking 34, 75 problems with 18–19 public key encryption 107
Leibniz, Gottfried 84, 99 rational 33 pulleys 41
71, 76, 84, 109 Lemaire, Alex 38 sequences 50–51 pyramids 78, 81
Cretan labyrinth 86, 87 Leonardo da Vinci 51 systems of 27, 28–29, 31 Pythagoras 26, 32–33
number system 28–29 Leonardo of Pisa see Fibonacci very large 41, 43
see also Archimedes; lie detectors 43 very small 43 theorem 32
Eratosthenes; Pythagoras light 98 visualizing 12, 13, 19
Greenwich Meridian 94, 95 light-years 95 numerophobia 18 Q
Gudder, Stan 117 logic 104–105, 114–115
lucky numbers 62–63 O quadrilaterals 72
H
M octagon 72 R
hackers 107 octahedron 33, 78, 81, 84
Halley, Edmond 98 magic numbers 51, 54–55 optical illusions 23, 88–91 radius 76
heptadecagon 58 magic squares 54–55 optics 98 Ramanujan, Srinivasa 16
heptagon 72 magic tricks 64–65 orbits 77, 98 ratio 51
hexagons 72, 76 magnetism 59 rational numbers 33
Hipparchus 71 Malthus, Thomas 50 P rational thought 10
Hippasus 33 maps 96–97 rectangle 72
hobo scale 47 math packaging 79 rhombus 72
hockey stick sums 53 panel puzzle 6 right angle 70
honeycombs 73 learning 14–15, 18, 26–27 paper, walking through 83 right triangle 70
Hopper, Grace 21 skills 10, 12–13 parabola 77 Roman measurements 44
horsepower 47 Mayan number system 28–29 paradoxes 104 Roman numerals 28–29, 31,
human body mazes 86–87 parallelogram 72
measurements 42–47, 71 Pascal, Blaise 52 62
body clock 95 memory 13, 15 Pascal’s triangle 52–53 rotational symmetry 73
measuring with 44 Mendel, Gregor 116 patterns 7, 36–37, 50–53, 113 roulette 101
not symmetrical 73 Möbius strip 91 Penrose triangle 90 Russia 63, 84, 85, 94
Hypatia 21 Moon 59, 113 pentagon 72
hypotenuse 32 Moore, Gordon E. 50 perfect numbers 37 S
motion 98 perspective 88
I multiplication 38 phi 51 satellite dishes 77
music 11, 33 pi 76 savants 16
icosahedron 33, 78, 79, 84 pictograms 103 scale 97
infinity 60–61 N pie charts 21, 103 scalene triangle 70
insight 11 Pilot ACE 111 science 6, 116–117
International Date Line 94–95 nature 51, 67, 73, 116 pinball 100 scientific equations 113
intuition 34 network diagrams 87 place-value system 27, 31 scientific notation 43
irrational numbers 33, 76 neurons 11, 16 planetarium, first 40 scientific thinking 10
Islam 62 Newton, Isaac 98–99 planets 77, 79, 99 Scoville scale 47
isosceles triangle 70 Nightingale, Florence 21 seconds, counting 45
Noether, Amalie 20 dwarf planet 58 semiprimes 67
J nonagon 72 plants 51, 116 sequences 50–51
numbers Platonic solids 84 shadows 44
Jefferson, Thomas 106 poles 94 shape code 109
a world without 29 Polybius cipher 109 shapes 7, 13, 26, 70–77
K animals’ sense of 14–15 polygons 72
children’s sense of 14–15 polyhedra, regular 33 impossible 90–91
Kakuro 57 first written 27 powers 41, 43 3-D 23, 78–83
Kaprekar’s Constant 65 irrational 33, 76 practice 19 sieve system 66
King, Augusta Ada 21 predictions 101 smells, measuring 47
kite 72 lucky and unlucky 62–63 prime factors 67 snowflake 73
Königsberg bridge problem 85 magic 51, 54–55 prime numbers 66–67 soccer balls, shape of 79
Kovalevskaya, Sofia 20 misleading 19 sound, measuring 46
spatial skills 11

127

spheres 79 tetrahedron 33, 78, 81, 82, 84 W
spinning objects 20 3-D shapes 23, 78–83
spirals 51 thunderstorms 45 Walsingham, Francis 106
square numbers 36 tiling 73 water bomb 83
squares 71, 72, 75 time 94–95 web, spider’s 73
time zones 94–95 Wiles, Andrew 16
logical 104 Torino scale 46 women as mathematicians
magic 54–55 tornadoes, measuring 47
squaring 39 torus 81 20–21
standard units 42 trapezium 72
starfish 73 triangles 32, 70–71, 74 Z
statistics 21
string theory 117 Pascal’s triangle 52–53 zero 30–31
substitution cipher 108 Penrose triangle 90 probability of 100
Sudoku 56 triangular numbers 37
Sujiko 57 Turing, Alan 110–111
Sun 44, 45, 79, 99
supercomputers 117 U
superstitions 62–63
symmetry 73 Universe 61, 116–117

T unlucky numbers 62–63

tables of information 103 V
tallies 26, 102
Tammet, Daniel 16 Venn diagrams 103
tangrams 74 video games, benefits of 23
telegraph, electric 59 vision 11, 79, 90
telescopes 98 visualization 12, 13, 19, 22–23
temperature scales 31 volcanoes, explosivity of 46
ten 32 volume, measuring 40

decimal system 28, 31
tessellation 73

Credits

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