Chapters

### The importance of maths in everyday life

Since antiquity, mathematics has been an integral part of our everyday lives.

Our understanding of math, and the constant desire to enlarge the extent of our knowledge about the world has taken us far, so that today, the world as we know it would not be possible without maths.

**Mathematics is an application of matter** and contributes to all of our methodical and systematic behaviours.

It is Maths, for instance, that has brought order to the communities across this planet and prevented chaos and catastrophes. Many of **our inherited human qualities are nurtured and developed by Maths theories**, like our spatial awareness, our problem-solving skills, our power to reason (which involves calculated thinking) and even our creativity and communication.

Things that you wouldn't expect to bear any relation to Maths do in fact come down to an underlying need for mathematics and the structure it brings to our everyday lives.

**maths is ever-present in any single craft or profession**- whether you are an artist or a scientist.

**a need to budget for things**such as supplies, tools, studios, etc, and may additionally need to use their logical thinking and numbers to work out suitable prices for their artwork.

**in direct contact with numbers, measurements, equations, and formulas**in their everyday tasks and maths will reign over all the research that they do.

Wait, this all sounds a bit deep and rather daunting, doesn't it? Saying that **maths governs almost everything in this world** and that its importance to life is immeasurable? Don't worry, here comes the fun part!

### Turning maths problems into benefits

Maths has a reputation with many, usually those who encounter difficulty when faced with it, for being a boring subject. But that really isn't the case! What's more, by introducing some fun elements to **the way students are taught mathematical theories and concepts**, then far more pupils would be excelling in this area!

It's so easy to just switch off and omit to take in any information when we are just not that interested. But, the way to **retain and process that important information** and to not let it just go straight back out is to enjoy working things out.

Did you know, that you can **have loads of fun while learning maths **and the many tricks and puzzles it presents? Most of us can't resist a good riddle and solving difficult logic puzzles is a fantastic brain teaser.

Making subjects such as algebra, fractions, and probability more entertaining isn't as hard it sounds! **If you don't believe us, then try out these gratifying maths puzzles!**

That's right, it is possible to practice maths while enjoying yourself! Whether an optical illusion, a picture puzzle or logic games, figuring out the answer to a tricky puzzle is a great way to improve your problem solving skills.

## Test Your Maths Logic with a Tricky Maths Puzzle!

In a quest to understand the world around him, man has employed maths in an effort to tease out tangible proofs. The history of mathematics is punctuated by great minds wrestling with the great enigmas of their time.

**Mathematical puzzles and brainteasers **combine reasoning with numbers, calculations and figures.

To solve such puzzles, one needn't have the brain of a mathematical genius, but it is important to take a logical approach and apply the maths skills one has learnt during the curriculum of their maths revision GCSE, from simple multiplication and division, quadratic equations and calculus.

So get ready to channel your inner mathematician with these 5 challenging puzzles that may be encountered in **maths classes.**

But first!

Let's consider how and why these brainteasers are so useful to us.

### The benefits of maths brain teasers

It's not all in your head! **Brain training is equally good for your entire body and wellbeing**.

The NHS itself states that "keeping the mind active may have various benefits, including a reduced risk of dementia. In general, it would seem sensible to keep the mind as well as the body active."

There are various **things that keeping your mind active with cognitive training can do for you**, such as:

- Increase your ability to memorise
- Cut down on the risk, and slow the decline, of mental illnesses such as Dementia
- Improve your brain processing speed
- Prevent boredom
- Enhance concentration

Also, have you ever considered the fact that **everything changes your brain**? Each new person you meet, each new story you read, each new flower you smell... there are so many 'firsts' that continue to take place throughout our lives that we probably don't even give a second thought to. However, when you sit back and think about it, **your brain is constantly developing and being influenced** by surroundings.

So, if you are still dubious about how a bit of brain training can help you, then just think about this: if, like many others, one single image of a celebrity looking **your idea of perfection** can trigger a powerful sense of motivation to go to the gym/buy some new clothes/change the way you do your hair/etc, then can't one tiny mathematical game also have the force to change your mindset and make something inside click? Who knows **what regular brain training could do** in the sense of how you approach maths and how you think about things in general!

Here is a formula that you might like the look of which could apply to you if you learn math.

*Maths = Control = Self-Improvement = Satisfaction*

### Brain teaser questions: The 100 Prisoners Problem

There are 100 prisoners, sentenced to death, in a prison. Out of the blue, the prison's director proposes a challenge

He assigns each prisoner a number between 1 and 100, then installs in his office a cabinet with 100 drawers, each containing a random number between 1 and 100, corresponding to those assigned to the prisoners.

**Each number appears only once.**

He asks each prisoner to open 50 drawers and check the number in each.

Once each prisoner has entered the office, he is forbidden from communicating with his fellow prisoners, nor to change the sequence of draws or leave any clues.

**No prisoner will know what numbers the other inmates have seen.**

The prison director gives two possible outcomes:

- All of the prisoners find their respective numbers and all are pardoned.
- None find their numbers and they are all are executed.

**What is the chance that each prisoner finds the drawer corresponding to his number?**

According to the law of mathematical probability, the chance that all would be pardoned is (1/2)^{100}, or 0.0000000000000000000000000000008.

There is a clever strategy that offers the prisoners the chance to increase these odds, and live. What is it?

### The 3 Gods

Behind three characters called A, B and C hide 3 gods known as **'true', 'false' and 'random'**.

The 'true' god always responds with the truth, the 'false' god always lies and the 'random' one alternates unpredictably between the two.

The challenge is 'simple'!: Discover** the respective identities of A, B **and** C** by asking only three questions to which the answer is either true or false.

Each question can only be asked of one god, but if you decide to question a single god more than once (a maximum of three times), the other gods will not be able to answer.

Your questions may be unrelated to each other.

### The Triangular Cake

When preparing his class for a mathematics competition, a teacher decides to offer his students a cake in the form of a **triangle with three unequal sides**.

He places an order with a cake shop, giving the measurements of the cake's three sides.

The baker orders a box for the cake, giving the same measurements. When the cake is done, however, he finds that while the measurements have been respected, the shape is symmetrical, rather than identical, to that of its cake.

He calls the maths teacher to ask how he should cut the cake such that it fits in the box.

The teacher replies that **two cuts will suffice**.

How should these be made?

### The Cat and Mouse Problem

A cat and a mouse decide to play "heads or tails".

To liven up the game, they decide to change the rules: Each player must choose** a combination of 3 results** (e.g. heads, tails, heads).

They toss the coin many times, and the first to see one of his combinations appear in three consecutive coin tosses wins the game.

The **two players cannot choose the same combination**.

The cat, feeling himself to be the stronger player, starts first. The mouse, the smarter of the two, decides to let him go ahead.

How can the **chance of winning** be increased, for each player?

### The Duck and the Cat

There is a duck in the middle of a circular pond. At the edge of this pond is an impatient cat.

While the duck would like to taste the grass at the edge of the pond, the cat would very much like to taste the duck!

The cat doesn't know how to swim, and is too afraid of water to enter the pond.

The duck, meanwhile, has wings which are too small to let him fly away.

Knowing that the cat can run four times faster than the duck can swim, **is it possible for the duck to reach the edge of the pond without getting caught by the cat?**

## More Maths Brain Teasers to Challenge Your Maths Abilities

A brain teaser is a question whose answer **should be** fun to come by. Granted, no everyone enjoys brain teasers – in which case they would be frustrating rather than fun but, once you get the hang of them, you too will find them amusing.

There are all types of brain teasers; some involving conditions or situations; others predicated on a certain sequence. For example:

Mary’s father has five daughters: Nana, Nene, Nini and Nono. What is the name of the fifth daughter?

Some might follow the vowel pattern and proclaim Nunu to be the fifth daughter but, as so often is the case in brain teasers, the answer is in the question.

If Mary’s father has five daughters, then Mary must be one of them... right?

Maths brain teasers can also have such obvious answers but sometimes require more thought. For example:

If three birds lay three eggs in three days, how many eggs does one bird lay in one day?

The answer to this one is ambiguous. From **a mathematical perspective**, one bird lays 1/3 of an egg in one day. However, as** no bird known to man** can lay just one-third of an egg, the logical answer is either **one** or **none.**

*See what we mean by frustrating?*

**Brain teasers should be level-appropriate**; naturally, we would not want to torture any math students with puzzles that require some knowledge of higher algebraic functions when they’re not at that learning stage!

### Brain Teasers for Younger Learners

Some addition required: using only 8 eights, how can you add them together to equal 1000?

Answer: 888+88+8+8+8 = 1000

I’m thinking of a three-digit number. The second digit is four times as big as the third and the first is three less than the second.

Answer: 141

Which three numbers yield the same result whether they are added or multiplied together?

Answer: 1, 2 and 3

Liam made four snowballs on Monday. On Tuesday, he made nine and on Wednesday, 14. How many did he make on Thursday and Friday?

Answer: increase the amount by five each day.

### More Advanced Brain Teasers

The highest temperature recorded at Heathrow on January 1

^{st}was -1 degree. On January 2^{nd}, it was 4 degrees. What was the temperature on January 3^{rd }if the average temperature for the first three days of January was 2 degrees?Answer: 3 degrees

A hot dog vendor wants to buy equal amounts of sausages and buns but the buns come in packs of 10 and sausages in packs of 8. How many packs of each would the vendor need to buy to achieve his goal?

Answer: 10 packs of sausages and eight packs of buns.

All of these statements are true: AxB=12; A+B+C=12; and B-A=1. What sequential numbers are represented by A, B and C?

Answer: A=3, B=4, C=5

### The Difference Between Brain Teasers and Riddles

Many people make no distinction between these two types of mind benders. After all, they are both **questions designed to make you think**... right?

The difference between brain teasers and riddles lies in their intent.

A riddle sometimes uses **double meanings or other phrasing** that demand ingenuity and non-linear thinking to solve. Here is an excellent example of such:

What starts with T, ends with T and has T in it?

A teapot!

A brain teaser, on the other hand, generally does not employ tricky construction or multiple meanings:

I am an odd number. Take away one letter and I become even. What number am I?

The answer, seven, is twice true: 7-1=6, which is an **even number**, added to the fact that removal of the S leaves** the word even**!

Brain teasers are generally for amusement purposes so they would be effective **as a warm-up or icebreaker** at the start of a maths lesson.

If you are a tutor, you might keep a stock of these brain teasers in reserve to lighten the mood and **get your tutee mentally prepared** to learn.

**Caregivers:** you too can cultivate your charges’ thinking by launching the occasional brain teaser.

Stuck in traffic? Waiting at the doctor’s office? Perfect time to exercise the mind!

## Maths Logic Puzzles

Just like traditional puzzles whose interlocking pieces form a complete picture, logic puzzles embrace the same principle – minus the pieces.

Logic puzzles require the participant to put those figurative pieces together correctly using only their thought processes... and perhaps a pencil to make notes.

Some of the more **popular logic puzzles involving numbers** include Sudoku and KenKen. One particular logic puzzle that swept the globe and was even a central theme to the Hollywood blockbuster movie The Pursuit of Happyness is...

*Can you put the pieces together to come up with the right answer?*

**Rubik’s Cube**, the brainchild of architect Erno Rubik, captivated the world immediately upon its premiere in 1980. Since then, several iterations have evolved from the initial cuboid concept: Rubik’s Snake and Rubik’s Magic, as well as a host of custom-built puzzles and even digital versions.

Today, nearly 40 years after the cube’s debut, countries all around the world host** speed-cubing competitions**. Some of the challenges involve blind cubing, cubing with your feet and, of course, solving cube puzzles in the least amount of time. The World Cubing Association regulates and sponsors these events.

Mr Rubik devised the cube puzzle as a challenge for his students – he was a professor of architecture at the time.

The device, made of wood and held together by rubber bands, was meant to give his pupils hands-on experience manipulating a 3-dimensional object.

He had not intended to **create a logic puzzle** and, indeed, was unaware that it was one until he attempted to restore it to its initial condition – all colours on their respective sides.

*So, if you find yourself with idle time and your hands are just twitching for something to do, pick up a Rubik’s cube and see if you can devise a new algorithm to solve it!*

In the meantime, here are **a few other logic puzzles** to try your hand at. *Unfortunately, they are not the kind you can physically manipulate!*

You have a bag containing 10 apples. You encounter 10 friends, each who want an apple. You distribute the apples – one to each friend and have one left in the bag. How can that be?

Answer: you gave the last friend the bag containing the last apple.

You have three small bags, each containing two marbles. Bag 1 has two blue marbles, bag 2 has two green marbles and bag 3 has one green and one blue marble. Reaching into one bag, you extract a blue marble; how probable is it that the marble remaining in that bag is blue?

Answer: there is a 2/3 probability that the remaining marble will also blue.

How can you add two 3-digit numbers to always equal 1089?

Answer: pick a 3-digit number and then reverse it. Subtract your original number from the reversed number and reverse that result. Add that number to the subtraction result to obtain 1089

A 4-inch, solid cube of wood is painted red on all 6 sides. And then, it is cut 1-inch cubes. How many of those cubes will have 3 red sides, how many will have 2 red sides or 1 red side and how many will have no red at all?

Answer: 8 cubes will have 3 red sides, 24 will have 2 red sides, 24 will have 1 red side and 8 will have no red at all.

Some of these puzzles seem to defy logic, as in the case of #2: how can one have **a 2/3 probability** of the second marble being blue if green marbles outnumber the blue after your first draw?

This phenomenon is known as the Monte Hall Problem and relies on intuition and elimination.

1. You can** eliminate the bag** containing only green marbles; you obviously didn’t pick from it. *That leaves 2 bags, each containing a blue marble.*

2. Now that you have (figuratively) eliminated 2 green marbles, the number of **blue marbles still in the bags is double** the number of green marbles...

3. As the total number of marbles to select from is 3 and **two of them are blue**, you have a 2-in-3 chance of again selecting blue.

Most aver that you would have a 50% chance of choosing blue again but, **now that we’ve done the math**, we see that assumption is not correct!

Likewise with the red-painted cube. Many** seem to assume the cube is hollow** and neglect to add the eight unpainted cubes that make up the core of the larger cube.

Finally, the bag of apples and all of those greedy friends: many intuit that **each apple will be removed** from the bag to be given away but that is only necessary for the first nine apples.

*Let the tenth greedy friend dispose of the apple bag for you!*

As for the three digits always totalling 1089? It’s actually quite interesting to see how much fun playing with algebra can really be...

## Bonus: Three Maths Riddles

If you like fun maths games, word problems, mah-jong or brain-teasers, you'll love these puzzles:

### #1

### #2

### #3

## Man and Mystery

Man has always sought to understand the world in which he evolved.

He has researched untiringly, consulted countless tomes and debated with his contemporaries, to better understand the world of maths. Each answer has lead to more questions.

The desire to solve riddles is part of our genetic heritage:

We are born to seek answers.

Why are we here, on Earth? Is there life after death? Who were the first humans? How did they live?

Since antiquity, some of the **great mysteries rooted in maths** and physics have eluded our understanding:

- How were the Pyramids of Egypt constructed; for what purpose were they so arranged?
- How can we explain the mathematical genius of great men like Leonardo da Vinci, Archimedes, Newton, Henri Poincaré and Stephen Hawking?
- Other archaeological mysteries of mathematical importance which still confound our understanding include: The Sphinx in Egypt, the pillar of Delhi; an iron pillar more than 7 meters high and 1600 years old which has never rusted, and the Megalithic Spheres of Costa Rica; 300 spheres each 2 meters in diameter and weighing 16 tonnes, whose period of origin and purpose remain unknown.

Becoming absorbed in maths problems can be a very good way for someone to forget their troubles.

To do so, you may need to use:

- geometry
- mental arithmetic
- applied mathematics
- long division
- mathematical theorems
- trigonometry

## Maths puzzles and Lessons

### To Love Maths... or not

Should we all love maths and mathematical equations?

Why do they matter?

Why do some people turn off when it comes to this topic?

### What are some Reasons to Love solving Maths problems?

- Maths help can ascertain whether something is true or false.
- There is a certain elegance in mathematical theories. Because of their very conciseness and simplicity, you may find that you're able to gain new understanding with only a small amount of study.
**The important thing is to always seek to understand**, rather than learning dozens of formulas and theorems without grasping their ins and outs. - Maths can be very useful in poker when it comes to winning a bet!
- Maths is a very powerful tool: It is possible to achieve exceptional results and applications that at first seem beyond our reach.
- Thanks to maths, you will not only gain a deeper understanding of the world around you, but be
**better able to approach other disciplines**such as physics, chemistry and economics. - Maths is like a game in that it is logical, formal and
**stimulates your brain**as do games like chess, sudoku and even Candy Crush Saga! - Once you have grasped the main principles, maths becomes a kind of second nature that helps you understand and solve the problems around you.

### Why do many find Maths so Difficult?

**Mathematics is its own language**, and to use it well, you need to master its specific grammar, vocabulary and spelling. There are rules to be learned and those to be applied without question.- As a subject, it requires a great deal of
**self-discipline**. It's not good enough to settle for 'almost' in maths: You have to be**concise and methodical**.

- Finally, maths is a demanding discipline that requires
**regular and consistent practice**. Whether you are alone in front of a screen, or your textbook, whether you are taking lessons with a**private**maths tutor, you need to be hard-working and persevere, especially outside of traditional maths classes, if you are in such a program of study.

### What can you expect to study on maths courses?

**GCSE**

According to **the specifications set out by the government**, a GCSE in mathematics should enable students to:

1. develop fluent knowledge, skills and understanding of mathematical methods and

concepts

2. acquire, select and apply mathematical techniques to solve problems

3. reason mathematically, make deductions and inferences and draw conclusions

4. comprehend, interpret and communicate mathematical information in a variety of

forms appropriate to the information and context.

During a GCSE course, **the primary topics** that you will encounter in your maths studies are:

- Number
- Algebra
- Ratio, proportion and rates of change
- Geometry and measures
- Probability
- Statistics

Surely there's some room to have fun among these modules? Of course, much of it **depends on your teacher, their teaching style and the time you have to cover the content in lessons** so don't go giving your tutor a hard time of it when they have a class that won't settle quickly.

That said, there's no harm in asking if your teacher has heard of these cool math games and if you might be able to try some similar puzzles and games during class!

**A Level**

As for **A Level and AS students**, the government thinks it's important for them to:

- understand mathematics and mathematical processes in a way that promotes confidence fosters enjoyment and provides a strong foundation for progress to further study
- extend their range of mathematical skills and techniques
- understand coherence and progression in mathematics and how different areas of mathematics are connected
- apply mathematics in other fields of study and be aware of the relevance of mathematics to the world of work and to situations in society in general
- use their mathematical knowledge to make logical and reasoned decisions in solving problems both within pure mathematics and in a variety of contexts, and communicate the mathematical rationale for these decisions clearly
- reason logically and recognise incorrect reasoning
- generalise mathematically
- construct mathematical proofs
- use their mathematical skills and techniques to solve challenging problems which require them to decide on the solution strategy
- recognise when mathematics can be used to analyse and solve a problem in context
- represent situations mathematically and understand the relationship between problems in context and mathematical models that may be applied to solve them
- draw diagrams and sketch graphs to help explore mathematical situations and interpret solutions
- make deductions and inferences and draw conclusions by using mathematical reasoning
- interpret solutions and communicate their interpretation effectively in the context of the problem
- read and comprehend mathematical arguments, including justifications of methods and formulae, and communicate their understanding
- read and comprehend articles concerning applications of mathematics and communicate their understanding
- use technology such as calculators and computers effectively and recognise when such use may be inappropriate
- take increasing responsibility for their own learning and the evaluation of their own mathematical development

The **overarching themes** of an A-Level math course are, therefore, the following, with numerous sub-topics making up each section.

- mathematical argument, language and proof
- mathematical problem solving
- mathematical modelling

**Degree-level Math**

At degree level, you can choose to study towards **a basic Maths qualification** (no less demanding than a more specialised course, we'll add!) but you can also choose to **focus on a specialism** like mathematical finance, mathematical physics, mathematical biology, actuarial maths, history of maths, special relativity, quantum theory or medical statistics, though many of these are available as modules in later years of a Maths bachelor of science degree.

The** topics you'll study** during your introductory year on a Maths degree include:

- calculus
- algebra
- analysis
- mechanics
- probability
- statistics
- geometry
- vectors
- computational maths

Of course, even if you **like maths and think you are quite good at it** (and you certainly will after all these brain teaser games!) then you don't have to study Maths, per say.

Why not consider economics, engineering, computer science, physics or theoretical physics, another science subject such as chemistry, biology or psychology, or accounting or finance as ideas?

### Where can math lead you professionally?

If you're **seeking a career in Finance**, then some level of math qualification is required, but you may not necessarily need a degree. Areas you might want to consider are accountancy, actuarial work, investment management, investment banking, and retail banking.

Aside from the financial sector, there are industries such as engineering and information technology that can benefit from someone who is good with numbers.

Alternatively, different avenues that maths graduates follow include roles like defence and intelligence officer, statistician, operational researcher, academic mathematician, teacher of Maths in a primary or secondary school, or positions within the law, media, business or public sectors.

You can see what jobs are out there for maths enthusiasts by visiting a job website.

By now, hopefully, you'll have seen, that **maths lessons get your neurones firing and can take you in many places**! So, why not find an online math tutor with Superprof?

Equally as important, they provide us with a **better understanding of our world**.

Finally, when you know where to look, it's easy to see the imprint of maths in our daily lives!

Share your maths riddles and answers in the comments!

I’m thinking of a three-digit number that whose digits are all different. The second digit is four times as big as the third and the first is three less than the second. Answer: 582

A hot dog vendor wants to buy equal amounts of sausages and buns but the buns come in packs of 10 and sausages in packs of 8. How few packs of each would the vendor need to buy to achieve his goal?

Answer: five packs of sausages and four packs of buns.