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Parallelogram #1

Welcome to the first of four weekly Parallelograms, a collection of mathematical challenges designed to stretch your brain and make your neurons more squiggly. You can start and stop whenever you like, but you will need to complete all the challenges by 7pm on Sunday.

This week’s Parallelogram challenge is in seven parts… some Challenges will have a single theme, while others have sections that shoot off in wildly different directions. In short, these challenges are often going to be a random walk through the mysteries of mathematics. Be prepared to encounter all sorts of weird ideas.

  1. Maths jokes
  2. Mini cross-number
  3. Maryam’s inspiration
  4. Domino toppling
  5. Junior Maths Challenge question
  6. Weird optical illusions

Instructions

  • You will get points for every correct answer (or nearly correct answer).
  • Some questions have a hint (sometimes two hints). Please think hard and try hard before resorting to the hint. The hints will help you get the right answer, but you will lose marks.
  • It does not really matter what you score get, the main thing is that you think hard about the problems … and then learn where you went wrong when the answers appear.

1. Maths jokes

It sounds odd, but there are maths jokes. Lots of jokes. In fact, lots of mathematicians are very funny. Some of my favourite comedians (Dara O'Briain, Romesh Ranganathan and Dave Gorman) studied maths at university, and I have written a whole book about the mathematicians who write The Simpsons.

So, how good is your mathematical sense of humour? And how much maths do you know? Below are 4 jokes with 3 punchlines each. Can you spot the punchline? You will need to a bit of maths in order to work out the right answer.

½ mark

1.1. Why is 6 afraid of 7?

(a) Because 1, 2, 3
(b) Because 7, 8, 9
(c) Because 5 predates 6

The correct answer is (b), because it sounds like ”7 ate 9”.

½ mark

1.2 What are the 10 kinds of people in the world?

(a) Set 0, Set 1, Set 2, …, Set 9
(b) α, β, γ, δ, ε, ζ, η, θ, ι, κ
(c) Those who understand binary, and those who don’t

In binary 10 = 2, so 10 kinds of people could mean 2 kinds of people.

½ mark

1.3 What does the “B” in Benoit B Mandelbrot stand for?

(a) Benoit B Mandelbrot
(b) Binomial
(c) Breviation

This was a tough one. Benoit B Mandelbrot helped developed the mathematics of fractals, which are structures or shapes that appear similar when you zoom in or zoom out. Trees, clouds and certain mathematical shapes are fractal, as shown on the right. The bit of the tree in the red box looks just like the whole tree… and you zoom in even further and see more and more micro-replicas of the whole tree.

This video shows what happens if you zoom into the so-called Mandelbrot Set, and you see the same pattern (a sort of seahorse’s tail) reappearing again, and again, and again. So, returning to the joke, the B represents Benoit B Mandelbrot, because zooming into the name gives you the whole name again.

½ mark

1.4 Why did the chicken cross the Möbius strip?

(a) To get to the other… er…?
(b) To reach a finite conclusion
(c) To integrate itself into the tarmac

Another toughie – a Möbius strip is a strange mathematical shape which only has one side, so you can’t get to the other side. As shown here, it is just a loop with a single twist, but if you imagine running your finger along one side of the loop you will find that your finger covers every bit of the loop … so it only has one side. Möbius strips are fantastic, one of my favourite bits of geometry, and we’ll return to them in a future Parallelogram.

2. Mini cross-number

2 marks

2.1 What is the digit x in this cross-number?

Across
1. A cube
3. A cube

Down
1. One less than a cube
2. A number

The correct answer is x = 4.

1 across, 3 across & 1 down are all cubes (or one less than a cube) and they are all 2-digit numbers, which means they are between 10 and 99. The only cube numbers between 10 and 99 are 27 and 64. So 1 across and 3 across must be 27 and 64, or 64 and 27. Meanwhile, 1 down must be 26 or 63. The cross-number only works if 1 down is 26, because 3 across can then be 64, which means x = 4.

HINT 1

1 across, 3 across & 1 down are all cubes (or one less than a cube) and they are all 2-digit numbers, which means they are between 10 and 99. There are not very many cube numbers between 10 and 99, so start by writing them down and then see how they might fit into the grid.

3. Maryam’s inspiration

You might be surprised to know that there is no Nobel Prize for maths. However, there is an even bigger prize for mathematicians, which is called the Fields Medal. It is only given every 4 years, so it is very rare and precious.

In 2014, Maryam Mirzakhani became the first woman to win a Fields Medal. She was born in Iran and now works in America, and you can find out about life and work in the short video below.

Not surprisingly, Maryam talks about some incredibly complex maths, so don’t worry if you find it confusing. In fact, if you think you understand what Maryam is describing, then you probably don’t. Watch it carefully and answer the question below.

1 mark

3.1 What indoor sport does Maryam mention in one of her mathematical examples?

(a) Chess
(b) Darts
(c) Billiards
(d) Table tennis

Maryam talks about a problem involving the path of a ball bouncing around a billiard table.

Maryam has said that she first became fired up about mathematics when her big brother told her a story about the great mathematician Carl Friedrich Gauss. When Gauss was a schoolboy, his teacher tried to keep the class busy by asking them to add up all the numbers from 1 to 100. I suspect that he wanted to nip out for a 15-minute nap.

However, before the teacher had even left the room, Gauss’s hand shot up and he announced that the answer was 5,050.

Gauss was right, but how did he add up the first 100 numbers in just a few seconds?

Here is Gauss's trick in three stages:

  1. Pair up all the numbers in the following way and add them up:

    1 + 100 = 101,
    2 + 99 = 101,
    3 + 98 = 101

    50 + 51 = 101

  2. So, you have 50 pairs of numbers, which all add up to 101.

  3. So, the result is 50 x 101 = 5,050!

Another way to think about this is:

  • (n+1) represents step 1, because you pair up all the numbers, so that all the pairs add up to (n+1).
  • (n/2) represents step 2, because the number of pairs is equal to the total number of numbers divided by 2.
  • (x) represents step 3, because you then multiply the sum of each pair by the number of pairs.

Let’s see if you can apply Gauss’s trick.

1 mark

3.2 Add up all the numbers from 1 to 50

The correct answer is 1275.

1 mark

3.3 Add up all the numbers from 1 to 1,000

The correct answer is 500500.

If you got either of these questions wrong, then take a look at this website, which goes into more detail about ways of adding the numbers from 1 to 100, and is worth looking at.

On 15 July 2017, two weeks after this Parallelogram was created, I was shocked and saddened to hear about the death of Maryam Mirzakhani. The Iranian-born NASA scientist, Firouz Naderi, wrote: “A light was turned off today, it breaks my heart…. Gone far too soon.” Commenting on her brilliant work with Alex Eskin, she commented: “If we knew things would be so complicated, I think we would have given up … I don’t know; actually, I don’t know. I don’t give up easily.”

I wanted Mirzakhani to be part of the first Parallelogram because her story is so inspirational. In future years, I hope that students working through this Parallelogram will continue to be inspired by her achievements.

4. Domino toppling

I watched this terrific video last week. Take a look and answer the question afterwards.

1 mark

4.1 Starting with 5mm tall domino, how many dominoes would you need to knock down the Empire State Building?

(a) 30
(b) 100
(c) 30,000
(d) 100,000,000

Each domino can knock over a domino that 1.5 times bigger, which does not seem big, but when this happens over and over again then the dominoes get bigger and bigger very quickly. This is known as exponential growth and we will return to this in a future Parallelogram. In the meantime, here is an even more extraordinary example of domino toppling where the dominos get bigger and bigger. In fact, it was a world record at the time.

5. Junior Maths Challenge question

1 mark

5.1 Which of these three statements are true?

(i) Doubling a positive number always makes it larger.
(ii) Squaring a positive number always makes it larger.
(iii) Taking the positive square root of a positive number always makes it smaller.

The correct answer is just (i).

Statement (i) is true since 2x > x, if x > 0.
Statement (ii) is not true. For example, (½)2 = ¼, which is not larger than ½.
Statement (iii) is also not true. For example, √¼ = ½, which is not smaller than ¼.

6. Weird optical illusions

The amazing, mind-warping objects in this video have been created by mathematical artist and engineer Professor Kokichi Sugihara. These incredible illusions literally mess with your brain.
Sometimes the geometry that you study at school might seem ordinary – what could be more ordinary than the angles of a triangle or the symmetry of a square? – but your teachers are providing you with the first stepping stones towards gaining an understanding of the nature of space, and how you can build the sort of flabbergasting structures in this video. There is one question afterwards and then you can hit the submit button and your first Parallelogram has been completed.

1 mark

6.1 Professor Sugihara’s work could help:

(a) Drivers
(b) Architects
(c) Surgeons

The correct answer is (a), because drivers can make errors if their brains are fooled by an optical illusion caused by the geometry of the road.

Find out more

You can read more about mathematician Maryam Mirzakhani in this interview published in the Guardian: 'The more I spent time on maths, the more excited I got'.

This article from the Telegraph newspaper provides some history about the Fields Medal, the biggest prize in mathematics.

This website goes into more detail about ways of adding the numbers from 1 to 100, and is worth looking at.

Credits

  • Portrait of Carl Gauss by Christian Albrecht Jensen, 1840 (public domain)
  • Image of Maryam Mirzakhani taken from Alchetron

The problems in this Parallelogram come from a variety of places and we will list them in the fourth and final Parallelogram.


Before you can finish your Parallelogram, you need to take 30 seconds to answer a few questions that will tell us what you thought about this week’s challenges.

Too easyToo difficult
Too shortToo long
HorribleLots of fun
Very boringVery interesting

Only hit the submit button if you have finished the whole
Parallelogram and checked your answers.

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