Parallelogram 6 Year 10 17 Oct 2019Heroic Hidden Figures

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Noun: Parallelogram Pronunciation: /ˌparəˈlɛləɡram/

  1. a portmanteaux word combining parallel and telegram. A message sent each week by the Parallel Project to bright young mathematicians.

These challenges are a random walk through the mysteries of mathematics, most of which will be nothing to do with what you are doing at the moment in your classroom. Be prepared to encounter all sorts of weird ideas, including a few questions that appear to have nothing to do with mathematics at all.

  • Tackle each Parallelogram in one go. Don’t get distracted.
  • When you finish, remember to hit the SUBMIT button.
  • Finish by Sunday 27 Oct if your whole class is doing parallelograms.

IMPORTANT – it does not really matter what score you get, because the main thing is that you think hard about the problems... and then examine the solution sheet to learn from your mistakes.

As most of you will have a half-term coming up, this Parallelogram is longer than usual and has a couple of extra questions. And your next parallelogram will be in two weeks time on 31st October, and it will be appropriately terrifying.

1. Heroic Hidden Figures

Back in year 7, Parallel featured a clip from the film “Hidden Figures”, but it is such an incredible TRUE story that we are revisiting it. Below is the trailer, but first here’s a reminder of what the film is all about. And, remember, it is a TRUE story.

Hidden Figures tells the incredible story of Katherine Jonson, Dorothy Vaughan and Mary Jackson – brilliant African-American women working at NASA who served as the brains behind the launch into orbit of astronaut John Glenn, a stunning achievement that turned around the Space Race. The visionary trio broke all the barriers put in front of them because of their gender and race, and inspired generations.

2 marks

1.1. When Mary Jackson complains about the car being a pile of junk, Katherine Johnson says that she is welcome to ride in the bus... in particular, the back of the bus. She is probably referring the fact that it had been the rule in some parts of America for black men and women to sit at the back of segregated buses. It was one woman in particular who fought this injustice by refusing to give up her seat to a white passenger in Montgomery (Alabama) on 1 December 1955.

In her autobiography, she explained why she refused to give up her seat: “People always say that I didn't give up my seat because I was tired, but that isn't true. I was not tired physically, or no more tired than I usually was at the end of a working day. I was not old, although some people have an image of me as being old then. I was forty-two. No, the only tired I was, was tired of giving in.”

The surname of the woman was Parks, but was her first name?

Correct Solution: Rosa

2. Euler’s Method

In this clip from “Hidden Figures”, Kathryn Johnson is trying to solve a difficult problem, and realises that some maths from centuries ago might offer a solution. In case it is not clear from the clip, solving this problem is a matter of life and death, as it might decide whether or not the astronaut John Glenn returns to Earth.

Kathryn Johnson relied on Euler’s Method, which allows mathematicians to solve a difficult equation by finding an answer that is not perfectly accurate, but which is accurate enough for the purpose in question.

One approximation that is often used by mathematicians is to say that:

1+a21+a×2

So, 1+0.0121+2×0.011.02

The actual answer is 1.0201, so it is a very good approximation.

So, 1+0.0521+2×0.051.1

The actual answer is 1.1025, so it is a reasonably good approximation.

Now try the approximation for yourself.

2 marks

2.1 We know that 1.12=1.21, but what is the approximate answer using the method above?

  • 1.0
  • 1.11
  • 1.12
  • 1.2
  • 1.22

1.12=1+0.121+2×0.11.2

2 marks

2.2 We know that 1.22=1.44, but what is the approximate answer using the method above?

  • 1.0
  • 1.22
  • 1.4
  • 1.44
  • 1.444

1.22=1+0.221+2×0.21.4

2 marks

2.3 We know that 1.52=2.25, but what is the approximate answer using the method above?

  • 1.5
  • 2.0
  • 2.5
  • 2.55
  • 3.0

1.52=1+0.521+2×0.51+12

2 marks

2.4 We have looked at approximations using this technique for numbers ranging from 1.01 to 1.5.

What happens to the approximation as the number gets bigger and bigger?

  • The approximation gets better and better
  • The approximation has a similar level of accuracy
  • The approximation gets worse and worse

If we fully expand the bracket 1+n2, then we get 1+2n+n2.

Our approximation is just 1+2n, as we are ignoring the n2 bit. So, the approximation works very well if n2 is very small, which is true when n is much smaller than 1.

So, this approximation method works very well for numbers that are only a tiny bit bigger than 1, but very badly for numbers that are much bigger than 1, say 1.2 and bigger.

3. Intermediate Maths Challenge Problem (UKMT)

3 marks

3.1 When travelling from London to Edinburgh by train, you pass a sign saying “Edinburgh 200 miles”. Then, 312 miles later, you pass another sign saying “Half way between London and Edinburgh”.

How many miles is it by train from London to Edinburgh?

  • 393
  • 396 12
  • 400
  • 403 12
  • 407

After travelling a further 3 12 miles towards Edinburgh after passing the "Edinburgh 200 miles" sign, the train is (200 − 3 12) miles = 196 12 miles from Edinburgh. As the train is now half way between London and Edinburgh, the distance from London to Edinburgh is (2 × 196 12) miles = 393 miles.

4. Intermediate Maths Challenge Problem (UKMT)

4 marks

4.1 The diagram shows a heptagon with a line of three circles on each side. Each circle is to contain exactly one number. The numbers 8 to 14 are distributed as shown and the numbers 1 to 7 are to be distributed to the remaining circles. The total of the numbers in each of the lines of three circles is to be the same.

What is this total?

  • 18
  • 19
  • 20
  • 21
  • 22
Show Hint (–1 mark)
–1 mark

Each of the integers from 1 to 7 occurs in exactly two of the seven lines of three circles. Each of the integers from 8 to 14 occurs in exactly one of these lines.

Show Hint (–1 mark)
–1 mark

Therefore, if we add up all the totals of the numbers in the seven lines, we are adding each of the integers from 1 to 7 twice, and each of the integers from 8 to 14 once.

Each of the integers from 1 to 7 occurs in exactly two of the seven lines of three circles. Each of the integers from 8 to 14 occurs in exactly one of these lines.

Therefore, if we add up all the totals of the numbers in the seven lines, we are adding each of the integers from 1 to 7 twice, and each of the integers from 8 to 14 once. It follows that the sum of the seven line totals is

2×1+2+3+4+5+6+7+8+9+10+11+12+13+14=2×28+77
=56+77
=133.

Since the total for each of the seven lines is the same, the total of the numbers in one line is 133÷7=19.

We have shown that, if the numbers can be placed so that the total of the numbers in each of the lines of three circles is the same, then this common total is 19.

So, assuming, as we are entitled to, that one of the options is correct, we deduce that B is the correct option. For a complete solution, however, it is necessary to show that the numbers can be placed so that the total of the numbers in each line is 19.

5. The world’s top thinker

The magazine Prospect recently asked its readers to vote on the world’s top thinker.

The winner was a mathematician who had won the world’s most prestigious prize for mathematics, the Fields Medal … arguably harder to win than a Nobel Prize.

Caucher Birkar grew up in war-torn western Iran in the 1980s, where his family were farmers: “My parents were farmers, as all my ancestors were farmers. We had a piece of land around the village and we—including myself and my brother—were charged with going to work on these fields, and grow all kinds of vegetables… wheat, and barley… My mother never attended any school. My father attended school up to primary school, and then he did not continue because it just wasn’t really practical.”

You can read his remarkable story here – read the article and answer the two questions below.

2 marks

5.1 Caucher Birkar is a name he adopted when he came to Britain. What does it mean?

  • magnificent mathematician
  • modest mathematician
  • miracle mathematician
  • magical mathematician
  • migrant mathematician
2 marks

5.2 According to Birkar, what is mathematics?

  • part science, part art
  • part science, part engineering
  • part science, part philosopher
  • party science, party alchemy
  • part science, party mystery

6. Intermediate Maths Challenge Problem (UKMT)

4 marks

6.1 Anna has the same number of brothers as she has sisters. Each one of her brothers has 50% more sisters than brothers.

How many children are in Anna's family?

  • 5
  • 7
  • 9
  • 11
  • 13
Show Hint (–1 mark)
–1 mark

It’s not very mathematical, and it's not very productive, but you could try trial and error to reduce the possibilities. For example, is 5 possible? 5 sons & 0 daughters, or 4 & 1, or 3 & 2, or 2 & 3, or 1 & 4, or 0 & 5? No, clearly it is not.

You could try the same approach with the other options, but it will take you a while… but you would get an answer eventually.

Show Hint (–1 mark)
–1 mark

Suppose that Anna has n brothers. Since she has equal numbers of brothers and sisters, she also has n sisters.

Consider one of Anna's brothers, say Ben. All of Anna's sisters are also Ben’s sisters, and Ben also has Anna as one of his sisters. So Ben has n+1 sisters. All of Anna brothers, other than Ben himself, are also Ben’s brothers. So Ben has n1 brothers.

Since Ben has 50% more sisters than brothers, it follows that

n+1n1=150100=32

Suppose that Anna has n brothers. Since she has equal numbers of brothers and sisters, she also has n sisters.

Consider one of Anna's brothers, say Ben. All of Anna's sisters are also Ben’s sisters, and Ben also has Anna as one of his sisters. So Ben has n+1 sisters. All of Anna brothers, other than Ben himself, are also Ben’s brothers. So Ben has n1 brothers.

Since Ben has 50% more sisters than brothers, it follows that

n+1n1=150100=32

and therefore

2n+1=3n1,

that is,

2n+2=3n3,

from which it follows that

n=5.

Therefore Anna has 5 brothers and 5 sisters. So, including Anna, there are 5 + 5 + 1 = 11 children in her family.

There will be more in TWO weeks, so check your email or return to the website on Thursday at 3pm on October 31st.

In the meantime, you can find out your score, the answers and go through the answer sheet as soon as you hit the SUBMIT button below.

When you see your % score, this will also be your reward score. As you collect more and more points, you will collect more and more badges. Find out more by visiting the Rewards Page after you hit the SUBMIT button.

It is really important that you go through the solution sheet. Seriously important. What you got right is much less important than what you got wrong, because where you went wrong provides you with an opportunity to learn something new.

Cheerio, Simon.