Parallelogram 13 • Level 5 • 28 Nov 2024Gravitational Waves

Noun: Parallelogram Pronunciation: /ˌparəˈlɛləɡram/

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

Tackle each Parallelogram in one go. Don’t get distracted.
When you finish, remember to hit the SUBMIT button.
Finish by Sunday night 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.

1. Writing one hundred

Can you write the number one hundred on a piece of paper, but without taking your pen off the paper?

Have a think about the problem, and perhaps take a lateral approach. That means, instead of taking a headlong run at the problem, think about it from different angles, examine what the question might mean, or what it might not mean. Be creative.

The answer is in the short video below, but have a hard think before you click play. You really need to come up with some sort of answer, even if it is not brilliant, or even correct.

(If you have problems watching the video, right click to open it in a new window)

2 marks

1.1 How well did you do?

I got the correct answer as shown in the video, or I got a different answer, but one that was interesting and perhaps equally good.
I had no idea.
(Not answered)

For example, you could write “c”, which is the Roman numeral for 100 and does not require you to you take you pen off the paper

2. Gravitational Waves

One of the greatest scientific discoveries this century was the detection of gravitational waves, which were predicted by Albert Einstein at the start of the previous century.

Take a look at the video below, presented by Professor Kelly Holley-Bockelmann. It is longer than the usual videos we have in Parallel, but it is a brilliant lecture, so it is well worth watching all 18 minutes.

(If you have problems watching the video, right click to open it in a new window)

2 marks

2.1 When you hug someone, is that hug written on spacetime?

Yes
No
(Not answered)

2 marks

2.2 When you smile, will those spacetime ripples travel to distant stars?

Yes
No
(Not answered)

2 marks

2.3 Do your actions matter on a cosmic level?

Yes
No
(Not answered)

3. Intermediate Maths Challenge Problem (UKMT)

3 marks

3.1 Which of the following is not the sum of two primes?

5
7
9
11
13
(Not answered)

The sum of two odd numbers is even. Therefore, no odd number is the sum of two odd primes. The only even prime is 2. It follows that if any of the odd numbers 5, 7, 9, 11 and 13 is the sum of two primes, it is the sum of an odd prime and 2.

We see that 5 = 3 + 2, 7 = 5 + 2, 9 = 7 + 2, 11 = 9 + 2 and 13 = 11 + 2.

The numbers 3, 5, 7 and 11 are all primes, so each of 5, 7, 9 and 13 is the sum of two primes. However, 9 is not a prime and therefore 11 is not the sum of two primes.

As we showed above, the only odd prime numbers that are the sum of two primes are those of the form p + 2, where p is an odd prime. In this case the numbers p, p + 2 form what is called a twin prime, that is, they are consecutive odd numbers that are both prime. It is an unsolved problem as to whether there are infinitely many twin primes.

According to the Goldbach Conjecture every even number greater than 2 is the sum of two primes. This has not been proved but, at the time of writing, it has been verified for all even numbers up to 4×1018. Some partial results are known. For example, Chen’s Theorem (proved by Chen Jingrun in 1973) says that all but a finite number of even numbers can be expressed as the sum of a prime and a number which is either prime or the product of two primes.

4. Prime sums

1 mark

4.1 Is 63 the sum of two primes?

Yes
No
(Not answered)

63 = 61 + 2.

1 mark

4.2 Is 95 the sum of two primes?

Yes
No
(Not answered)

For the answer to be yes, the two primes would have to be an odd number and an even number. The only even prime is 2, so the odd number would have to be 93, which is not a prime (because 93 = 3 × 31).

5. Intermediate Maths Challenge Problem (UKMT)

5 marks

5.1 The diagram shows a regular octagon.

What is the ratio of the area of the shaded trapezium to the area of the whole octagon?

1 : 4
5 : 16
1 : 3
2 : 2
3 : 8
(Not answered)

Show Hint (–2 mark)

2 mark

We let P, Q, R, S, T, U, V and W be the vertices of the regular octagon. Let K, L, M and N be the points where the diagonals PS and WT meet the diagonals QV and RU, as shown in the diagram.

Because PQRSTUVW is a regular octagon, QKP, WNV, SLR and UMT are congruent right-angled isosceles triangles. We choose units so that the shorter sides of these triangles have length 1.

Show Hint (–1 mark)

1 mark

Then, by Pythagoras’ Theorem, the hypotenuse of each of these triangles has length 12+12=2. Since the octagon is regular, it follows that each of its sides has length 2.

Each of the triangles QKP, WNV, SLR and UMT, forms half of a square with side length 1, and so has area 12.

We let P, Q, R, S, T, U, V and W be the vertices of the regular octagon. Let K, L, M and N be the points where the diagonals PS and WT meet the diagonals QV and RU, as shown in the diagram.

Because PQRSTUVW is a regular octagon, QKP, WNV, SLR and UMT are congruent right-angled isosceles triangles. We choose units so that the shorter sides of these triangles have length 1.

Then, by Pythagoras’ Theorem, the hypotenuse of each of these triangles has length 12+12=2. Since the octagon is regular, it follows that each of its sides has length 2.

Each of the triangles QKP, WNV, SLR and UMT, forms half of a square with side length 1, and so has area 12.

The shaded trapezium is made up of the two triangles QKP and WNV, each of area 12, together with the rectangle KNWP which has area 2×1=2. Therefore the area of the shaded trapezium is 2×12+2=1+2.

The octagon is made up of the four triangles QKP, WNV, SLR and UMT, each with area 12, the four rectangles KNWP, RLKQ, MUVN and STML, each with area 2, and the central square KLMN which has area 2×2=2. Therefore the area of the octagon is 4×12+4×2+2=4+42=41+2.

It follows that the ratio of the area of the shaded trapezium to the area of the octagon is 1+2:41+2=1:4.

6. Intermediate Maths Challenge Problem (UKMT)

5 marks

6.1 In a particular group of people, some always tell the truth, the rest always lie. There are 2016 in the group. One day, the group is sitting in a circle. Each person in the group says, "Both the person on my left and the person on my right are liars."

What is the difference between the largest and smallest number of people who could be telling the truth?

0
72
126
288
336
(Not answered)

Each truth teller must be sitting between two liars. Each liar must be sitting next to at least one truth teller and may be sitting between two truth tellers.

The largest number of truth tellers occurs when each liar is sitting between two truth tellers and each truth teller is sitting between two liars. In this case the truth tellers and liars alternate around the table. So half (that is, 1008) of the people are truth tellers and half (1008) are liars. This arrangement is possible because 2016 is even.

The smallest number of truth tellers occurs when each liar is sitting next to just one truth teller and so is sitting between a truth teller and a liar. In this case as you go round the table there is one truth teller, then two liars, then one truth teller, then two liars and so on. So one-third (672) of the people are truth tellers and two-thirds (1374) are liars. This arrangement is possible because 2016 is divisible by 3.

Therefore, the difference between the largest and smallest numbers of people who could be telling the truth is 1008 − 672 = 336.

There will be more next week, and the week after, and the week after that. So check your email or return to the website on Thursday at 3pm.

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.