Parallelogram 9 • Level 5 • 31 Oct 2024Map of Mathematics

Noun: Parallelogram Pronunciation: /ˌparəˈlɛləɡram/

a portmanteau 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 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. Map of Mathematics

If you have been earning badges after completing the first few Parallelograms, then you might have noticed that the badges are all based on images from the Map of Mathematics diagram designed by Dominic Walliman.

Here is a short video about Dominic’s Map of Mathematics.

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

2 marks

1.1. Which of the following statements is not true?

There is an infinite number of integers.
There is an infinite number of real numbers.
There are more integers than real numbers.
There are more real numbers than integers.
(Not answered)

2. Earthquakes versus nuclear tests

Here is a clip from YouTuber Steve Mould, all about how geologists can tell the difference between vibrations from an earthquake compared to those from a nuclear test... and how you spot the location of a seismic event.

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

2 marks

2.1 Imagine you have seismometers (vibration detectors) at the North Pole and the South Pole, which detect P-waves at the same time... and then detect S-waves at the same time. What do you know?

There was an earthquake
There was a nuclear test
(Not answered)

2 marks

2.2 Where did the event causing the vibrations occur?

At the North Pole
At the South Pole
In the northern hemisphere
In the southern hemisphere
Somewhere on the equator
(Not answered)

If the vibrations arrived simultaneously, then the seismic event must be equidistant between the North and South Poles, so this means it must have occurred somewhere on the equator.

1 mark

2.3 For every 1 second delay between the P-waves arriving and the S-waves arriving, we can add 8 km to the distance of the seismic event. In this example, what was the rough delay between the P-waves and the S-waves arriving?

20 seconds
20 minutes
1,250 minutes
(Not answered)

If the event took place on the equator, then it took place 10,000 Km from the North and South Poles. (The circumference of the Earth is 40,000 km, and the distance between the poles and the equator is one quarter of this.)

10,000/8 ~ 1,250 seconds or roughly 20 minutes.

3. Intermediate Maths Challenge Problem (UKMT)

3 marks

3.1 In the diagram, what is the value of x?

23
24
25
26
27
(Not answered)

Show Hint (–1 mark)

1 mark

The sum of the interior angles of a polygon with n sides is n−2×180°. So the sum of the interior angles of the hexagon in the diagram is 6−2×180°=4×180°=720°.

The sum of the angles at a point is 360°. Therefore the interior angle of the hexagon adjacent to the exterior angle of 75° is 360−75°, and the interior angle adjacent to the exterior angle of x° is 360−x°. Since the sum of the interior angles of the hexagon is 720°, we have 24+27+360−75+26+23+360−x=720.

We can rearrange the left hand side of this equation to obtain 24+27+26+23−75+360+360−x=720.

It follows that 100−75+720−x=720. Hence 25−x=0. Therefore x=25.

4. Intermediate Maths Challenge Problem (UKMT)

4 marks

4.1 How many of the following positive integers are divisible by 24?

22×32×52×73
22×32×53×72
22×33×52×72
23×32×52×72

0
1
2
3
4
(Not answered)

An integer n is divisible by 24 if, and only if, n=24×m, for some integer m. Since the prime factorization of 24 is 23×31, it follows that n is divisible by 24 if, and only if n=23×31×m, for some integer m.

Another way to put this is that an integer n is divisible by 24 if, and only if, in the prime factorization of n, the exponent of 2 is at least 3, and the exponent of 3 is at least 1.

It follows from this that, of the given numbers, there is just one which is divisible by 24, namely 23×32×52×72.

5. Intermediate Maths Challenge Problem (UKMT)

4 marks

5.1 The shaded region in the diagram, bounded by two concentric circles, is called an annulus. The circles have radii 2cm and 14cm.

The dashed circle divides the area of this annulus into two equal areas.

What is its radius?

9cm
10cm
11cm
12cm
13cm
(Not answered)

Show Hint (–1 mark)

1 mark

In answering this question we use the fact that the area of a circle with radius r is πr2.

The area of the shaded annulus is the area inside the outer circle of radius 14 cm minus the area inside the inner circle of radius 2 cm. Therefore this area is, in cm²,

π142−π22=196π−4π=192π.

Show Hint (–1 mark)

1 mark

Let the radius of the dashed circle be x cm.

Similarly, the area between the dashed circle and the inner circle is, in cm²,

πx2−π22=x2−4π.

In answering this question we use the fact that the area of a circle with radius r is πr2.

Let the radius of the dashed circle be x cm.

The area of the shaded annulus is the area inside the outer circle of radius 14 cm minus the area inside the inner circle of radius 2 cm. Therefore this area is, in cm²,

π142−π22=196π−4π=192π.

Similarly, the area between the dashed circle and the inner circle is, in cm²,

πx2−π22=x2−4π.

Since the dashed circle divides the annulus into two equal areas, the area between the dashed circle and the inner circle is equal to half the area of the annulus. Half the area of the annulus is, in cm², 192π÷2=96π. Therefore

x2−4π=96π.

It follows that

x2−4=96,

and hence that

x2=100.

The number x is positive as it is the radius, in centimetres, of the dashed circle. It follows that x=10. Hence the radius of the dashed circle is 10 cm.

If you want to properly explore Dominic Wallimans’ “Map of Mathematics”, then make sure you visit his webpage after you hit the SBUMIT button.

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.