Parallelogram 40 • Level 5 • 6 Jun 2024Classic Countdown

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.
Don’t worry if you score less than 50%, because it means you will learn something new when you check the solutions.

1. Classic Countdown

Take a look at this classic number game from Countdown.

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

Here is an explanation of what happened by Junaid Mubeen, Director of Education, at Whizz Education, who was also a Countdown octochamp.

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

If you want to practice some Countdown problems, then you can visit this interactive website.

2.

3 marks

2.1 Think of a number, double it, then add 3. Multiply your answer by 4 and from this take away 5. Now also take away the number you first thought of. No matter what your first number was, your answer will always be a multiple of:

2
3
5
7
11
(Not answered)

Let the number thought of be x. Then the final number is 42x+3−5−x=7x+7=7x+1.

3.

4 marks

3.1 It has been estimated that the mass of insects caught by spiders in a year in the UK is equal to the mass of the human population of the UK. Assuming this population is 60 million and the average mass of a human is 70kg, what is the mass, in tonnes, of insects caught by spiders per year in the UK?

4.2
42
4200
420000
4200000
(Not answered)

60 million × 70 kg = 4200 million kg = 4.2 million tonnes, since 1 tonne = 1000 kg.

4.

4 marks

4.1 The diagram shows three semicircles, each of radius one.

What is the size of the total shaded area?

π+2
3
32π+1
4
2π−1
(Not answered)

Show Hint (–2 mark)

–2 mark

Redraw the diagram as shown below and the problem becomes much easier.

The shaded area may be divided into a 2 × 1 rectangle plus a semicircle and two quarter circles, all of radius 1.

Hence the total area is that of the rectangle plus that of a circle of radius 1, ie 2 + π.

5.

5 marks

5.1 Suppose that e, i, n and t represent different positive whole numbers, n+i+n+e=9, t+e+n=10 and i=1.

What is t?

2
3
4
5
6
(Not answered)

We are given that 2n+e=8 and t+e+n=10. Subtracting the first equation from the second gives t−n=2.

As n cannot equal 1, the minimum value of t is 4 but this gives n=2, e=4 which is impossible.

If t=5 then n=3 and e=2, which is allowed. If t>5 then n>3 and e is not a positive whole number, so 5 is the only possible value of t.

Before you hit the SUBMIT button, here are some quick reminders:

You will receive your score immediately, and collect your reward points.
You might earn a new badge... if not, then maybe next week.
Make sure you go through the solution sheet – it is massively important.
A score of less than 50% is ok – it means you can learn lots from your mistakes.
The next Parallelogram is next week, at 3pm on Thursday.
Finally, if you missed any earlier Parallelograms, make sure you go back and complete them. You can still earn reward points and badges by completing missed Parallelograms.

Cheerio, Simon.