Parallelogram 41 • Level 5 • 12 Jun 2025A concrete transfer

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. An unusual football transfer

3 marks

1.1 A number of years ago, a newspaper reported that Turkish football team Sarigol Municipality transferred four of its players in return for a fee of 225 sacks of cement, needed to repair their stadium. At the same rate of exchange, how many sacks of cement would the transfer fee for a full team of eleven players and one reserve?

233
450
675
900
2700
(Not answered)

The transfer fee for a total of 12 players would be three times that for 4 players.

2. Liar Numbers

Here's a video by the great James Grime:

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

4 marks

2.1 Numbers that pass this prime number test, but which are not actually prime, are known as what?

Carmichael numbers
Vanmichael numbers
Articulatedlorrymichael numbers
Wagonmichael numbers
Busmichael numbers
(Not answered)

4 marks

2.2 The third number that passes the prime test, but which is not a prime, is 1,729. It is known as what?

An uber number
A taxicab number
A minicab number
A chauffer number
A bus number
(Not answered)

3.

4 marks

3.1 After a year's training, Minnie increased her average speed in the London Marathon by 25%.

By what percentage did her time decrease?

50
30
20
10
5
(Not answered)

Show Hint (–1 mark)

–1 mark

It helps to think about this problem in terms of fractions, rather than percentages. If the new speed is 5/4 of the original speed, then what is the new time?

Minnie's average speed this year is 54 of last year's value. Hence her time this year will be 45, ie 80% of last year's time.

4.

5 marks

4.1 What is the area of the pentagon shown?

12ab−c
12ba+c
12ab+c
12bc−a
12ca+b
(Not answered)

The area of the pentagon is that of a rectangle of length b and breadth a plus that of a triangle of base b and height c−a, ie ab+12bc−a=12b2a+c−a=12ba+c.

The area may also be considered to be the sum of the areas of two trapezia which have parallel sides a and c and whose heights have total length b.

5.

5 marks

5.1 The width:height ratio of television screens is changing from the traditional 4:3 to the widescreen 16:9. If a traditional screen and a widescreen have the same area, then what is the ratio widescreen width:traditional width?

Assume that television screens are rectangles.

2:3
3:2
3:2
4:3
none of these
(Not answered)

Show Hint (–2 mark)

–2 mark

Let the widescreen width and traditional width be w and W respectively. Then the respective heights are 9w16 and 3W4.

As the areas are equal:

w×9w16=W×3W4, ie w2=43W2. Hence w : W = 2:3

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.