Parallelogram 14 • Level 5 • 5 Dec 2024Chrismaths - Part 1

Noun: Parallelogram Pronunciation: /ˌparəˈlɛləɡram/

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

It’s nearly Christmas. Not very many people realise that the original name for this holiday was Chrismaths. For centuries, children traditionally completed mathematics problems during the festive period. The amount of effort that children put into their mathematics problems allowed Santa to decide who had been naughty and who had been nice.

With this in mind, this next few week’s of Parallelogram contain a WHOLE maths challenge paper, to stretch your brain over Chrismaths.

Good luck and happy Chrismaths. Don’t eat too many mince pie charts!

Simon.

PS: I want to say thank you to the UK Mathematics Trust, who own the copyright to these questions.

1.

3 marks

You are told that 30 pupils have 25 different birthdays between them.

What is the largest number of pupils who could share the same birthday?

2
3
4
5
6
(Not answered)

There must be 25 pupils who all have different birthdays. If the remaining pupils all have the same birthday as one of these pupils, then 6 pupils will share the same birthday.

2.

4 marks

Which of the following straight lines cuts the shaded area in half?

XA
XB
XC
XD
XE
(Not answered)

The shaded area is a trapezium of area 123+7×5=25.

Line XD forms one side of a trapezium of area 12.5, since 121+4×5=12.5.

3.

4 marks

This figure is made from a straight line 16cm long and two quarter circles, one with its centre at the midpoint of the straight line.

What is the area of the figure (in cm²)?

64
16π
32 + 16π
32π
16 + 8π
(Not answered)

As the diagram shows, the figure may be cut into two parts which fit together to form a square measuring 8cm × 8cm.

4.

4 marks

A, B, C, D, E, P and Q are points on the number line as shown.

One of the points represents the product of the numbers presented by P and Q. Which is it?

A
B
C
D
E
(Not answered)

As P and Q both lie between 0 and 1, their product will be greater than 0 but smaller than P and smaller than Q. Of the options available, only B satisfies these conditions.

Furthermore, its position is correct since P is approximately equal to 12, which means that the product of P and Q lies approximately halfway between 0 and Q.

5.

5 marks

In a maths exam with N questions, you score m marks for a correct answer to each of the first q questions, and m+2 marks for a correct answer to each of the remaining questions.

What is the maximum possible score?

m+2N−2q
Nm
mp+m+2q
Nm+1
Nm+qm+2
(Not answered)

If you answer all questions correctly, you receive m marks for each of the N questions plus an extra 2 marks for the last N−q questions.

So the maximum possible score =m×N+2×N−q=mN+2N−2q=m+2N−2q.

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.