Parallelogram 16 • Level 5 • 21 Dec 2023Chrismaths - Part 3

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, so time for another instalment of your Christmaths challenge paper.

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

What is the value of a+b+c+d+e+f?

360
540
720
900
It depends on the triangle
(Not answered)

The angles marked a°, c° and e° may be considered to be the exterior angles of the triangle, and therefore have a total of 360°. As b°=a°, d°=c° and f°=e° (all pairs of vertically opposite angles), b°+d°+f°=360°. So a°+b°+c°+d°+e°+f°=720°.

2.

4 marks

The cuboids below all have the same volume. Which of them has the greatest surface area?

A)
B)
C)
D)
E)
(Not answered)

The surface areas of the cuboids are: A 68; B 70; C 56; D 52; E 76.

3.

4 marks

In this addition sum, each letter represents a different non-zero digit.

What is the value of a+w+a+y?

13
15
16
17
18
(Not answered)

We note first that y = 5 is the only non-zero digit which, when it is multiplied by 3, has itself as the units digit. So there is a carry of 1 into the tens column. We note also that a = 1 or a = 2 as “fly” < 1000 and therefore 3 × “fly” < 3000.

We now need 3 × l + 1 to end in either 1 or 2 and the only possibility is l = 7, giving a = 2 with a carry of 2 into the hundreds column.

As a = 2, f must be at least 6. However, if f = 6 then w = 0 which is not allowed. Also the letters represent different digits, so f ≠ 7 and we can also deduce that f ≠ 9 since f = 9 would make w = 9.

Hence f = 8, making w = 6 and the letters represent 875 × 3 = 2625.

4.

4 marks

Three rectangular-shaped holes have been drilled passing all the way through a solid 3 x 4 x 5 cuboid. The diagrams show the front, side and top views of the resulting block.

What fraction of the original cuboid remains?

1330
715
12
815
1730
(Not answered)

Consider the cuboid to be made up of 60 unit cubes. The front and side views show that top and bottom layers consist of the same number of cubes and from the top view we see that this number is 14. The front and side views indicate that only the 4 corner cubes remain in the middle layer, so the total number of cubes remaining is 2 × 14 + 4 = 32. The required fraction, therefore, is 3260=815.

5.

5 marks

If p, q and p−q are all positive integers, which of the following is least?

q2p2
p2q2
qp
qp
pq
(Not answered)

We may deduce that p>q, so, as p and q are both positive, p2q2 > pq > 1.

We may also deduce that 0 < q2p2 < qp < qp < 1.

Hence q2p2 is the least of the five numbers.

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.