Parallelogram 47 • Level 2 • 24 Jul 2025Fire and water

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.
Your score & answer sheet will appear immediately after you hit SUBMIT.
Don’t worry if you score less than 50%, because it means you will learn something new when you check the solutions.

1. Can you light a match with water?

Watch the video below - there will be a question to follow it:

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

3 marks

1.1 The Statue of Liberty is coated in...

copper oxide
chlorophyll
malachite
shredded Shrek
jade
(Not answered)

2.

2 marks

2.1 If the net shown is folded to make a cube, which letter is opposite X?

A
B
C
D
E
(Not answered)

Let X be on the top face of the cube. If the base is placed on a horizontal surface, then A, B, C, E will all be on vertical faces of the cube and D will be on the base, opposite X.

3.

3 marks

3.1. In the diagram on the right, ST is parallel to UV.

What is the value of x?

46
48
86
92
94
(Not answered)

As ST is parallel to UV, ∠PRT=132° (corresponding angles).

So ∠PRQ=48° (angles on a straight line).

From the exterior angle of a triangle theorem, ∠SQP=∠QPR+∠PRQ, so x=134−48=86°.

4.

4 marks

4.1 In the following expression each ▲ is to be replaced with either + or - in such a way that the result of the calculation is 100.

123 ▲ 45 ▲ 67 ▲ 89

The number of + signs used is p and the number of - signs used is m. What is the value of p - m ?

-3
-1
0
1
3
(Not answered)

One approach is trial and error, as there are only a few combinations. For example, if it is (+ 45) then it is hard to see how we obtain a result of 100. So, we can then explore (- 45). Following this approach quickly leads you to the correct answer, but there is also a more rigorous and mathematical approach.

The sum is made up of 123 and ± 45, ± 67and ± 89. Suppose that the total of the positive terms in the calculation is x and the total of the negative terms is y. So x>0 and y<0. We need to have that:

x+y=100.

We also have that:

x−y=123+45+67+89=324.

Adding these equations, we obtain 2x=424. So x=212 and hence y=−112. It is readily seen that 45+67=112 and that no other combination of 45, 67 and 89 gives a total of 112. So the correct calculation must be 123−45−67+89=100 with 1 plus sign and 2 minus signs. So p=1 and m=2, giving p−m=−1.

5.

5 marks

5.1 A 6 by 8 and a 7 by 9 rectangle overlap with one corner coinciding as shown.

What is the area (in square units) of the region outside the overlap?

6
21
27
42
69
(Not answered)

The two shaded regions measure 3 by 7 and 1 by 6, so the total area outside the overlap is 27 units².

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.