Parallelogram 8 • Level 4 • 26 Oct 2023Mighty Magic Square

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.
Finish by midnight on Sunday if your whole class is doing parallelograms.
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. Prime magic square

This is a terrific little video about about a mighty PRIME magic suare from the YouTuber Vsauce. Take a look and answer the questions below.

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

1 mark

1.1 Kevin admits that one of the elements in this magic square is NOT prime. Which number in the magic square is not prime?

Correct Solution: 1

2 marks

1.2 The rows, columns and diagonals add up to 9,635. How many prime factors does 9,635 have? Remember, 1 is not a prime number.

Correct Solution: 3

3 prime factors, namely 5×41×47.

2.

3 marks

How many of the following four numbers are prime?

3
33
333
3333

A) 0
B) 1
C) 2
D) 3
E) 4
(Not answered)

The number 3 is prime, but the other numbers listed are not prime:

33 = 3 × 11
333 = 3 × 111
3333 = 3 × 1111.

3.

3 marks

Triangle QRS is isosceles and right-angled.

Beatrice reflects the P-shape in the side QR to get an image.

She reflects the first image in the side QS to get a second image.

Finally, she reflects the second image in the side RS to get a third image.

What does the third image look like?

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

The effect of the successive reflections is shown in the diagram.

4.

3 marks

In the diagram, PQRS is a parallelogram; ∠QRS=50°; ∠SPT=62° and PQ=PT.

What is the size of ∠TQR?

A) 84°
B) 90°
C) 96°
D) 112°
E) 124°
(Not answered)

Because PQRS is a parallelogram, ∠SPQ=∠QRS=50°. Therefore ∠TPQ=62+50°=112°.

Therefore, as the angles in a triangle add up to 180°, ∠PQT+∠PTQ=180°−1120=68°.

Because PQ=PT, the triangle QPT is isosceles, and so ∠PQT=∠PTQ. Therefore ∠PQT=∠PTQ=34°.

Because PQRS is a parallelogram, ∠PQR+∠QRS=180°, and therefore ∠PQR=180°−50°=130°.

Therefore, ∠TQR=∠PQR−∠PQT=130°−34°=96°.

5.

4 marks

The diagram shows a large equilateral triangle divided by three straight lines into seven regions.

The three grey regions are equilateral triangles with sides of length 5 cm and the central black region is an equilateral triangle with sides of length 2 cm.

What is the side length of the original large triangle?

A) 18 cm
B) 19 cm
C) 20 cm
D) 21 cm
E) 22 cm
(Not answered)

Let P,Q,R,S,T,U and V be the points shown. All the angles in all the triangles are 60°. So ∠QRT=∠PSU and hence RT is parallel to SU.

Similarly, as ∠RSV=∠TUV, RS is parallel to TU. Therefore RSUT is a parallelogram.

Therefore RS has the same length as TU, namely, 2 + 5 = 7 cm. Similarly PQ has length 7 cm.

So the length of PS which is the sum of the lengths of PQ, QS and RS is 7 + 5 + 7 = 19 cm.

6.

5 marks

The parallelogram PQRS is formed by joining together four equilateral triangles of side 1 unit, as shown.

What is the length of the diagonal SQ?

A) 7
B) 8
C) 3
D) 6
E) 5
(Not answered)

Let T be the foot of the perpendicular from Q to the line SR extended. Now RQT is half of an equilateral triangle with side length 1. Hence the length of RT is 1/2 and hence ST has length 1+1+12=52.

By Pythagoras’ Theorem applied to the right angled triangle RQT,12=122+QT2.

Therefore QT2=12−122=1−14=34

Hence, by Pythagoras’ Theorem applied to the right angled triangle SQT,SQ2=ST2+QT2=522+34=254+34=7.

Therefore, SQ=7.

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.