Parallelogram 8 Level 4 26 Oct 2023Mighty Magic Square

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Noun: Parallelogram Pronunciation: /ˌparəˈlɛləɡram/

  1. 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.


3 marks

How many of the following four numbers are prime?


  • 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 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.


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=PQRPQT=130°34°=96°.


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.


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=12122=114=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.