Parallelogram 31 Level 3 4 Apr 2024In the art gallery

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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.


5 marks

1.1 Which of the following points is not at a distance of 1 unit from the origin?

  • (0, 1)
  • (1, 0)
  • (0, -1)
  • (-1, 0)
  • (1, 1)
  • (Not answered)

Each of points (0, 1), (1, 0), (0, -1) and (-1, 0) is 1 unit from the origin, but the point (1, 1) is at a distance of 2 units from the origin.


5 marks

2.1 The diagram on the right shows the positions of four people (each marked x) in an Art Gallery.

In the middle of the room is a stone column.

Ali can see none of the other three people. Bea can see only Caz. Caz can see Bea and Dan. Dan can only see Caz.

Who is at position p?

  • Ali
  • Bea
  • Caz
  • Dan
  • More information needed.
  • (Not answered)

The person in position p is the only one who can see two of the others.

We are told that Caz can see Bea and Dan but everyone else can see just one other person or no-one. So it must be Caz who is at position P.

We can also see that Ali, who can see no-one, must be in the position shown. Bea and Dan must be in the other two positions, but the information we are given doesn't enable us to work out which is where.


6 marks

3.1 Talulah plants 60 tulip bulbs. When they flower, she notes that half are yellow; one third of those which are not yellow are red; and one quarter of those which are neither yellow nor red are pink. The remainder are white.

What fraction of the tulips are white?

  • 124
  • 112
  • 16
  • 15
  • 14
  • (Not answered)

Since half the 60 bulbs are yellow, 30 are yellow and 30 are not yellow. One third of the 30 bulbs that are not yellow are red, so 10 are red and 20 are neither yellow nor red. One quarter of the 20 that are neither yellow nor red, are pink. So 5 bulbs are pink.

This leaves 15 bulbs which are neither yellow, nor red nor pink. So there are 15 white bulbs. Therefore the fraction of bulbs that are white is 1560, that is, 14.


6 marks

4.1 Jack and Jill played a game for two people. In each game, the winner was awarded 2 points and the loser 1 point. No games were drawn. Jack won exactly 4 games and Jill had a final score of 10 points. How many games did they play?

  • 5
  • 6
  • 7
  • 8
  • impossible to determine
  • (Not answered)

Jack won exactly 4 games. So Jill lost 4 games and gained 4 points from these losses. Jill gained 10 points altogether and so gained 10 − 4 = 6 points from her wins. Since there are 2 points for a win, Jill won 3 games. So Jack won 4 games, Jill won 3 games and altogether 4 + 3 = 7 games were played.


6 marks

5.1 The interior angles of a triangle are 5x+3y°, 3x+20° and 10y+30°, where x, y are positive integers.

What is the value of x+y?

  • 15
  • 14
  • 13
  • 12
  • 11
  • (Not answered)

Crucially, the question said that x and y are positive integers.

The angles of a triangle add up to 180°. So 5x+3y+3x+20+10y+30=180. This gives 8x+13y+50=180. Therefore 8x+13y=130, and so


It follows that 8x is a multiple of 13 and hence, as 8 and 13 have no common factors, it must be that x is a multiple of 13. So x=13a, where a is a positive integer. It then follows that


and so, cancelling the factor 13


and hence 10y is a positive number which is a multiple of 8. The only positive value of y for which 10y>0 and 10y is a multiple of 8, is y=2, which gives a=1 and hence x=13. So x+y=13+2=15.

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.