Parallelogram 32 Level 2 11 Apr 2024Chocolate algebra

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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.
  • When you finish, remember to hit the SUBMIT button.
  • Finish by Sunday night if your whole class is doing parallelograms.
  • Don’t worry if you score less than 50%, because it means you will learn something new when you check the solutions.


6 marks

1.1 Box P has p chocolates and box Q has q chocolates, where p and q are both odd and p>q.

What is the smallest number of chocolates which would have to be moved from box P to box Q so that box Q has more chocolates than box P?

  • qp+22
  • pq+22
  • q+p22
  • pq22
  • q+p+22
  • (Not answered)

Because p and q are both odd, p+q is even. If the chocolates were shared equally between the boxes there would be 12p+q chocolates in each box.

So the least number of chocolates there must be in box Q if it is to have more chocolates than box P is 12p+q+1.

Because box Q starts with q chocolates in it, to end up with 12p+q+1 in box Q the number of chocolates we need to transfer from box P to box Q is



5 marks

2.1 The diagram shows two arrows drawn on separate 4 cm × 4 cm grids. One arrow points North and the other points West.

When the two arrows are drawn on the same 4 cm × 4 cm grid (still pointing North and West) they overlap. What is the area of overlap?

  • 4 cm2
  • 4 12 cm2
  • 5 cm2
  • 5 12 cm2
  • 6 cm2
  • (Not answered)

By drawing one arrow on top of the other, as shown, we see that the region of overlap covers the whole of 4 of the 1 cm × 1 cm squares into which the grid is divided, and 4 halves of these squares. So the area of the overlapping region is 4 + 4 (12) = 6 cm2.


5 marks

3.1 Which of the following numbers is divisible by 7?

  • 111
  • 1111
  • 11111
  • 111111
  • 1111111
  • (Not answered)

The problem may be solved by dividing each of the alternatives in turn by 7, but the prime factorisation of 1001, i.e. 1001=7×11×13, leads to the conclusion that 111 111, which is 111×1001, is a multiple of 7.


5 marks

4.1 What is the remainder when the square of 49 is divided by the square root of 49?

  • 0
  • 2
  • 3
  • 4
  • 7
  • (Not answered)

Because 49=7×7, it follows both that 49=7 and that 7 is a factor of 49.

Therefore 7 is also a factor of 492.

Hence the remainder when the square of 49 is divided by the square root of 49 is 0.


6 marks

5.1 After playing 500 games, my success rate at Spider Solitaire is 49%. Assuming that I win every game from now on, how many extra games do I need to play in order that my success rate increases to 50%?

  • 1
  • 2
  • 5
  • 10
  • 50
  • (Not answered)

Since I have won 49% of my first 500 games, so far I have won 49100×500=49×5=245 games. So I have lost 500245=255 games. I need now to win enough games so that I have won as many as I have lost.

So, assuming I win every game from now on, I need to win 255245=10 more games.

Another approach is to say that I am going play and win W more games. At that point, the number of games that I win will be:


Therefore 245+W500+W=0.5


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.