Parallelogram 40 Level 4 6 Jun 2024The pigeonhole principle

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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. Intermediate Maths Challenge Problem (UKMT)

2 marks

1.1 Three consecutive positive integers less than 20 are, in ascending order, prime, even and triangular.

What is the product of these three integers?

  • 6
  • 60
  • 990
  • 1786
  • 2730
  • (Not answered)
Show Hint (–1 mark)
–1 mark

The triangular numbers below 20 are 1, 3, 6, 10 and 15.

2. Intermediate Maths Challenge Problem (UKMT)

3 marks

2.1 Which of the following is equal to 3993?

  • 3
  • 9
  • 27
  • 81
  • 243
  • (Not answered)
Show Hint (–1 mark)
–1 mark


3 marks

2.2 Find the integer n such that 27818127=3n.

Correct Solution: 135

Show Hint (–1 mark)
–1 mark


Now do the same for the denominator, writing it as a power of 3.

Then simplify.

3. Some hairy problems

The Pigeonhole Principle basically says that if you are trying to fit, say, 5 pigeons into 4 pigeonholes, then at least two pigeons must occupy the same space.

There’s nothing special about pigeons, of course - this principle can be applied to any group of objects.

The Pigeonhole Principle is hardly a revelation - in fact it seems so obvious at first that you might wonder why we even bother with it.

But as this video explains, the Pigeonhole Principle can be applied in creative ways to solve problems that might otherwise prove difficult.

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

2 marks

3.1 To use the Pigeonhole Principle to demonstrate that two (non-bald) people in London have the same number of hairs on their head, what are the ‘pigeonholes’?

  • The actual pigeonholes located in London
  • The houses in London
  • The numbers of possible hairs on the head of a Londoner
  • The people of London
  • (Not answered)
2 marks

3.2 What two bits of information do you need to apply the Pigeonhole Principle to this problem?

  • Number of thirty-year olds in London, number of sixty-year olds in London
  • Average number of hairs on a human head, population of London
  • Total number of hairs on all the heads in London, capital of London
  • Average height of a Londoner, size of London
  • (Not answered)
2 marks

3.3 What does the Pigeonhole Principle reveal about data compression?

  • There’s a computer program that can compress every file to a smaller file without any loss of data
  • There’s a computer program that can compress every file into a larger file without any loss of data
  • There’s no computer program that can decompress every file into a smaller file without any loss of data
  • Files can never be compressed
  • (Not answered)

4. Intermediate Maths Challenge Problem (UKMT)

3 marks

4.1 Which of these is equal to 0.88888888... + 0.077777777...?

Note: ... signifies that the decimal is recurring.

  • 0.877777777...
  • 0.88
  • 0.955555555...
  • 0.966666666...
  • 0.988888888...
  • (Not answered)
Show Hint (–1 mark)
–1 mark

0.88888888...= 8/9 and 0.077777777.. = 7/90

5. A weekday puzzle

3 marks

5.1 If tomorrow is Saturday, what is the day after the day before yesterday?

Correct Solution: Thursday

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.