Parallelogram 38 Level 1 23 May 2024Hilbert Hotel

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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. Welcome to the Hilbert Hotel

You have just arrived at the Hilbert Hotel and are ready to start your richly deserved vacation.

You soon realise this is no ordinary hotel - it has infinitely many rooms and infinitely many guests!

Things are about to get even stranger when a new guest arrives.

Watch as Emily Riehl explains the strange arithmetic of infinity using the example of the Hilbert Hotel.

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

1 mark

1.1 How is the new guest accommodated at the Hilbert Hotel?

  • They have to share a room with a randomly chosen guest
  • A new room is built for them on the spot
  • Every guest moves to the next room along
  • They have to sleep in the lounge
  • (Not answered)
1 mark

1.2 What is infinity + 1,000,000?

  • Still the same infinity
  • A larger infinity
  • A smaller infinity
  • It is impossible to say
  • (Not answered)
2 marks

1.3 When the new guests arrive from the second Hilbert Hotel, they can all still be accommodated.

Where does the guest in room 123 end up?

Correct Solution: 246

Show Hint (–1 mark)
–1 mark

To make space this time, every guest goes to twice their current room number.

2. Some puzzles!

2 marks

2.1 What fraction of the rectangle is shaded?

  • 13
  • 14
  • 25
  • 38
  • 29
  • (Not answered)
Show Hint (–1 mark)
–1 mark

The fraction that is not shaded is 12 + 16.

Consider the parts that are not shaded.

There are two such parts - the first is a triangle taking up half the rectangle, and the second is a smaller triangle taking up a sixth of the rectangle.

So the total unshaded fraction is 12 + 16 = 23 .

This means the shaded part takes up 13 of the rectangle.

2 marks

2.2 Veronica rolls three six-sided dice.

The product of the three numbers is 90.

What is the sum of the three numbers?

  • 10
  • 14
  • 18
  • 90
  • More information is needed.
  • (Not answered)

The numbers in play are 1, 2, 3, 4, 5, 6.

The only combination of three numbers from this list that gives a product of 90 (even allowing for repeats) is 6 × 5 × 3.

So the sum of the three numbers is 6 + 5 + 3 = 14.

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.