Week 4The Infinite Hotel

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Noun: Parallelogram Pronunciation: /ˌparəˈlɛləɡram/

  1. a portmanteaux 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 midnight on Sunday if your whole class is doing parallelograms.
  • Make sure you check the solution sheet, celebrate your successes and (most important of all) learn from your mistakes.

1. Hilbert’s Hotel

You all know the word infinity, but do you really understand what it means? The video below explains some of the weird mathematics behind the concept of infinity.

WARNING! This video will make your brain ache. Concentrate and watch it twice if necessary. Then answer the questions below.

First, to give you a head start, here are some key points about what you are about to watch:

  • Mathematician David Hilbert imagined a hotel with an infinite number of rooms.
  • The good news is that it is full with an infinite number of guests.
  • A visitor enters the hotel and asks for a room… but the hotel is full.
  • How can the hotel fit the visitor into the hotel?

3 marks

1.1. How does the hotel night manager fit in the new guest?

  • He puts them in the cellar.
  • He throws out the guest currently in Room 1 and puts the new guest there.
  • He asks every current guest to move to the room that is 1 number higher.
  • He puts the new guest in Room 1, so that he shares with the current guest.
2 marks

1.2. How does the hotel night manager find rooms for an infinite number of new guests?

  • Every current guest moves to a room number double their current room number.
  • The night manager builds a second infinite hotel dedicated to the new guests.
  • The night manager asks every new guest to share a room with an existing guest.
  • The night manager logs onto the Airbnb website.
2 marks

1.3. You will not find the answer to this question in the video, so you will have to trust your instinct or start googling.

If the infinite hotel can deal with infinity-plus-one-more guests (indeed infinity-plus-infinity-more guests), then does this mean that all infinites are the same?

  • Yes
  • No
  • Nes
  • Yo

2. Even more chaos

For the last two weeks, Parallelograms have included a brief look at mathematical chaos, first via a clip from Jurassic Park and then by looking at the motion of a chaotic pendulum.

This clip will be our last time of dipping our toes into chaos. It shows three dots moving around the screen (and plotting three curves in the process) according to the exact same formula. They start off in almost the same position – they are pretty much right on top of each other – so as they move they seem to follow a single path and leave the same trace. But, as you should remember, chaos describes things that start off in a similar way, but which end up behaving very differently. After you watch the clip, there are three questions to answer.

1 mark

2.1. The three dots are different colours – red, blue and green. When do the dots start to diverge (which means they have started to move apart)?

  • Roughly 40 seconds
  • Roughly 43 seconds
  • Roughly 47 seconds
  • Roughly 50 seconds
  • Roughly 53 seconds
1 mark

2.2. After the three dots start split apart, when does the red dot fly off in a different direction?

  • Roughly 56 seconds
  • Roughly 58 seconds
  • Roughly 60 seconds
  • Roughly 62 seconds
  • Roughly 64 seconds
2 marks

2.3. Which of these statements best describes the settled motion of the three dots?

  • The motion of the dots is totally random.
  • The dots tend to orbit a single fixed point.
  • The dots tend to orbit two separate fixed points.
  • The dots tend to orbit three separate fixed points.

3. Junior Maths Challenge Problem

As I mentioned in previous Parallelograms, if you are a Year 8 student, then it is likely that you will be taking part in the United Kingdom Maths Trust (UKMT) competition known as the Junior Maths Challenge (JMC). If you do particularly well, you might earn yourself a gold, silver or bronze certificate, but you will have to work hard as you will be competing against students from across the country.

Your teachers will help you prepare for this national maths competition, but in each week's Parallelogram we will always include one UKMT Junior Maths Challenge question.

3 marks

3.1 Each side of an isosceles triangle is a whole number of centimetres. Its perimeter has length 20 cm. How many possibilities are there for the lengths of its sides?

  • 3
  • 4
  • 5
  • 6
  • 7

(We will have a similar question next week, so make sure you check the answer thoroughly.)

Let the length of the two equal sides of the isosceles triangle be a cm. Since the triangle has perimeter 20cm, the third side will have length 202a cm. Since this must be a positive length, 202a>0 and so a<10.

In a triangle, the length of one side must be less than the sum of the lengths of the other two sides. So 202a<2a.

This gives 20<4a and hence 5<a. So, we have 5<a<10.

Therefore, as a is a whole number, there are just four possible values for a, namely 6, 7, 8 and 9.

So there are four possibilities for the side lengths of the triangle:

  • 6, 6, 8;
  • 7, 7, 6;
  • 8, 8, 4; and
  • 9, 9, 2.

If you missed any of the earlier Parallelograms, then try to go back and complete them. After all, you can earn reward points and badges by completing each Parallelogram. Find out more by visiting the Rewards Page after you hit the SUBMIT button.

There will be another Parallelogram next week, and the week after, and the week after that. So check your email or return to the website on Thursday at 3pm.

In the meantime, you can find out your score, the answers and go through the answer sheet as soon as you hit the SUBMIT button below.

It is really important that you go through the solution sheet. Seriously important. What you got right is much less important than what you got wrong, because where you went wrong provides you with an opportunity to learn something new.

Cheerio, Simon.