Parallelogram 3 Year 10 26 Sep 2019The Kessel Run

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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.

Welcome to the third of our Parallelograms designed for Year 10 students, a collection of mathematical challenges designed to stretch your brain and make your neurons more squiggly.

These challenges are a random walk through the mysteries of mathematics, most of which will be nothing to do with what you are doing at the moment in your classroom. Be prepared to encounter all sorts of weird ideas, including a few questions that appear to have nothing to do with mathematics at all.

  • 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.

IMPORTANT – it does not really matter what score you get, because the main thing is that you think hard about the problems... and then examine the solution sheet to learn from your mistakes.

1. The Kessel Run

Take a look at this clip from “Star Wars IV” (A New Hope). In particular, pay attention to the line about the Kessel Run.

“No questions asked” – but this is a Parallelogram – there has to be a question, maybe several. And here is the first one.

2 marks

1.1. This clip is controversial, because Han Solo brags that the Falcon made the Kessel Run in "less than twelve parsecs". Normally, when we talk about completing races, we would brag about completing a race in less TIME, but the parsec is a unit of DISTANCE.

So, this is probably a blunder in the original script, but George Lucas has explained it away by claiming that Solo modified "the navigational system to get through hyperspace in the shortest possible distance". In other words, the skill in completing the Kessel Run is all about finding the shortest route. Now the question.

What is a parsec?

  • 30 million billion metres
  • 30 billion metres
  • 30 million metres
  • 30 million million metres
  • 30 billion billion metres
2 marks

1.2. What is the very approximate distance from the Earth to the Sun in parsecs?

  • 1
  • one tenth
  • one hundredth
  • one thousandth
  • one millionth
  • one billionth
2 marks

1.3. You ought to google this, unless you are completely sure... the parsec is based on the arcsecond. The arcsecond is a unit of...

  • time
  • distance
  • angle
  • area
  • volume
2 marks

1.4. The parsec might seem like an odd unit of distance, but it makes a lot of sense to astronomers. If you want to see a genuinely odd unit of distance, then meet the Smoot. The Smoot is equal to:

  • the length of Smoot the cat
  • the height of Oliver Smoot
  • the width of a Smoot sack of coal
  • the length of the printed word “Smoot”
  • the length of a Suzuki motorbike

The Smoot is named after Oliver R. Smoot, who in 1958 lay down repeatedly on the Harvard Bridge (between Boston and Cambridge, Massachusetts) so that his fellow students could use his height to measure the length of the bridge. One Smoot is 1.702 metres.

2. 3s and 8s

3 marks

2.1 Using some or all of the operations of addition, subtraction, multiplication and division and using the digits 3, 3, 8 and 8 each once and only once make an expression equal to 24.

This is a seriously tricky problem. So, just think hard about it and see if you can figure out the answer. If you can solve it, then well done. If not, then the answer will be available when you hit submit.

Either way, everyone gets 3 marks.

  • Please can I have 3 marks

This problem is from the terrific NRICH website, a treasure of mathematical ideas and questions. Go and explore.

8383=24, or

3. Intermediate Maths Challenge Problem (UKMT)

Every Parallelogram will contain two or more UKMT Intermediate Maths Challenge problems. Remember, NO CALCULATORS!

3 marks

3.1 How many of these five expressions give answers which are not prime numbers?

12+22
22+32
32+42
42+52
52+62

  • 0
  • 1
  • 2
  • 3
  • 4

We have

12+22=1+4=5,
22+32=4+9=13,
32+42=9+16=25,
42+52=16+25=41, and
52+62=25+36=61.

Of these answers 5, 13, 41 and 61 are prime numbers, but 25 is not a prime number.

Therefore just one of the given options does not work out to be a prime number.

4. Intermediate Maths Challenge Problem (UKMT)

4 marks

4.1 The angles of a quadrilateral taken in order are x°, 5x°, 2x° and 4x°.

Which of the following is the quadrilateral?

  • kite
  • parallelogram
  • rhombus
  • arrowhead
  • trapezium
Show Hint (–1 mark)
–1 mark

As the sum of the interior angles in a quadrilateral is 360°, we know that x+5x+2x+4x=360°.

Show Hint (–1 mark)
–1 mark

Sketch out the quadrilateral with the angles at vertices P, Q, R, S in the order of the question. If x+5x+2x+4x=360°, then x+5x=2x+4x=180°, so the side connecting vertices S and R and the side connecting P and Q must be parallel. (Think about why this is.)

We let the vertices of the quadrilateral be P, Q, R, S with the angles as shown in the diagram.

The sum of the interior angles of a quadrilateral is 360°. Therefore, x+5x+2x+4x=360. That is, x+5x+2x+4x=360. Since x+5x=2x+4x, it follows that x+5x=2x+4x=180.

Hence SPQ+RSP=180°. Therefore the sides PQ and SR of the quadrilateral are parallel. Sincex+4x=5x180, we have SPQ+PQR180°. Therefore the sides PS and QR are not parallel.

We therefore see that the quadrilateral has one pair of opposite sides that are parallel and one pair of opposite sides that are not parallel. Therefore the quadrilateral is a trapezium.

I hope you enjoyed the third Parallelogram of the year. There will be more 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.

When you see your % score, this will also be your reward score. As you collect more and more points, you will collect more and more badges. Find out more by visiting the Rewards Page after you hit the SUBMIT button.

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.