Parallelogram 3 Year 11 24 Sep 2020Mars Ingenuity

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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.

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. Mars Ingenuity

On July 30th 2020, the rover, Perseverance, started its six and a half month journey to Mars. On its belly it carried Ingenuity, a lightweight helicopter specially designed for flight on Mars. Watch this video from the YouTube channel Veritasium about Ingenuity, and see how ingenious it really is!

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

2 marks

1.1. What makes powered flight on Mars easier than on Earth?

  • Its gravity is weaker
  • Its gravity is stronger
  • It has a thinner atmosphere
  • It has a denser atmosphere
  • It is colder, so the instruments don’t need cooling
  • (Not answered)
2 marks

1.2. What makes powered flight on Mars more difficult than on Earth?

  • It has a denser atmosphere
  • It has a thinner atmosphere
  • Its gravity is weaker
  • Its gravity is stronger
  • (Not answered)
2 marks

1.3. Order the planets Earth, Venus and Mars, starting with the planet with the thinnest atmosphere, ending with the planet with the densest atmosphere.

  • Earth, Venus, Mars
  • Earth, Mars, Venus
  • Venus, Mars, Earth
  • Mars, Earth, Venus
  • (Not answered)
3 marks

1.4. Perseverance itself weighs one metric tonne, and Ingenuity weighs about 1.8kg. What percentage increase in weight does Perseverance experience carrying Ingenuity?

Correct Solution: 0.18%

Since one metric tonne is 1000kg, the percentage increase is 1.8kg ÷ 1000kg x 100 = 0.18%.

2. Intermediate Maths Challenge Problem (UKMT)

3 marks

2.1 The perimeter of the three shapes shown are made up of straight lines and semi-circular arcs of diameter 2.

They will fit snugly together as in a jigsaw.

What is the difference between the total perimeter of the three separate pieces and the perimeter of the shape formed when the three pieces fit together?

  • 0
  • 2+2π
  • 8+4π
  • 22+2π
  • 30+6π
  • (Not answered)

The difference in perimeters is the total length of the edges which are hidden when the pieces are fitted together.

These are eight straight edges of length 1 and four semicircular arcs of radius 1.

So the required difference is 8×1+412×2×π×1=8+4π.

3. An index of digits

4 marks

3.1 Below, the integers from 0 to 8 have been placed in a sequence which follows a rule. If we added the digit 9 to the sequence, where would it go?

8, 5, 4, 1, 7, 6, 3, 2, 0

  • At the start
  • Between 8 and 5
  • Between 4 and 1
  • Between 7 and 6
  • At the end
  • (Not answered)
Show Hint (–1 mark)
–1 mark

If you read the last digit as “Nought” instead of “Zero”, the sequence would be 8, 5, 4, 0, 1, 7, 6, 3, 2.

Show Hint (–1 mark)
–1 mark

If you read the numbers in French (zéro, un, deux, trois, quatre, cinq, six, sept, huit, neuf) the sequence would be 5, 2, 8, 9, 4, 7, 6, 3, 1, 0.

Show Hint (–1 mark)
–1 mark

Consider the first letter of each number when written as a word.

If you write out each integer as a word, and then alphabetise them, they fall into our sequence.

4. Intermediate Maths Challenge Problem (UKMT)

4 marks

4.1 Supergran walks from her chalet to the top of the mountain. She knows that if she walks at a speed of 6 mph she will arrive at 1 pm, whereas if she leaves at the same time and walks at 10 mph, she will arrive at 11 am.

At what speed should she walk if she wants to arrive at 12 noon?

  • 7.5 mph
  • 7 17 mph
  • 7.75 mph
  • 60 mph
  • 8 mph
  • (Not answered)
Show Hint (–1 mark)
–1 mark

We do not know the start time or the distance of Supergran’s journey. Let’s choose to call the distance x. Calculate an expression for the time taken for arrival at 1pm and another for arrival at 11am. (Note that this is a difference of 2 hours)

Show Hint (–1 mark)
–1 mark

At 6mph, Supergran would take x6 hours, whereas at 10mph she would take x10 hours. x6x10=2.

Show Hint (–1 mark)
–1 mark

x6x10=2 simplifies to 5x3x=60 (by multiplying both sides by 30), so x=30 miles. Hence Supergran’s departure time is 8am.

Let the distance from the chalet to the top of the mountain be x miles. Then, at 6 mph Supergran would take x6 hours, whereas at 10 mph she would take x10 hours.

So x6x10=2, that is 5x3x=60, so x=30.

Hence Supergran's departure time is 8 am and to arrive at 12 noon she should walk at 304 mph, that is 7 12 mph.

5. Intermediate Maths Challenge Problem (UKMT)

5 marks

5.1 The diagram shows a pattern of eight equal shaded squares inside a circle of area π square units.

What is the area (in square units) of the shaded region?

  • 113
  • 135
  • 123
  • 179
  • 2
  • (Not answered)
Show Hint (–2 mark)
–2 mark

Let the centre of the circle be O and let A and B be corners of one of the shaded squares, as shown.

As the circle has area π units², its radius is 1 unit.

So OB is 1 unit long.

Let the length of the side of each of the shaded squares be x units.

Show Hint (–2 mark)
–2 mark

By Pythagoras' Theorem: OB2=OA2+AB2, that is 12=2x2+x2.

Let the centre of the circle be O and let A and B be corners of one of the shaded squares, as shown. As the circle has area π units², its radius is 1 unit. So OB is 1 unit long.

Let the length of the side of each of the shaded squares be x units.

By Pythagoras' Theorem: OB2=OA2+AB2, that is 12=2x2+x2.

So 5x2=1. Now the total shaded area is 8x2=8×15=135 units².

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.