Parallelogram 7 Level 2 19 Oct 2023Build your own plane

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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. Palindromic products

This is a cute pattern.

1 × 1 = 1
11 × 11 = 121
111 × 111 = 12321
1111 × 1111 = 1234321
11111 × 11111 = 123454321
111111 × 111111 = 12345654321
1111111 × 1111111 = 1234567654321
11111111 × 11111111 = 123456787654321
111111111 × 111111111 = 12345678987654321

The product, the result of multiplying numbers with repeating 1s, has this beautiful symmetry. The products are palindromes, which means that they are the same whether written forwards or backwards.

2 marks

1.1. What happens if we add one more 1?

What is 1111111111 × 1111111111?

Of course, you can type this into a calculator, but first try to follow the pattern and work out what the answer might be… then check it with a calculator.

Correct Solution: 1234567900987654321

If you follow the pattern, the result should be:


The 10 fits in the middle, but we can’t write 10 where there is only space for 1 digit. Instead, we could put a 0 where the 10 is, and carry the 1 over to the column on the left, which gives us:


But we can’t write (9 + 1), which is 10, where there is only space for 1 digit. Instead, we could put a 0 in that column and carry the 1 over to the column on the left, which gives us:


2. Building brilliant paper planes

Here is an interview with John Collins, who broke the world distance record for paper planes. Although John designs and builds the planes, he does not throw them.

Listen carefully, as I will be asking a question about the thrower.

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

1 mark

2.1 The thrower is an ex-sportsman, but which sport did he play?

  • American football
  • Baseball
  • Cricket
  • Darts
  • Elephant Polo (yes, it’s a real sport, and more importantly it means that the answers start with an A, B, C, D and E).
  • (Not answered)

The answer is American football, because the paper plane engineer keeps talking about the thrower being a ‘quarterback’.

If you didn’t know that a quarterback is an American footballer, then a quick google would have solved the problem.

3. Junior Maths Challenge Problem (UKMT)

3 marks

3.1 A 5 cm × 5 cm square is cut into five pieces, as shown.

Each cut is a sequence of identical copies of the same shape but pointing up, down, left or right.

Which piece has the longest perimeter?

  • A
  • B
  • C
  • D
  • E
  • (Not answered)
Show Hint (–1 mark)
–1 mark

If we look at pieces B and C, they have roughly the same perimeter – let’s call it 12 lengths. However, B has a longer perimeter than C, because more of B’s sides are crooked, and crooked sides are longer than straight sides.

We let the length of each crooked line be x cm. Since this crooked line is longer than the side length of one of the squares, 1 < x.

The perimeters of the five pieces are, in centimetres:

  • A: 4 + 6x
  • B: 2 + 10x
  • C: 7 + 5x
  • D: 6 + 6x
  • E: 1 + 11x

As 1 < x, E has the longest perimeter.

4. QI

The BBC TV show “QI” is full of interesting facts and ideas. Here are a few interesting bits from “QI” and some follow-up questions.

2 marks

4.1 “It takes a million cloud droplets to make one raindrop.”

Clouds consist of microscopic droplets of liquid water, which gather together to form raindrops, which in turn fall to Earth. If the cloud droplets have a radius of 0.002 mm, then what is the radius of a raindrop (based on the idea that a million droplets make a drop)?

  • 0.002 mm
  • 0.2 mm
  • 2 mm
  • 20 mm
  • 2,000 mm
  • (Not answered)

If it takes one million droplets to make a drop, then you can imagine that the drop is made of 100 droplets across, 100 droplets up and 100 droplets deep, because 100 × 100 × 100 = 1 million.

And, 100 × 0.002 mm = 0.2 mm.

2 marks

4.2 You might need to do some research to answer this question: what is the weight of a cumulus cloud?

  • The same as a mouse.
  • The same as a cat.
  • The same as a lion.
  • The same as an elephant.
  • The same as a blue whale.
  • (Not answered)

A typical cumulus cloud weighs about 500 tons, and a blue whale is a couple of hundred tons or more.

2 marks

4.3 “A million seconds is 11.6 days”

How long is a billion seconds?

  • A few days
  • A few weeks
  • A few months
  • A few decades
  • A few centuries
  • (Not answered)

1,000,000,000 / (365 × 24 × 60 × 60) = 31.7 years

2 marks

4.4 What happened roughly a million days ago?

  • Universe is formed
  • Earth is formed
  • Atlantic Ocean is formed
  • Roman kingdom is formed
  • United States is formed
  • (Not answered)

1,000,000 / 365 = 2,721 ago, or roughly 720 BC

4 marks

4.5 “The Sun gets 4 billion kilogrammes lighter every second.”

Einstein’s famous formula is:


It explains how mass (m) can be turned into energy (E) at the rate of (c2) per kilo.

c is the speed of light, which is 3×108 m/s.

So, 1 Kg creates 1×3×1082 Joules of energy, or 9×1016 Joules of energy.

Bearing in mind the QI quote, roughly how much energy does the Sun generate every second?

  • 4×1016 Joules
  • 4×1017 Joules
  • 4×1020 Joules
  • 4×1026 Joules
  • 4×1035 Joules
  • (Not answered)
Show Hint (–1 mark)
–1 mark

Energy per second =m×c2

Energy per second =m×c2

#5. Maths Jokes

Can you match the mathematical joke to the correct punchline?

  • Punchline A: Because seven, eight, nine!
  • Punchline B: When I got back, he’d only done jobs one, three, five, and seven.
  • Punchline C: The farmer says, “But I’ve counted them and I’ve only got 36!” The sheepdog replies, “I know, but I rounded them up.”
2 marks

5.1 Why was six afraid of seven?

  • Punchline A
  • Punchline B
  • Punchline C
  • (Not answered)
2 marks

5.2 A talking sheepdog rounds up all the sheep into the pen for his farmer. He comes back and says, “Okay, Chief — all 40 sheep accounted for”.

  • Punchline A
  • Punchline B
  • Punchline C
  • (Not answered)
2 marks

5.3 I hired an odd man to do eight jobs for me.

  • Punchline A
  • Punchline B
  • Punchline C
  • (Not answered)

Finally, if you want to find out more about making paper planes, then watch this video about paper plane guru John Collins.

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 time.
  • 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.