Parallelogram 5 Level 5 5 Oct 2023The Philosophy of Mathematics

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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. The philosophy of mathematics

According to Wikipedia, philosophy (from the Greek, meaning "love of wisdom") is “the study of general and fundamental questions about existence, knowledge, values, reason, mind, and language.”

There are many links between mathematics and philosophy ranging from logic to questions around the reality of mathematics. Are numbers real, do they exist out there in the universe, and did humans discover them. Or are they just an invention of the human brain?

In this clip, from a documentary in the American Nova series, the mathematicians interviewed are all on the “discovered” side of the argument, but there are many who would disagree, and the truth is that nobody really knows.

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

2 marks

1.1. Which of the following proofs has yet to be discovered (or invented)? Some googling might be required. Or maybe not.

  • Fermat’s Last Theorem
  • Fermat’s Little Theorem
  • Riemann Hypothesis
  • Pythagorean Theorem
  • Godel’s Incompleteness Theorem
  • (Not answered)

All the options, except one, were labelled ‘theorems’, which means that they have been proved. A ‘hypothesis’ is an idea in mathematics that has yet to be proved, but which might be true. So, the Riemann Hypothesis has yet to be proved. In fact, it is probably the most important question in mathematics, and there is a $1 million prize for whoever can find a proof.

2. How to make a paper popper

When I was a kid, I used to make paper poppers or paper crackers – here’s how to make them.

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

It is not easy to build a paper popper, so you might need to check out this list of instructions if your first attempt fails. My main advice is to make sure your initial folding goes up much more than half-way:

My friends and I told each other that the paper popper created such a loud sound, because somehow something was breaking the sound barrier (or going faster than the speed of sound). In fact, Wikipedia gives the right explanation: “There are many variations of a paper popper, but they all involve a folded sheet of paper being gripped and thrust down. This causes air to be forced into the paper's folded flaps, making the paper flaps pop out in the opposite direction at a high velocity, thus making a loud popping noise. According to popular misconception, the sound is supposed to be caused by the paper breaking the sound barrier. What actually happens is that air rushes in a blast wave to fill the vacuum created in the pocket that opens.”

Sound is a wave or vibration in the air, which has a frequency, wavelength and speed. These combine in the following way:

Frequency (Hertz) × Wavelength (metres) = Speed (m/s)

The speed of sound in air is typically 340 m/s.

2 marks

2.1 If the crack from a paper popper has a frequency of 100 Hertz, then what is its wavelength in meters?

Correct Solution: 3.4

100 × wavelength = 340

So, wavelength = 340/100 = 3.4 metres

2 marks

2.2 If the crack from a paper popper has a wavelength of 1.7 metres, then what is its frequency in Hertz?

Correct Solution: 200

Frequency × 1.7 = 340

So, frequency = 340/1.7 = 200 Hertz.

3. Intermediate Maths Challenge Problem (UKMT)

3 marks

3.1 Amrita makes jam and chutney today. She makes jam every 12 days and makes chutney every 20 days. After how many days will she again make both jam and chutney?

Correct Solution: 60

Amrita makes jam again after 12, 24, 36, 48, 60, 72, 84 days and so on. She makes chutney after 20, 40, 60, 80, 100 days and so on. The first number in both of these lists is 60. So Amrita next makes both jam and chutney after 60 days.

4. Intermediate Maths Challenge Problem (UKMT)

4 marks

4.1 The sum of two numbers a and b is 7 and the difference between them is 2.

What is the value of a×b?

  • 814
  • 914
  • 1014
  • 1114
  • 1214
  • (Not answered)

We have a+b=7. Also, depending on whether a>b or b>a, the value of ab is either +2 or −2. In either case, ab2=4, and so a+b2ab2=494=45. Now:


Hence 4ab=45. It follows that a×b=454=1114.

In the above solution we were able to find the value of ab without first finding the values of a and b. An alternative method would be to first find a and b and then ab.

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.