Parallelogram 19 • Level 3 • 11 Jan 2024Lou Costello, Pi and Pendula

Noun: Parallelogram Pronunciation: /ˌparəˈlɛləɡram/

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. Weird film about π

Back in 1998, I went to see a film at the cinema called π. To be honest, it was fairly dreadful, so I would not recommend watching the whole film. However, take a look at this snippet from the trailer.

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

I like the comment that “mathematics is the language of nature”, and you will see that more and more as you study different areas of maths. For example, take a look at this image of the bottom of a pine cone you will see mathematical spirals. There are green spirals going in one direction and red spirals going in the opposite direction.

1 mark

1.1. How many green spirals are there?

Correct Solution: 8

1 mark

1.2. How many red spirals are there?

Correct Solution: 13

These numbers come from a sequence in which you add the previous two numbers to get the next number, e.g, 1 + 1 = 2, then 1 + 2 = 3, then 2 + 3 = 5, and so on.

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377...

1 mark

1.3. This series of numbers is called:

The triangular sequence
The additive series
The 112 order
The Fibonacci sequence
The Leibniz series
(Not answered)

The sunflower below also has two sets of spirals, and the numbers of spirals are also part of this number series.

1 mark

1.4. How many anticlockwise (x) spirals?

Correct Solution: 34

1 mark

1.5. How many clockwise (y) spirals?

Correct Solution: 21

2. Junior Maths Challenge Problem (UKMT)

The Junior Maths Challenge is a national maths competition run by the UK Maths Trust, and many of you may have entered last year and might be entering this year. Ask your teacher, and he or she will know whether you or your class will be entering.

The Junior Maths Challenge is aimed at confident young mathematicians who want to stretch their brains with questions that are bit spicier than those you might encounter in the classroom. Each week, the Parallelograms will contain one or two Challenge questions from the tests given in previous years.

These will be good practice if you are entering this year, and even if you are not entering they will still be good ways to stretch your mathematical mind. To get you warmed up, here is a relatively easy question... followed by another one that most of you should be able to attack.

2 marks

2.1. What is the sum of the marked angles in the diagram?

90°
180°
240°
300°
360°
(Not answered)

The six marked angles are the interior angles of 2 triangles. The interior angles of 1 triangle add up to 180°. So the marked angles add up to 2x180° = 360°.

3. Junior Maths Challenge Problem (UKMT)

2 marks

3.1. Peter Piper picked a peck of pickled peppers. 1 peck = 1/4 bushel and 1 bushel = 1/9 barrel. How many more pecks must Peter Piper pick to fill a barrel?

12
13
34
35
36
(Not answered)

We have 1 barrel = 9 bushels and 1 bushel = 4 pecks, and so 1 barrel = 9 × 4 = 36 pecks. Peter Piper has already picked 1 peck, so he needs to pick 35 more pecks to fill a barrel.

4. Calculating π with a pendulum

Last week’s Parallelogram looked at how the length of a pendulum is linked to the time it takes to swing, via a formula involving π. The video below from the brilliant Matt Parker (the Stand-up Mathematician) shows how you can use this formula to work out the value of π. Watch carefully and answer the question below.

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

1 mark

4.1 How long did it take in Matt’s experiment for 1 swing?

3.1 seconds
3.128 seconds
3.141 seconds
31.28 seconds
31.4 seconds
(Not answered)

2 marks

4.2 If it was a perfect experiment, how long should it have taken for 10 swings of the pendulum?

3.128 seconds
3.14 seconds
10.0 seconds
31.28 seconds
31.4 seconds
(Not answered)

Matt’s experiment was imperfect (in fact all experiments are imperfect), so the answer to question 4.1 and Matt’s result in the video was 3.128 seconds. However, if it was a perfect experiment, then each swing would have been π seconds long, or 3.14 seconds. So 10 swings would have been 31.4 seconds, and that is the answer.

5. Infinite expressions for π

Matt’s pi-endulum video starts with something astonishing, namely that there are some beautiful ways of calculating π. (BTW, I use the word ‘beautiful’ because some things in maths are beautiful. They strike your heart and make you go, wow!)

About 350 years ago, mathematicians proved that:

Why on earth should adding half the “odd” fractions and subtracting the other half of “odd” fractions and multiplying by 4 be equal to π? When you’re older, you might find out, but for now just be gobsmacked.

This equation is very slow, by which I mean that you need to take into account a large number of terms before you get an answer that becomes accurate. If you work out the first 300 terms of the formula, then you will eventually work out that π ≈ 3.14. (BTW the squiggly equals sign “≈” means “approximately equal to”.)

In 1748, Leonard Euler proved this recipe for calculating π

We can rearrange this to obtain:

Let’s see what happens if we use this recipe to calculate π, using just one term, then two terms, then three, four and five terms.

π = 2.45

π = 2.74

π = 2.86

π = 2.92

π =

The result gets closer and closer to the real value of π as we include more and more terms.

2 marks

5.1 What is the value of π when you include the 1/5² term?

3.00
2.99
2.98
2.97
2.96
(Not answered)

Rohit Kumar has created some graphs that show how these two recipes get better and better at approximating π as you include more and more terms. Take a look at the graphs below and answer two questions.

1 mark

5.2 Which series always underestimates the value of π?

Neither of them
Both of them
The one on the left
The one on the right
(Not answered)

Because the series on the left has positive and negative terms (addition and subtraction), it keeps jumping above the value of π, and then below it, and then above it, and then below it, and so on. By contrast, the series on the right only has positive terms, so it starts at a very low value and gradually approaches the true value from below. Therefore the series on the right always underestimates the true value of π.

1 mark

5.3 Both series make the biggest improvements in approximating π...

During the first few terms
During the last few terms
During the middle few terms
When they put in the most effort
(Not answered)

Initially, both series are wildly inaccurate, but within the first 25 terms they both begin to home in on the true value of π. Thereafter the curve is almost flat, suggesting that there are only minor improvements in getting close to π. Hence, the biggest improvements take place during the first few terms.

6. Lou Costello’s masterclass

Last week’s Parallelogram had some disturbing arithmetic courtesy of Ma and Pa Kettle. Here is a similar comedy routine from the great Lou Costello, one of America’s most famous comedians. Watch carefully and see if you can work out why the maths is fundamentally wrong.

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

2 marks

6.1 If Lou Costello were to work out 3 × 15 or (15 + 15 + 15), what would he think the result is?

Correct Solution: 18

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 week.
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.

Additional Stuff

From now on, Parallelograms will often contain this Additional Stuff section, which carries no mark, but which you might find interesting. Why not take a look? However, it is optional, so you can also just skip to the SUBMIT button and click.

This website has lots of great material about “Fibonacci Numbers and Nature”.

You can find lots of other videos by Matt Parker at his YouTube channel.