Parallelogram 11 • Level 6 • 16 Nov 2023Hanging a Picture

Noun: Parallelogram Pronunciation: /ˌparəˈlɛləɡram/

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. Banned numbers

Did you know that throughout history, and even to this day, some numbers are banned? Watch this TED-Ed video explaining why.

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

2 marks

1.1 According to the video, what reasons have been given throughout history for banning numbers?

Because numbers can encode just about any information.
Because the existence of some numbers was perceived as a threat to the notion of a perfect universe.
Because of symbolic meaning.
All of the above.
(Not answered)

2. Intermediate Maths Challenge Problem (UKMT)

3 marks

2.1 Old Martha has 5 children, each of whom has 4 children, each of whom has 3 children, each of whom is childless. How many descendants does Old Martha have?

12
20
25
60
85
(Not answered)

Martha has 5 children, 20 grandchildren and 60 great grandchildren, ie 85 descendants.

3. Knot theory

In the following video, physicist Jade Tan-Holmes solves a puzzle about hanging a picture in a particular way using a branch of mathematics called knot theory. Enjoy (or knot!) – by the way, it contains some quite complex concepts, but stick with Jade until the end and you will definitely understand the core mathematics in this video.

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

3 marks

3.1 Jade said that a common technique mathematicians use is to strip away everything but the most important features of a problem. Which of these ideas did she NOT strip away from the problem?

The chicken
The pins
The strings
(Not answered)

4 marks

3.2 Using the notation from this video, why will the link XYY−1X−1 come undone?

Because first the Xs will cancel, leaving the Ys, which will then cancel.
Because first the Ys will cancel, leaving the Xs, which will then cancel.
Because there are the same number of Xs as X−1s and there are the same number of Ys as Y−1s.
(Not answered)

Show Hint (–1 mark)

–1 mark

Only an X directly next to an X−1 cancels. XYX−1Y−1 would not cancel, but XX−1Y−1Y would completely cancel.

So YY−1 cancels, leaving XX−1, which then cancels.

4 marks

3.3 What does the link XX−1YXY−1YY−1Y−1Y reduce to?

Nothing, because everything cancels out.
Y
X
YXY−1
(Not answered)

The XX−1 and Y−1Y at each end cancel, leaving YXY−1YY−1.

Then the YY−1 on the right cancels, leaving YXY−1.

The last Y and Y−1 can't cancel because the X is in the middle.

4. Intermediate Maths Challenge Problem (UKMT)

4 marks

4.1 A long-sleeve shirt has 8 front buttons and 2 cuff buttons; a short-sleeve shirt has 6 front buttons and no cuff buttons. The factory which makes 'Slimboy Shirts' uses 10 times as many front buttons as cuff buttons. What is the ratio of long-sleeve shirts to short-sleeve shirts produced by the factory?

4 : 1
4 : 3
2 : 3
3 : 4
1 : 2
(Not answered)

Every long-sleeve shirt requires 2 cuff buttons; hence for every one long-sleeve shirt produced by the factory, a total of 20 front buttons are used. The long-sleeve shirt requires 8 of these and therefore the remaining 12 will be the front buttons on 2 short-sleeve shirts. Thus the required ratio is 1:2.

5. Intermediate Maths Challenge Problem (UKMT)

5 marks

5.1 In how many different ways can seven different numbers be chosen from the numbers 1 to 9 inclusive so that the seven numbers have a total which is a multiple of 3?

fewer than 10
10
11
12
more than 12
(Not answered)

Show Hint (–2 mark)

–2 mark

Note that the sum of the whole numbers from 1 to 9 inclusive is 45, a multiple of 3.

Thus, the stated problem is equivalent to finding a pair of numbers (the ones to exclude from our set of seven) whose sum is a multiple of three.

Note that the sum of the whole numbers from 1 to 9 inclusive is 45, a multiple of 3.

Thus the stated problem may be reduced to finding the number of ways of choosing two of these numbers whose sum is a multiple of 3.

There are 12 ways of doing this: 1, 2; 1, 5; 1, 8; 2, 4; 2, 7; 3, 6; 3, 9; 4, 5; 4, 8; 5, 7; 6, 9; 7, 8.

I hope you enjoyed this Parallelogram. 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.