Parallelogram 21 • Level 5 • 25 Jan 2024Monkey Magic

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.
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. A card trick

Here is a lovely mathematical card trick by James Grime, from the terrific Numberphile YouTube channel.

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

2 marks

1.1 The trick is based on the...

The Proizvolov constant
The Proizvolov equality
The Proizvolov identity
The Proizvolov variable
(Not answered)

3 marks

1.2 James performed the trick with 10 cards (labelled 1 to 10). What number would you end up with (the sum of the difference for each pair) if you performed the trick with only 8 cards (labelled 1 to 8)?

Correct Solution: 16

James explained that the sum was always the square of half of the biggest number, if the numbers were consecutive. So for cards labelled 1 to 8, the sum is 822=16.

2. Monkey Magic

Young babies, a few months old, are not impressed by magic. The entire world is magical and without rules, so objects disappearing is nothing special. Older infants, who are beginning to learn the laws of the universe, are mildly shocked when a magician uses trickery to make a sweet disappear. And, of course, once we reach school age, we all appreciate that magic is all about tricks that appear to do the impossible. But what about monkeys and apes?

When you watch this video, please bear in mind that you can sometimes see how the trick is done, but from the animal’s point of view the trick is amazing.

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

3. Intermediate Maths Challenge Problem (UKMT)

4 marks

3.1 The number ‘tu’ is the two-digit number with units digit u and tens digit t. The digits a and b are distinct, and non-zero.

What is the largest possible value of ‘ab’ − ‘ba’?

81
72
63
54
45
(Not answered)

Show Hint (–1 mark)

–1 mark

‘ab’ − ‘ba’ =10a+b−10b+a

The place-value notation means that ‘ab’ =10a+b and ‘ba’ =10b+a. Therefore

‘ab’ − ‘ba’ =10a+b−10b+a
=10a−a+b−10b
=9a−9b
=9a−b.

The largest possible value of 9a−b occurs when a−b has its largest possible value. Because a and b are different non-zero digits, the largest value of a−b is 8, when a=9 and b=1. In this case ‘ab’ − ‘ba’ =91−19=72.

Therefore the largest possible value of ‘ab’ − ‘ba’ =72.

4. Intermediate Maths Challenge Problem (UKMT)

4 marks

4.1 Each edge of a cube is coloured either red or black. If every face of the cube has at least one black edge, what is the smallest possible number of black edges?

2
3
4
5
6
(Not answered)

Show Hint (–1 mark)

–1 mark

The answer is NOT 4.

Each edge of the cube borders two faces. As there are 6 faces, a minumum of three black edges will be required.

The diagram shows that the required condition may indeed be satisfied with three black edges.

5. Intermediate Maths Challenge Problem (UKMT)

5 marks

5.1 Brachycephalus frogs are tiny – less than 1 cm long – and have three toes on each foot and two fingers on each ‘hand’, whereas the common frog has five toes on each foot and four fingers on each ‘hand’. All of the frogs have two hands and two feet.

Some Brachycephalus and common frogs are in a bucket. Each frog has all its fingers and toes. Between them they have 122 toes and 92 fingers.

How many frogs are in the bucket?

15
17
19
21
23
(Not answered)

Let b be the number of Brachycephalus frogs in the bucket and let c be the number of common frogs in the bucket.

A Brachycephalus frog has three toes on each foot and two fingers on each ‘hand’. Therefore, in total, it has 6 toes and 4 fingers. A common frog has, in total, 10 toes and 8 fingers.

Therefore, between them b Brachycephalus frogs and c common frogs have 6b+10c toes and 4b+8c fingers. Hence, from the information given in the question, we have 6b+10c=122, and 4b+8c=92.

Subtracting the second equation from the first, we obtain 2b+2c=30.

Dividing the last equation by 2, we deduce that b+c=15.

Therefore, the total number of frogs in the bucket is 15.

6. Words without “a”

Here’s a game you can play.

How many words can you say in 15 seconds that do NOT contain the letter A? As soon as you say a word with an “a” in it, your turn is over.

How well did you do? 10 words is okay, 15 is good, 20 is very good, but here’s a way that you can get 50 words... all you need to do is count quickly from 1 to 50... or maybe even 100 if you can count that quickly. If you remember from last week, the first number containing an “a” is 101.

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.