Parallelogram 24 • Level 3 • 15 Feb 2024A curious bear and Doctor Who

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 What colour is the bear?

Here is a classic riddle. A hunter leaves her camp and walks 1 km south. She sees a bear and is about to shoot it, when her gun jams. She escapes by running 1 km east and then heads 1 km north back to camp. Huh! That doesn’t seem to make sense. Here is the question.

2 marks

1.1. What colour is the bear's fur?

Black
White
Brown
Blue
Red
(Not answered)

The answer is white because it is a polar bear, because you are at the North Pole. Why? Normally moving south-east-north does not get you back to where you started, but imagine that you are at the North Pole. On the surface of a sphere (like the Earth), the rules of geometry are different, south-east-north forms a complete triangle!

Also, the triangle has three right angles, so the internal angles of a triangle on a sphere can add up to 270°.

By the way... this riddle has hidden depths, and so far you have only heard half the story. More next week.

2. Junior Maths Challenge Problem (UKMT)

3 marks

2.1 Starting at the square containing the 2, you are allowed to move from one square to the next either across a common edge, or diagonally through a common corner.

How many different routes are there passing through exactly two squares containing a 0 and ending in one of the squares containing a 9?

7
13
15
25
32
(Not answered)

In order to reach a 9 in three steps, the first zero must be one of the three adjacent to the 2 and the second zero must be one of the five adjacent to a 9.

The table shows the number of such routes to that point.

So the total number of different routes is 25.

3. Doctor Who mathematics

If you are fan of Doctor Who, then you might already know that the show regularly features mathematics, and many of the characters are mathematicians (e.g., Romulus and Remus Sylvest (The Twin Dilemma), Danny Pink (Into the Dalek), Jackson Lake (The Next Doctor), Alistair Gordon Lethbridge-Stewart (Mawdryn Undead), Adric (Full Circle) and Zoe Heriot (The Wheel in Space).

This is my favourite maths clip from Doctor Who.

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

So the Doctor spots that the pattern involves happy prime numbers. You will already know about prime numbers, but here Dr Ria Symonds is from the University of Nottingham explains what makes a number happy.

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

You can think about a happy number in the following way. Take a positive integer, then replace the number by the sum of the squares of its digits. Take the new number, and repeat the process again and again. If the process eventually takes you to 1, then the original starting number is a happy number. If the process avoids 1 and goes into a cycle of numbers, then it is an unhappy number.

For example, 97 is happy, because it leads to the following sequence.

9² + 7² = 130
1² + 3² + 0² = 10
1² + 0² = 1 = happiness

Because 97 is prime, it is not just a happy number, but also a happy prime.

I am going to define a quantity called “the sum of all happiness” for a number, which is the total of all the numbers on the happiness path added together.

So, the sum of all happiness for 97 is (97 + 130 + 10 + 1), which equals 238.

2 marks

3.1. 32 is a happy number. What is the sum of all happiness for 32?

Correct Solution: 56

2 marks

3.2. 79 is a happy number. What is the sum of all happiness for 79? Is there a shortcut you can use?

Correct Solution: 220

Because 79 has the same digits as 97, it will follow the same happiness path. Because 97 was the example given above, you can cheat and start with the answer given above, which is 238. However, your path started with 79, not 97, so you need to subtract 97 and add 79, which gives you a total of 220.

3 marks

3.3. 4 is an unhappy number. If you follow the usual process (ie, 4² = 16, and then square the digits of 16 to get a new number, and so on) you end up creating a loop, as opposed to heading towards 1. Including the number 4 once, how many numbers are in the loop.

Correct Solution: 8

You should have found the following loop: 4, 16, 37, 58, 89, 145, 42, 20, 4, ... It consists of eight numbers if you only include 4 once.

3 marks

3.4. In 2010, mathematicians discovered the largest known happy prime, which is 2^42,643,801 – 1. Write down the complete path and prove that this number is indeed happy. Only kidding! That would take you several lifetimes.

Instead, if you expressed 2^42,643,801 in the normal way (ie, not as a power of 2), how many digits would it contain?

2 digits
42,643,801 digits
roughly 7 million digits
more than 10 million digits
less than 5 million digits
(Not answered)

My calculator screamed ERROR when I tried to be lazy. Instead, I started by noticing 2¹⁰ ≈ 1,000.

You might not have covered how to manipulate exponentials in detail, so don’t worry if you cannot follow this completely, but you can then say:
2^42,643,801 ≈ (2¹⁰)^4,264,380, because 2^ab = (2^a)^b
≈ (1,000)^4,264,380
≈ (10³)^4,264,380, because 1,000 = 10³
≈ 10^{3 × 4,264,380}
≈ 10^12,793,140, so the number has about 12,293,140 digits.

4. Junior Maths Challenge Problem (UKMT)

This is one of the toughest Junior Maths Challenge questions that you will come across, so I have given a hint. Try and solve the problem on your own – and be tenacious – but the hint is there if you need it.

5 marks

4.1. The points S, T, U lie on the sides of the triangle PQR, as shown, so that QS = QU and RS = RT.

∠TSU = 40°. What is the size of ∠TPU?

60°
70°
80°
90°
100°
(Not answered)

Show Hint (–1 mark)

–1 mark

Let ∠RST = x° and ∠QSU = y°. As RS = RT, the triangle RST is isosceles and hence ∠RTS = ∠RST = x°. Hence, as the angles in triangle RST add up to 180°, we have that ∠SRT = (180 - 2x)°.

Show Hint (–1 mark)

–1 mark

∠SQU = (180 - 2y)°. Hence from triangle PQR, we have that ∠TPU = (180 - (180 - 2x) - (180 - 2y))° = (2 (x + y) - 180)°.

The question was hard, and the answer is pretty hard, too. Go through it slowly, step by step, and you really will understand it. If you are flummoxed by one particular step, then ask your teacher to explain it.

Let ∠RTS = x°. Then ∠RST = x as RS = RT.

Let ∠QSU = y°. Then ∠QUS = y° as QS = QU.

As RSQ is a straight line, x + y + 40 = 180; so x + y = 140.

Now ∠TPU = 180° - ∠TRS - ∠SQU
= 180° - (180 - 2x)° - (180 - 2y)°
= 180° - 180° + 2x° - 180° + 2y°
= 2(x + y)° - 180°
= 2 × 140° - 180°
= 100°

5. An odd square.

Here is something curious.

You could pick any number, but let’s try it with the number 6.

First, write down a 6 by 6 grid of the first 6 × 6 odd numbers.

1	3	5	7	9	11
13	15	17	19	21	23
25	27	29	31	33	35
37	39	41	43	45	47
49	51	53	55	57	59
61	63	65	67	69	71

If you add up the numbers to the top row, then you get:

1 + 3 + 5 + 7 + 9 + 11 = 36 = 6²

If you add up the numbers in either diagonal, then you get:

1 + 15 + 29 + 43 + 57 + 71 = 216 = 6³

11 + 21 + 31 + 41 + 51 + 61 = 216 = 6³

Finally, if you add up all the numbers in grid, then you get 6⁴.

The crazy thing is that you can do this this with any number all the way up to infinity. But, for simplicity, let’s try it with the number 3 and create a 3 by 3 grid of the first nine odd numbers.

1	3	5
7	9	11
13	15	17

Top row: 1 + 3 + 5 = 9 = 3²

Diagonals: 1 + 9 + 17 = 5 + 9 + 13 = 27 = 3³

All: 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 = 81 = 3⁴

Why does this happen? Let’s look at why the top row adds up to the square of the number.

A 3 x 3 grid of 9 blocks consists of 1 block in the top left, 3 blocks wrapped around it, and 5 blocks wrapped around those. So a 3² grid consists of (1 + 3 + 5) blocks, and 1, 3, 5 are the first three odd numbers and fill up the top row of the grid. That’s why 1 + 3 + 5 = 9 = 3².

And why do the diagonals of an n × n grid add up to n³? And why do all the numbers in an n × n grid add up to n⁴? That’s something for you to think about.

6. Monty Python

And now for something completely different. Monty Python’s Flying Circus was one of the BBC’s most anarchic and revolutionary comedy shows. The sketches were based on a whole series of absurd premises, such as the Ministry of Silly Walks and a customer complaining about a dead parrot.

As far as I know, they only once touched on mathematics. Maths and philosophy overlap when it comes to the subject of logic, so take a look at this sketch titled “The Philosophers’ Football Match” and try to answer these two questions. You might need to do some googling. (By the way, as you watch this video, does it remind you of something that might appear on "Horrible Histories"? I suspect that some of the "Horrible Histories" silliness was inspired by Monty Python.)

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

2 marks

6.1. Which of the Greek footballing philosophers was killed by a Roman soldier, because he was so busy doing mathematics that he refused to obey the soldier’s orders?

"Chopper" Sophocles
Archimedes
Plotinus
(Not answered)

2 marks

6.2. Which of the German footballing philosophers invented calculus (separately from Isaac Newton), one of the most important topics in mathematics? He also invented the first mass-produced mechanical calculator.

Gottfried Leibniz
Immanuel Kant
Georg “Nobby” Hegel
(Not answered)

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

If you are interested in studying physics (or maths or engineering) at university, then you might want to watch this video by Physics Girl, in conversation with Jabril Ashe. She talks about her path into physics. Jabril, by the way, is a computer programmer, as well as a YouTuber:
For more Doctor Who stuff, here is the BBC Doctor Who fun and games website.

1	3	5	7	9	11
13	15	17	19	21	23
25	27	29	31	33	35
37	39	41	43	45	47
49	51	53	55	57	59
61	63	65	67	69	71

1	3	5	7	9	11
13	15	17	19	21	23
25	27	29	31	33	35
37	39	41	43	45	47
49	51	53	55	57	59
61	63	65	67	69	71

1	3	5	7	9	11
13	15	17	19	21	23
25	27	29	31	33	35
37	39	41	43	45	47
49	51	53	55	57	59
61	63	65	67	69	71