Parallelogram 14 Year 7 9 Jan 2020Codes and Physics

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Noun: Parallelogram Pronunciation: /ˌparəˈlɛləɡram/

  1. a portmanteaux 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. Secret codes

Over the next few weeks, one of the mathematical themes that we will be exploring is secret codes. We’ll be making and breaking codes. The proper word for this is cryptography, from the Greek words for “secret” and “writing”.

You might think that cryptography is all about language, because we are messing around with messages made of words. Actually, cryptography is 99% maths, and numbers are at the heart of secrecy, particularly today in the age of the Internet. You might be surprised to know that some of the biggest employers of mathematicians are the secret services in countries around the world. And, of course, big companies employ mathematicians to protect their secrets (and maybe steal the secrets of rival companies).

The cryptography theme starts properly next week, but in the meantime here is a quick puzzle about a code just to warm you up.

Here is a (5x5) grid containing all the letters of the alphabet, except Z:

A B C D E
F G H I J
K L M N O
P Q R S T
U V W X Y

The word CODES is coded with the following series of symbols.

Try to work out how the code works.

2 marks

1.1. Which Star Wars character is represented by symbols below?

Show Hint (–1 mark)
–1 mark

In the example, CODES, the first symbol above represents C. It has one line on the left side, and three lines on the right side. In the grid, how does this pattern relate to the row and column containing C?

  • Rey
  • Han
  • Rex
  • Poe
  • Maz

By looking at the letters in CODES and the symbols that represent them, you should have noticed that the rule is “the letter in row x and column y is represented with a symbol which has x branches on the left and y branches on the right”. So the answer is Rey.

Alternatively, you might have noticed that the symbol for E in CODE also appears as the second symbol of the mystery name, which narrows down the possible answers to Rey or Rex. However, to decide which is correct, you really need to have worked out the general rule that links rows/columns with branches on the left/right.

2. More codes

As one final teaser for the cryptography ahead, here is a trailer for the film The Imitation Game, released in 2014. The film stars Benedict Cumberbatch playing the real-life codebreaker and mathematician Alan Turing, who cracked the German Enigma code in the Second World War. This true story shows how mathematicians can change history.

We’ll be looking at Turing and the Enigma code in a few weeks time, but first take a look at this clip and answer the two questions. You will find one answer in the film, and the other answer you will have to Google (or you might know it).

1 mark

2.1. During the course of a brief conversation lasting just a few seconds, how many people have died because of the Enigma cipher?

  • 0
  • 1
  • 2
  • 3
  • 4
1 mark

2.2. Before Britain tried to break the Enigma code, which country used its top mathematicians to make some significant breakthroughs in cracking Enigma?

  • USA
  • Poland
  • France
  • Sweden
  • Netherlands

The Polish mathematicians who cracked the Enigma code changed history and laid the foundations for the British codebreakers at Bletchley Park, who went on to help win the Second World War. You can read more about the Polish contribution to cracking Enigma in this Daily Telegraph article.

3. Wangle the angles

2 marks

3.1. Look at the diagram below and wangle (obtain by some devious manner) the angles x and y.

The triangles A and B are both equilateral.

It will probably help if you redraw the diagram – a quick sketch will make it much easier.

Angle x = Correct Solution: 155°
Angle y = Correct Solution: 85°

First, you can fill in the six 60° angles of the 2 equilateral triangles. Angle a1 is obvious (85°), and so is angle a2 (80°). Once you know a2, then a3 must be 35°.

The rest is easy.

Angle y must be (180 – a3 – 60)°, which is 85°.

Angle x must be (360 – 60 – 60 – a1)°, which is 155°.

4. A maths teacher’s age

A student asked her maths teacher if he would reveal his age. Seeing an opportunity to test his student’s mental skills, the teacher replied with a riddle: “My age in years is not a prime but is odd. Also, when the digits in my age are reversed and the new number added to my age, the result is a perfect square. Of course, if you prefer, you can take my age and reverse the digits and subtract that from my age and still get a perfect square.”

Although 51 is odd and not a prime, the teacher cannot be 51 years old, because neither (51 + 15) nor (51 -15) are perfect squares, and they both need to be perfect squares.

3 marks

4.1 How old is the teacher?

Correct Solution: 65

Show Hint (–1 mark)
–1 mark

It might be reasonable to assume that the teacher is more than 20 and less than 80. Moreover, the age is odd but not a prime, so you can start by listing all the odd numbers and striking out the primes, so you have: 21, 23, 25, 27, 29, 31, 33, 35, 37, 39,…

Show Hint (–1 mark)
–1 mark

If the second digit in the age is larger than the first digit, then when you reverse the digits you will have a larger number and you will not be able to subtract the reversed age from the actual age. Hence, you cross out all those numbers AB where B is bigger than A, so you have the strikeouts from Hint 1, plus the ones shown in bold: 21, 23, 25, 27, 29, 31, 33, 35, 37, 39,…

It might be reasonable to assume that the teacher is more than 20 and less than 80. Moreover, the age is odd but not a prime, so you can start by listing all the odd numbers and striking out the primes, so you have: 21, 23, 25, 27, 29, 31, 33, 35, 37, 39,…

If the second digit in the age is larger than the first digit, then when you reverse the digits you will have a larger number and you will not be able to subtract the reversed age from the actual age. Hence, you cross out all those numbers AB where B is bigger than A, so you have the first set of strikeouts plus the ones shown in bold: 21, 23, 25, 27, 29, 31, 33, 35, 37, 39,…

If you test whatever is left over, then you will eventually realise that one possible answer is 65. It is odd, but not prime, and 65+56=121=112, and 6556=9=32.

5. Dangerous physics

If you become a strong mathematician (which just requires lots of practice), then you will be well prepared to become a physicist, engineer or computer scientist. I think it is largely true to say that you cannot be a top class physicist, engineer or computer scientist unless your maths is strong.

Because of this link between maths and other subjects, we will occasionally explore topics beyond maths in our Parallelograms, and today I want to show you one of my favourite videos about physics.

Walter Lewin is one the world’s best physics lecturers, and this is one of his most famous demonstrations. One of the laws of physics says that if you drop a ball then it cannot bounce higher than the height from which you dropped it – imagine the consequences if the opposite was true – a ball would drop and bounce higher than its starting point, then fall and bounce even higher still, then fall and bounce higher still, and so on until the ball flew off into space. In short, objects tend to lose energy, often due to friction, so they never quite get back to their start position. This is also true of objects that swing back and forth, as demonstrated in the video below.

3 marks

5.1 There are different types of energy, and Professor Lewin says that the ball’s gravitational potential energy is transformed into:

  • Chemical energy
  • Kinetic energy
  • Elastic energy
  • Solar energy
  • Swing energy

Perhaps the professor is exaggerating when he implies that the demonstration is potentially deadly, but there is no doubt that it is dangerous, as demonstrated in the clip below. Before you laugh, bear in mind that the bright young student was brave and dedicated, but made just one small mistake, and paid a painful price.

Why did this experiment go so badly wrong? The ball followed the laws of physics, and it swung forward, and then it swung all the way back, but very slightly short of where it started. The problem was that the student, after releasing the ball, moved forward a tiny amount. That tiny movement was enough to result in a painful lesson. :::

6. A New Year reminder

As it is the start of the year, I thought it would be good just to remind you that Parallelograms are really tough, they cover topics that you might not have seen before and they are designed to confuse you. If too many people get 100%, then I will just have to make the Parallelograms tougher, weirder and more confusing. In short, if you only get 50%, don’t worry. If you get less than 50%, don’t worry. Just make sure you go through the solution sheet and try to find out why you got a question wrong. I really do mean it when I say that I am overjoyed if a student gets 25%, as long as they did their best, looked at the solution sheet, and then learned something. That is far better than getting 100% and learning nothing... although well done if you do get 100%.

Does that make sense? I just wanted to remind students, teachers and parents that Parallelograms are a major challenge, and to a large extent the scores don’t matter.

By the way, it does say at the start of every Parallelogram: “Don’t worry if you score less than 50%, because it means you will learn something new when you check the solutions.”

3 marks

6.1 Are you cool about Parallelograms being hard and therefore not necessarily getting a high score every week?

  • Yes, I am very cool about it. Positively chilled. Or maybe negatively chilled.
3 marks

6.2 Are you going to promise to go through the solution sheet every week and learn from your mistakes?

  • Yes, I promise to go through the solution sheet every week. Really. Seriously. I mean it. If not, then may the ghosts of dead mathematicians haunt me during maths lessons.

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.

If you want to find out more about codebreaker and mathematician Alan Turing, then this BBC article reveals what he was like as a schoolboy. It contains his school report, recently displayed at the Fitzwilliam Museum in Cambridge.

Professor Lewin is a superstar and a hero of mine. He has dedicated his life to exploring physics and explaining it to the rest of the world. In the next clip he shows some of the weird behaviour of spinning objects. Behind the experiment is a blackboard full of mathematics, and these equations predict exactly the strange behaviour of the spinning wheel.

And in the second clip, the professor explains some very advanced ideas about the death of stars, black holes and marshmallows. Interestingly, some of the physics he discusses is the same as the physics needed to describe the dangerous swinging ball demonstration. (Don’t worry if you only understand some of this video – just let the ideas blow your mind.)