Parallelogram 15 Year 7 16 Jan 2020Madam Curie’s square riddle

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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. Junior Maths Challenge Problem (UKMT)

The Junior Maths Challenge is a national maths competition run by the UK Maths Trust, and many of you 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 a 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.

1 mark

1.1. What is the value of 2×0×1+1?

  • 0
  • 1
  • 2
  • 3
  • 4

This is simply a matter of doing the sum... and remembering the order of operations. So, multiplications take priority over addition, so 2×0×1+1=2×0×1+1=0+1=1.

2. Junior Maths Challenge Problem (UKMT)

1 mark

2.1. How many of the integers 123, 234, 345, 456, 567 are multiples of 3?

  • 1
  • 2
  • 3
  • 4
  • 5

We could simply divide each of the given numbers by 3, and check that there is no remainder in each case. However, it is quicker to use the fact that an integer is divisible by 3 if and only if the sum of its digits is a multiple of 3.

123: 1 + 2 + 3 = 6
234: 2 + 3 + 4 = 9
345: 3 + 4 + 5 = 12
456: 4 + 5 + 6 = 15
567: 5 + 6 + 7 = 18

These are all multiples of 3, so all five numbers are multiples of 3, so the answer is 5.

In fact, the sum of any 3 consecutive numbers is a multiple of 3, as one of those numbers will already be a multiple of 3, one will be exactly 1 less than a multiple of 3, and one will be exactly one more than a multiple of 3. So, when added together the +1 and –1 will cancel, and give us a multiple of three.

We can show this algebraically for the three consecutive numbers n1, n, and n+1: n1+n+n+1=3n.

3. The tale of Marie Curie

Marie Curie is best known as a physicist and a chemist. Indeed, she won two Nobel prizes, one in physics and one in chemistry. Only four scientists in all of history have ever won two Nobels. However, as well as being a scientist, Curie had a strong interest in maths. Her father taught mathematics, and she also took mathematics courses at university, which certainly helped her win her Nobel Prizes.

Also, before marrying Pierre Curie, she fell in love with Kazimierz Żorawski, who would become a professor of mathematics. Tragically, his family refused to let him marry Curie, a decision that haunted him for the rest of his life. As an old man and a retired mathematician, he would regularly visit the statue of Curie that had been erected in Warsaw. She was still the great love of his life, and now one of the world’s most famous scientists.

Curie is one of the most heroic and inspiring scientists in history, and below is a short video that tells her story. Watch carefully and answer the question that follows.

1 mark

3.1. Curie’s 1911 Nobel Prize was for the discovery of an element that was named after a country. What is the name of the element?

  • Francium
  • Francesium
  • Polandium
  • Polonium
  • Newzealandium

Polonium was discovered in 1898 by Marie and Pierre Curie, and it was named after Marie Curie's homeland of Poland.

4. A rotten riddle

Can you solve the following riddle?

1 mark

4.1 If you break me, then I become better than I was before, and I become even harder to break.

What am I?

The answer is made up of exactly 6 of these 7 letters: BCRRDEO

Correct Solution: Record

The answer is Record.

5. Caesar cipher

Last week, I said that cryptography (the maths of codes and codebreaking) would be a theme in the next few Parallelograms, so here is this week’s lesson and puzzle relating to the writing of secret messages.

The Caesar cipher, which supposedly dates back to the Romans, involves shifting every letter by the same amount. For example, below is the standard alphabet, and we can imagine a shift of 2, which means encoding each letter with the letter that is 2 places further up.

A B C D E F G H I J K L M
00 01 02 03 04 05 06 07 08 09 10 11 12
N O P Q R S T U V W X Y Z
13 14 15 16 17 18 19 20 21 22 23 24 25

A is encoded as C... B is encoded as D... C is encoded as E, and so on.
CAT become ECV.

So far, so good, but where is the maths? In the alphabet above, each letter is matched with a number from 0 to 25, and instead of thinking of encoding as shifting, we can think of encoding as adding.

CAT = 2¦0¦19
CAT + 2 = 4¦2¦21 = ECV

Decoding is the opposite of addition, namely subtraction.

ECV = 4¦2¦21
ECV – 2 = 2¦0¦19 = CAT

So far, so good, but what happens if we encode a word such as YES with a shift of 3?

YES = 24¦4¦18
YES + 3 = 27¦7¦21 = ?HV

27 does not appear in our alphabet. How do we represent 27 as a letter? What letter does Y turn into?

The solution is modulo arithmetic, which operates by arranging all the allowable numbers in a circle, so that three odd things happen.

  1. In modulo (26) arithmetic, we replace 26 with 0.
  2. Addition normally means moving along the number line, but now we move around the number circle.
  3. That means, for example, 24 + 3 = 1 (not 27).

Therefore, YES is encoded as 27¦7¦21, which in modulo (26) is 1¦7¦21, which is BHV.

You encounter modulo arithmetic every day, when you look at a clock (and sometimes it is called clock arithmetic). Two hours after 11pm (or 23.00) is not 13pm (or 25.00), but rather 1am (or 01.00).

So, if we encoded the word WORD with a shift of 10, then:

WORD = 22¦14¦17¦3
WORD + 10 = 32¦24¦27¦13

The numbers need to be between 0 and 25, so we subtract 26 from the encoded number until it is ok.

WORD + 10 = 6¦24¦1¦13 = GYBN

A B C D E F G H I J K L M
00 01 02 03 04 05 06 07 08 09 10 11 12
N O P Q R S T U V W X Y Z
13 14 15 16 17 18 19 20 21 22 23 24 25

Now it is also time to try some simple encoding problems, but first here is an example: encode the word BOX with a shift of 1 into a new 3-letter word:

BOX in numbers = 1¦14¦23
BOX + 1 in numbers = 2¦15¦24
BOX + 1 in letters = CPY

1 mark

5.1 Encode the word BOX with a shift of 2 into a new 3-letter word. In other words, translate BOX into numbers, add 2 to each number and convert the result back into letters:

  • ZMV
  • ANW
  • CPY
  • DQZ
  • CQW
1 mark

5.2 Encode the word BOX with a shift of 10 into a new 3-letter word. In other words, translate BOX into numbers, add 10 to each number and convert the result back into letters:

  • MAG
  • LYH
  • PEJ
  • OEI
  • MBH

BOX is 1¦14¦23, and if we add 10 we get 11¦24¦33, which is L¦Y¦?.

33 is bigger than 26, so we subtract 26 to get 7, which means that '?' = 'H'.

So the answer is LYH.

1 mark

5.3 Encode the word BOX with a shift of 25 into a new 3-letter word. In other words, translate BOX into numbers, add 25 to each number and convert the result back into letters:

  • ZMV
  • ZNY
  • ANW
  • CPY
  • AMZ

Another way of adding 25 in modulo arithmetic is to subtract 1. Instead of going forward 25 spaces, you can just do one step back, which is a shortcut to getting the answer.
0¦13¦22 = ANW

1 mark

5.4 Encode the word BOX with a shift of 26 into a new 3-letter word. In other words, translate BOX into numbers, add 26 to each number and convert the result back into letters:

  • ANW
  • ZNY
  • CPY
  • APY
  • BOX

A shift of 26 gets you back to where you started, so no change.
1¦14¦23 = BOX

1 mark

5.5 Encode the word BOX with a shift of 52 into a new 3-letter word. In other words, translate BOX into numbers, add 52 to each number and convert the result back into letters:

  • APY
  • BOX
  • CPY
  • ANW
  • ZNY

Again, because this is a multiple of 26, no change.
1¦14¦23 = BOX

1 mark

5.6 Encode the word BOX with a shift of 260 into a new 3-letter word. In other words, translate BOX into numbers, add 260 to each number and convert the result back into letters:

  • APY
  • ANW
  • ZNY
  • CPY
  • BOX

Again, because this is a multiple of 26, no change.
1¦14¦23 = BOX

1 mark

5.7 Encode the word BOX with a shift of 261 into a new 3-letter word. In other words, translate BOX into numbers, add 261 to each number and convert the result back into letters:

  • CPY
  • BOX
  • ZMV
  • ANV
  • ZNY

A shift of 261 is the same as a shift of 260 and then a shift of 1. A shift of 260 makes no difference as it is a multiple of 26, so we are looking simply at a shift of 1.
2¦15¦24 = CPY

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.