Parallelogram 19 • Level 2 • 11 Jan 2024Madam Curie's square riddle

• 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 maths competition run by the UK Maths Trust, and if you're in the UK 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.

2. Junior Maths Challenge Problem (UKMT)

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.

4. A rotten riddle

Can you solve the following riddle?

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 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

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

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.

• You can find out more about Marie Curie and why reading her notebooks could seriously damage your health by visiting this website.
• This website allows you to encode and decode messages using the Caesar shift code