PG 19 14 Feb 2019Pie charts and speed-reading

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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.
  • 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.
  • As it is half-term, you have a bit of extra time to complete this Parallelogram. You should finish it by Sunday 24. And the next Parallelogram will appear on Thursday 28 February.

1. Pie Chart

You will have come across lots of pie charts, and they are a useful way of representing information. However, they are also a useful way of making jokes. These are some of my favourite funny pie charts. I have no idea who created the following pie charts, but well done to whoever invented them.

2. Junior Maths Challenge Problem (UKMT)

In case you have forgotten, or are new to Parallelograms, these Junior Maths Challenge questions come from an annual maths challenge that you might already have been part of or which you might be taking part in this year. The questions are sometimes tough, but that is the definition of a challenge.

3 marks

2.1 The 2010 match at Wimbledon between John Isner and Nicolas Mahut, which lasted 11 hours and 5 minutes, set a record for the longest match in tennis history. The fifth set of the match lasted 8 hours and 11 minutes.

Approximately what fraction of the whole match was taken up by the fifth set?

  • 15
  • 25
  • 35
  • 34
  • 910

We have that 8 hours and 11 minutes is 8 x 60 + 11 = 491 minutes and 11 hours and 5 minutes is 11 x 60 + 5 = 665 minutes. Therefore the exact fraction taken by the fifth set is 491665.

So we need to decide which of the given fractions this is closest to. This can be done in more than one way. For example, as 491 is just below 500 and 665 is almost ⅓ of 2000, we have that:

491665 ~ 5002000 = 14 = 34. (Here “~” means “is approximately equal to”.)

Another method is to say that:

491665 ~ 490665 = 98135 = 1419 ~ 1520 = 34

3. Speedreading

1 mark

3.1 According to this video from the Quirkology YouTube channel, what was your reading speed (if you were able to keep up with the video right to the end)?

  • 100 words/minute
  • 200 words/minute
  • 300 words/minute
  • 400 words/minute
  • 500 words/minute

4. Junior Maths Challenge Problem (UKMT)

3 marks

4.1 Peri the winkle leaves on Monday to go and visit Granny, 90m away. Except for rest days, Peri travels 1m each day (24-hour period) at a constant rate and without pause.

However, Peri stops for a 24-hour rest every tenth day, that is, after nine days travelling. On which day does Peri arrive at Granny’s?

  • Sunday
  • Monday
  • Tuesday
  • Wednesday
  • Thursday

Until Peri reaches Granny’s, he travels 9m in 9 days and then rests for a day. So he travels 9m in 10 days. So it takes him 90 days to travel the first 9m x 9 = 81m.

Then after a further 9 days he has travelled the final 9m and has reached Granny’s. So the journey takes him 99 days. Now 99 = 14 x 7 + 1, so the journey takes him 14 weeks and 1 day. Therefore he arrives on a Tuesday.

5. The monoalphabetic substitution cipher

In previous weeks, we have looked at codes that work by turning letters into numbers and then multiplying or adding to get a new number (and therefore a new letter). However, the most common way to send a code is to match each letter with a code-letter according to a random arrangement. For example, here is one such arrangement. The normal letter is in lower case, and the code letter is in upper case.

a b c d e f g h i j k l m
U Z G Q P O I V C D A B J
n o p q r s t u v w x y z
H F Y E M S R X N L T W K

So, according to this random rearrangement (or “key”), the word “hello” is encoded as “VPBBF”.

But according to the rearrangement below, the word “hello” would be encoded as “JQWWH”.

a b c d e f g h i j k l m
M P O B Q G C J I A D W V
n o p q r s t u v w x y z
Y H F S U Z R X T L N K E

The challenge for the codebreaker, who does not know the key, is to somehow break the code. For example, what on Earth does the following message mean? It is a famous quote about mathematics.

PAUCGN YH GFA XYBHG GAHG; GFABA YH TZ EABOUTATG EQUVA YT GFA LZBQR XZB CWQN OUGFAOUGYVH.

Such a coded message would have been unbreakable for centuries, until a 9th century mathematician called al-Kindi developed the oldest known method of code-breaking. “The Philosopher of the Arabs”, as he was known, realised that every letter has a personality, and if that letter is replaced with a different letter, then the personality is transferred to the new letter, so it should still be recognisable.

The most important part of a letter’s personality is how often it occurs. Some letters are more common than others, and the bar graph below shows the frequency of letters in English.

5 marks

5.1. Your challenge is to decipher the coded message about maths. Once you have worked out the quote, Google it to find out who said it, and then type his 5-letter surname into the answer box.

PAUCGN YH GFA XYBHG GAHG; GFABA YH TZ EABOUTATG EQUVA YT GFA LZBQR XZB CWQN OUGFAOUGYVH.

Correct Solution: HARDY

Tips:

  • Write a list of all the letters in the coded message and count how many times each one appears.
  • I created a code-breaking website, which has some tools that you might find useful. You can paste in the coded text in the ‘ciphertext’ box, click on ‘Frequency of individual letters’ and it will count the frequency of each letter.
  • From the frequency bar graph above, E and then T are generally the two most common letters in a message, so try to match these with the two most common letters in the coded message.
  • Once you have identified the two most common letters, try to guess/deduce some of the other letters. The code-breaking website can also help with this.
  • If you get stuck, be prepared to go backwards and try another letter.
  • If you get really stuck, then you can click on the hints below, but they will cost you 1 mark for each hint.
Show Hint (–1 mark)
–1 mark

CWQN = ugly

Show Hint (–1 mark)
–1 mark

Words (as well as letters) have frequencies. The most common 3-letter word in English is THE. Can you spot it in the message?

Show Hint (–1 mark)
–1 mark

Try to get inside the head of the person who wrote the message. We know this is a message about mathematics, so what word might you expect to find in the message?

The two most common letters in the coded message are A & G, which both account for 14% of the text. The two most common letters in English are E & T, so try out (A = E, G = T) and (A = T, G = E).

If you cannot decide which letter is E and which is T, then the second hint revealed that THE is the most common 3-letter word. The sequence GFA appears twice, so it probably means ‘the’. As well as knowing that (G = T, A = E), you also know that F = H. At this point, breaking the code involves trial and error. However, the third hint reminded you that this is a message about mathematics, so you could guess that OUGFAOUGYVH means ‘mathematics’.

The quote is "Beauty is the first test; there is no permanent place in the world for ugly mathematics," which was written the British mathematician G. H. Hardy. He was commenting on his belief (which other mathematicians share) that ugly maths is probably wrong, or at least there is a more beautiful and satisfying answer. Mathematicians follow logic, but they also have a gut reaction or an emotional reaction to some equations.

Before you hit the SUBMIT button, here are some quick reminders:

  • You will receive your score immediately, and 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.
  • Finally, if you missed any earlier Parallelograms, don't forget to go back and complete them.
  • You earn reward points and badges from completing missed Parallelogams.
  • As it is half-term, you have a bit of extra time to complete this Parallelogram. You should finish it by Sunday 24. And the next Parallelogram will appear on Thursday 28 February.

Cheerio, Simon.


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