PG 21 7 Mar 2019Computer music and techno ditties

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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. School competitions

Lots of schools offer students a weekly or monthly maths puzzle competition. If your school does not have such a competition, then you could suggest it to your maths teacher.

And if you are a student or teacher that already takes part in or runs a regular maths puzzle competition, then let us know if you have any questions that we could include in our Parallelograms by emailing us.

In the meantime, here is a question from a London school, posed to its students earlier this year.

3 marks

1.1. I have 4 digits. I am a perfect square. I look the same upside down. My square root is a prime number. Who am I?

Correct Solution: 6889

There are different ways to tackle this problem. For example, we could identify the 4-digit square numbers, but there are more than 60 of them. So, instead, let’s list all the 4-digit numbers that look the same upside down – they have to contain 0s, 1s, 6s, 8s and 9s. So, we have: 1001, 1111, 1691, 1961, 1881, 6009, 6119, 6889, 6699, 6969, 8008, 8118, 8888, 8698, 8968, 9006, 9116, 9886, 9966, 9696. The only perfect square (I think) is 6,889, and the square root is 83, which is indeed a prime. So, the answer is 6,889.

2. Delia Derbyshire

Following last week’s mini-exploration of the mathematics of Doctor Who, here is a short video (from a longer documentary) about Delia Derbyshire, who created the Doctor Who theme tune while she was working at the BBC Radiophonic Workshop.

Derbyshire, whose father was a sheet-metal worker, came from an ordinary family in Coventry, but her talent for mathematics took her on an extraordinary journey. She went to Cambridge University, where she studied both music and mathematics, and later she joined the BBC. Her love of mathematics and technology enabled her to create wholly new types of music, using entirely new electronics. Many credit her with playing a major role in the development of modern music, from techno to grime.

While you're watching this video, consider this question.

1 mark

2.1 According to Alan Sutcliffe, to do maths you need to be...

  • ...a perfectionist
  • ...obsessed
  • ...clever
  • ...confident
  • ...fast

3. Junior Maths Challenge Problem (UKMT)

2 marks

3.1. Two adults and two children wish to cross a river. They make a raft but it will carry only the weight of one adult or two children. What is the minimum number of times the raft must cross the river to get all four people to the other side? (N.B. The raft may not cross the river without at least one person on board.)

  • 3
  • 5
  • 7
  • 9
  • 1

We note that if an adult crosses the river, there has to be a child on the other side to bring the raft back, otherwise the adult has to bring the raft back and two journeys have been wasted.

So the first journey must involve two of the children crossing, and one of them bringing the raft back. Then an adult can cross the river, and the child on the other bank can bring the raft back.

In this way it has taken 4 journeys to get one adult across the river, with the other adult and the two children in their original position. With another 4 similar journeys, the second adult can cross the river. This leaves the two children in their original position. They can now cross the river in one journey.

In this way it takes 9 journeys to get everyone across the river.

4. Where do trees get their mass from?

Veritasium is a YouTube channel run by Derek Muller. He often asks apparently easy questions, and then explores the misunderstandings that many adults have on the topic. In this episode, Muller asks: “Where do trees get their mass from?”

Before watching the video, have a think about how you would answer this question.

5. An odd house

2 marks

5.1. A house has four walls. All the walls face south. Where is the house? The answer consists of a 5-letter word and a 4-letter word.

Correct Solution: North Pole

If you remember last week’s Parallelogram... the North Pole is an odd place, and geometry on a sphere does not obey the same laws as geometry on a flat piece of paper. A house on the North Pole would have all of its walls facing south.

6. Eight queens puzzle

0 marks

Can you place eight queens on a chess board, so that no queen can capture any other queen? If you don’t play chess, then all you need to know is that a queen can move forwards, backwards and diagonally. Hence, the queens need to be placed so that no queen is in the line of sight (straight or diagonally of any other queen). There are no marks for this question – I have included it just to stretch your brain.

You can use the following website to try different possibilities. Good luck. http://www.brainmetrix.com/8-queens/

By the way, there are 92 distinct solutions, and we’ll have one of them in the answer sheet.

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

  • This BBC webpage has a video about Delia Derbyshire and her friend Daphne Oram, who used mathematics and technology to invent a new type of music.

  • Last week’s “additional stuff” contained an interview with Physics Girl, and the interviewer was Jabrils, a YouTuber and software engineer. Here is a video by Jabrils, which recounts his first attempt at writing a machine learning game.