Parallelogram 27 Year 8 7 May 2020Ada Lovelace

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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. The mathematics of aging

An aunt is 40 years old and she has a niece who is 10 years old – aunty is 4 times older.

In 5 years from now, the aunt will be 45 and the niece will be 15 – aunty is 3 times older.

15 years later, the aunt will be 60 and the niece will be 30 – aunty is only 2 times older.

1 mark

1.1. Is the following statement true or false? The niece will eventually match and then overtake her aunt’s age.

  • True
  • False

Of course, the niece will always be 30 years younger than her aunt.

2. Junior Maths Challenge Problem (UKMT)

2 marks

2.1. Usain runs twice as fast as his mum. His mum runs five times as fast as his pet tortoise, Turbo. They all set off together for a run down the same straight path. When Usain has run 100 m, how far apart are his mum and Turbo the tortoise?

  • 5 m
  • 10 m
  • 40 m
  • 50 m
  • 55 m

When Usain has run 100 m his mum has run half this distance, that is, 50 m and Turbo has run one-fifth of his mum’s distance, that is, 10 m. So the distance between his mum and Turbo is (50 − 10) m = 40 m.

3. Double negatives

A double negative is something like: “I didn't not go to the park today.” This can be rewritten as: “I did not not go to the park today.”

This actually means: “I did go to the park today.” The “not not” cancel each other out. You can think about it mathematically, as (not = -1) and (not not = −1 × −1 = +1).

Sometimes a double negative works differently, such as in the phrase: “I didn't go nowhere today.” It has two NOTs in it, first in (didn’t = did not) and second in (nowhere = not somewhere)... so according to the previous rule this suggests that “I didn’t go nowhere” means “I went somewhere”. However, we all know that if someone says this then they didn’t go anywhere. If we think about it mathematically it is a negative added to a negative instead of a negative multiplied by a negative, i.e., (−1 + −1 = −2).

In the second example, the double negative means really, really negative, not positive.

My friend Len Fisher, an Australian physicist who lives in Bristol, once pointed out an example of something very unusual, namely a double positive.

In the early 1950′s, the esteemed Oxford philosopher J. L. Austin came to Columbia to present a paper about the structural analysis of language. He pointed out that, in English, although a double negative implies a positive meaning (i.e. “I’m not unlike my father…”), there is no language in which a double positive implies a negative. “Yeah, yeah,” scoffed Morgenbesser, a Professor of philosophy at Columbia University, who was sat at the back of the auditorium.

4. Ada Lovelace

This video tells the story of Ada Lovelace, one of history’s most important mathematicians. As you watch it, look out for the answer to the following question.

2 marks

4.1. Ada Lovelace worked alongside which famous mathematician?

  • Lord Kelvin
  • Lord Cavendish
  • Charles Babbage
  • Lord Byron
  • Michael Faraday

Before I forget, make sure you check out the Additional Stuff section, where you will find out about the Ada Lovelace comic book.

5. Ada’s puzzle

When historians went through the archive of Ada’s personal papers in the Bodleian Library at Oxford, they discovered a piece of paper with baffling doodles. Upon closer inspection, it became clear that Ada was tackling a problem in what we now call graph theory. Here is a related graph theory problem.

2 marks

5.1. The drawing below was made with a single line. In other words, the pencil was placed on the paper, it never left the paper until the drawing was complete, and at no time did the pencil go back over any line that had previously been drawn. Where might the drawing have begun?

  • A
  • B
  • C
  • D
  • None of the above

By trial and error, you should have worked out that the only place you can start from is C in the bottom right corner (or the unlabelled bottom left corner). Why? In fact, you have to start at one of the bottom corners and end at the other bottom corner. If you count the number of lines connected to each of the intersections, the total is always even apart from the two bottom right and left corners, which have 3 lines. This odd number of connections turns out to be crucial, and we’ll return to it in the next Parallelogram.

6. Junior Maths Challenge Problem (UKMT)

2 marks

6.1. How many hexagons are there in the diagram?

  • 3
  • 6
  • 9
  • 12

The twelve different hexagons are shown shaded in the diagram below.

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

  • The fantastic Sydney Padua has turned Ada into a comic book character and has written a graphic novel about her. This brief biographical sketch about Ada is available for free online.

  • Find out about Ada Lovelace Day. The next one is not for a while, but it is never too soon to starting planning how you are going to celebrate her life and achievements.