Parallelogram 26 Year 8 30 Apr 2020Snake Mathematics

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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.

This is the last Parallelogram before next week’s UKMT Junior Maths Challenge, so good luck if you are taking the test. Do the best you can and you might get a bronze, silver or gold certificate. And if you don’t get a certificate, then at least you had a go at taking the Junior Maths Challenge.

1. Vi Hart snake bonanza

This video shows the very amazing Vi Hart playing with flexible chain snakes. By the way, mathematicians love to just play with things, because mathematicians are bit odd and often interesting ideas emerge out of playing.

Watch the video and then have a go at the following questions.

1 mark

1.1. Which ‘slither’ is not valid?

  • RLRRLLRRL
  • RLRRLRRLL
  • RRRRRRRR
  • SRSSSRSSR
2 marks

1.2. If you start off with one snake, removing its head results in two new heads growing in its place. If you chop off all the existing heads, then two new heads will grow in each of their places. This carries on until you have a snake with 32 heads. How many heads did you have to cut off to get to this situation?

Correct Solution: 31

One approach is to draw a snake, then draw a line through it and draw two more heads near it. Then carry on until you get 32 heads and look back and you will see that you have crossed out 31 heads. Another way to think about it is to say: “Every time the snake loses 1 head, it actually ends up with 1 more head than it started with. So, to get from 1 to 32 heads, I will have to cut off 31 heads.”

2. Junior Maths Challenge Problem (UKMT)

3 marks

2.1. Peter has three times as many sisters as brothers. His sister Louise has twice as many sisters as brothers. How many children are there in the family?

  • 15
  • 13
  • 11
  • 9
  • 5

Suppose Peter has b brothers and hence 3b sisters. So, including Peter, there are b+1 boys and 3b girls in the family. So Louise has b+1 brothers and 3b1 sisters. Since Louise has twice as many sisters as brothers, 3b1=2b+1. This equation is equivalent to 3b1=2b+2. So b=3 and there are 4 boys and 9 girls in the family, making 13 children altogether.

3. Stephen Hawking (1942 – 2018)

When the cosmologist Stephen Hawking died, journalists discussed his incredible scientific achievements, and they also emphasised how his life had inspired people all over the world.

Here are three quotes from Stephen Hawking, that might inspire you.

“Try to make sense of what you see and wonder about what makes the universe exist. Be curious. And however difficult life may seem, there is always something you can do and succeed at. It matters that you don't just give up.”

**

“We are just an advanced breed of monkeys on a minor planet of a very average star. But we can understand the Universe. That makes us something very special.”

**

“The greatest enemy of knowledge is not ignorance, it is the illusion of knowledge.”

4. Homonyms

This is a relatively long video about words. It’s eleven minutes long and it is presented by the YouTube sensation Michael Stevens.

You might wonder, why have a video about words in a mathematics challenge? First, I think Stevens analyses words in very mathematical way, placing them within a Venn diagram. Second, if you are curious about mathematics, then I hope you are also curious about loads of other things, including words and language.

Watch, and have a go at the questions below.

2 marks

4.1. “Poecilonym” means the same as:

  • homonym
  • homophone
  • synonym
  • simile
  • metaphor
2 marks

4.2. Which two words are homographs?

  • record & record
  • blue & blew
  • buy & purchase
  • diarrhoea & diarroea
  • maths & mathematics

5. Junior Maths Challenge Problem (UKMT)

2 marks

5.1. On standard dice the total number of pips on each pair of opposite faces is 7. Two standard dice are placed in a stack, as shown, so that the total number of pips on the two touching faces is 5.

What is the total number of pips on the top and bottom faces of the stack?

  • 5
  • 6
  • 7
  • 8
  • 9

The total number of pips on the top and bottom faces of the two dice is 7 + 7 = 14. As there is a total number of 5 pips on the touching faces, there are 14 - 5 = 9 pips altogether on the top and bottom faces of the stack.

6. Pick a card

2 marks

6.1. This question was inspired by a video created by the psychologist Professor Richard Wiseman. I am holding two red cards and two black cards face down. You have to pick two of the cards. If you pick both of the black cards, then you win £4. By the way, it costs you £1 to play the game. Which of these statements is true?

  • This is a fair game, as you will win one time in four.
  • This game is great for you, as you will win money in the long run.
  • This game is a con, because you will lose money in the long run.

The game is a con, as explained in Professor Wiseman’s video 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

  • Here is a video entitled “7 amazing bets you will always win” by Professor Wiseman from his Quirkology YouTube channel.