Parallelogram 26 Year 10 30 Apr 2020Ignobel Prize

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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.
  • When you finish, remember to hit the SUBMIT button.
  • Finish by Sunday night if your whole class is doing parallelograms.

IMPORTANT – it does not really matter what score you get, because the main thing is that you think hard about the problems... and then examine the solution sheet to learn from your mistakes.

1. Ignobel Prize

The Ignobel Prizes are not offered for the greatest discoveries in science, but instead they are awarded to ideas that first make you laugh, and then make you think. Take a look at this video and answer the question below.

(If you have any problems watching the video then just right click and open it in a new window)

2 marks

1.1 In the past, when doctors performed therapeutic colonoscopies, there was a risk of explosion. What gas were the doctors introducing into the colon (the largest part of the large intestine) that was helping cause the explosion?

  • Hydrogen
  • Helium
  • Nitrogen
  • Methane
  • Oxygen

2. Intermediate Maths Challenge Problem (UKMT)

3 marks

2.1 What is the difference between 10% of one million and 10% of one thousand?

  • 9,900
  • 9,990
  • 90,900
  • 99,900
  • 999,900

10% of one million is 100,000; 10% of one thousand is 100; 100,000 - 100 = 99,900.

3. Intermediate Maths Challenge Problem (UKMT)

4 marks

3.1 One gallon of honey provides enough fuel for a bee to fly about seven million miles.

Roughly how many bees could fly one thousand miles if they had ten gallons of honey to share between them?

  • 7,000
  • 70,000
  • 700,000
  • 7,000,000
  • 70,000,000

Ten gallons of honey would provide enough fuel for one bee to fly about 70,000,000 miles.

So the number of bees which could fly 1000 miles is approximately 70,000,000 ÷ 1000, that is 70,000.

4. Line Rider

Line Rider is a fantastic website that allows you to build a sled track for a cartoon penguin called Bosh. It lets you explore the physics of Bosh’s universe, and take him on an adventure.

There's no question to go along with this, just have fun with it!

5. Intermediate Maths Challenge Problem (UKMT)

5 marks

5.1 What is the area (in square units) of the triangle formed by the three lines whose equations are: yx=6, x2y=3, x+y=6?

  • 55
  • 60
  • 65
  • 70
  • 75
Show Hint (–1 mark)
–1 mark

The equations of the three lines must be considered in pairs to find the coordinates of their points of intersection, i.e. the coordinates of the vertices of the triangle.

Show Hint (–1 mark)
–1 mark

If we take the first two lines, yx=6, x2y=3 then we can solve these simultaneous equations to show that they cross at (-15, -9).

You can find out where the other two pairs of lines cross to find the other corners of the triangle.

Show Hint (–1 mark)
–1 mark

The other two pairs of lines cross at (0, 6) and (5, 1).

Show Hint (–1 mark)
–1 mark

The area of the triangle may now be found by enclosing it in a rectangle measuring (20 × 15) and subtracting the areas of the three surrounding triangles.

The equations of the three lines must be considered in pairs to find the coordinates of their points of intersection, i.e. the coordinates of the vertices of the triangle.

It is left to the reader to show that these are (−15, −9), (0, 6) and (5, 1).

The area of the triangle may now be found by enclosing it in a rectangle measuring 20 × 15 and subtracting the areas of the three surrounding triangles from that of the rectangle.

This gives 300 - (112 12 + 12 12 + 100) = 75.

6. Senior Maths Team Challenge Problem (UKMT)

5 marks

6.1 This is a problem from the national final of the 2018-19 UKMT senior maths team challenge, so prepare yourself for some serious thinking.

Alex, Beatrice, Charmian and Denzil are four children. Each of them either always tells the truth or always tells lies.

One day they make the following statements:

  • Alex: ‘Beatrice always tells lies.’
  • Beatrice: ‘Alex and Denzil are either both truth tellers or both liars.’
  • Charmian: ‘Beatrice and Denzil are either both liars or both truth tellers.’
  • Denzil: ‘I always tell the truth.’

Who is definitely lying?

Correct Solution: Denzil

Show Hint (–2 mark)
–2 mark

Start by assuming that Alex is telling the truth and then try to fill in the table below. Then assume that Alex is lying and fill in another copy of the table. Is anyone lying in both versions of the table?

Child Truth or Lie How do we know?
Alex Truth Starting assumption
Beatrice
Charmian
Denzil

Let’s assume that Alex is telling the truth and then fill in the table below in the order indicated by the number in brackets.

Child Truth or Lie How do we know?
Alex Truth (1) Starting assumption
Beatrice Lie (2) Because Alex says so
Charmian Truth (4) Because we now know B & D are liars
Denzil Lie (3) Because B says A & D are the same, but we know she’s a liar

Let’s assume that Alex is telling a lie and then fill in the table below in the order indicated by the number in brackets.

Child Truth or Lie How do we know?
Alex Lie (1) Starting assumption
Beatrice Truth (2) Because Alex says the opposite and he lies
Charmian Lie (4) Because we now know B tells the truth & D lies
Denzil Lie (3) Because B says A & D are the same, and we now know she’s honest

Denzil is the only person who is a liar in both scenarios, so he is the liar.

(As a double check, it is worth looking at what happens if Denzil tells the truth. From there we can see what happens if Beatrice tells the truth... but you will find out that there is a contradiction because Beatrice also ends up being a liar. OR, if you see what happens if Beatrice is a liar, then it turns out there is another contradiction, because Beatrice also ends up telling the truth.)

There will be more next week, so check your email or return to the website on Thursday at 3pm.

In the meantime, you can find out your score, the answers and go through the answer sheet as soon as you hit the SUBMIT button below.

When you see your % score, this will also be your reward score. As you collect more and more points, you will collect more and more badges. Find out more by visiting the Rewards Page after you hit the SUBMIT button.

It is really important that you go through the solution sheet. Seriously important. What you got right is much less important than what you got wrong, because where you went wrong provides you with an opportunity to learn something new.

Cheerio, Simon.