Week 9Weird Shapes

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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. Square-wheeled stunt

Last week we look at how well square wheels work… as long as you have the right sort of surface. Here is a clip from the BBC series “Bang Goes the Theory”, in which Jem Stansfield attempts to become the first person ever to jump a forty-five degree ramp on a square wheeled motorbike. He is not going to jump over any buses (or even rhombuses), but it is still quite an achievement.

2. Shapes of Constant Width

If you ask a random person on the street to name a shape that rolls smoothly, then they will all say a circle, or perhaps a disc.

However, you now know that a square will roll smoothly, as long as the surface it rolls on has the correct shape.

But what if we demand that the surface is smooth and flat? Are we restricted to circles? The answer is “no”, as this video by science explorer Steve Mould explains.

2 marks

2.1 The 50p coin and the 20p coin are both 7-sided shapes of constant width? Why is that better than the coins being regular heptagons?

  • They roll better in vending machines.
  • They avoid corners, so they cause less pain to fingers.
  • They appear to be round, so look more like all the other coins.
  • They flip better and are less likely to land on their edges.
  • I really couldn’t think of a fourth wrong answer.
2 marks

2.2 The 2017 £1 coin has 12 sides. Because it has an even number of sides, it cannot be a shape of constant width... and you might want to have a think about why an even number of sides is incompatible with having a constant width.

However, to earn your two points, you just have to measure the difference between the narrowest and widest “diameters” of a £1 coin.

(You can measure one coin with a ruler and estimate, or you could try putting several of them end to end and measuring them when the points touch (wider) and when the edges touch (narrow). This should give you a more accurate answer than just measuring one coin.)

What is the difference between the narrowest and widest “diameters” of a £1 coin?

  • 0.4 mm
  • 0.8 mm
  • 1.2 mm
  • 1.6 mm
  • 2.0 mm

The new £1 coin has a variable width, between 23.03mm and 23.43mm.

1 mark

2.3 Which country (very appropriately) had a coin in the shape of a Reuleaux triangle?

  • Bahamas
  • Barbados
  • Barbuda
  • Bermuda

It is very appropriate that Bermuda has a triangular coin, because of the infamous Bermuda Triangle, where ships were said to disappear. Interestingly, there are lots of reasons to doubt the superstitious beliefs about the Bermuda Triangle, as ships and planes are no more likely to go missing in that area than in any other part of the ocean. If you’re interested, you can read more about rational explanations for the Bermuda Triangle here.

2 marks

2.4 Take a look at this clip showing a bicycle with unusually shaped wheels. In relation to the bike in the video, which of these statements is false?

  • Only one wheel is a Reuleaux triangle.
  • The axles move up and down relative to the ground.
  • The axles move up and down relative to the cyclist.
  • The ride is smooth because the frame rests on the wheels, not the axles.
  • The effect of the odd wheel shape is cancelled by the uneven road to give a smooth ride.

3. Junior Maths Challenge Problem

3 marks

3.1 Only one of the following statements is true. Which one?

  • A) ‘B is true’
  • B) ‘E is false’
  • C) ‘Statements A to E are true’
  • D) ‘Statements A to E are false’
  • E) ‘A is false’
Show Hint (–1 mark)
–1 mark

Statements C and D cannot be true, so that narrows down the options to A, B and E.

There are different ways to approach this question, but here is one way to tackle it.

For statement A to be true, B would also have to be true. But we are told that only one statement is true, so this is not possible. We deduce that statement A is false.

Therefore, statement E is true.

We could stop here, because we have found one statement that must be true. However, we really ought to satisfy ourselves that statement E is the only one that is true.

As statement E is true, statement B is false.

Statement D can never be true, because if statement D were true, all the statements, including D, would be false, and we would have a contradiction.

For similar reasons, statement C cannot be true.

So, statement E is true and all the other statements are false.

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 Parallelogams.

Cheerio, Simon.