Parallelogram 10 • Level 3 • 9 Nov 2023Weird Shapes

Noun: Parallelogram Pronunciation: /ˌparəˈlɛləɡram/

a portmanteau 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.

(If you have problems watching the video, right click to open it in a new window)

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.

(If you have problems watching the video, right click to open it in a new window)

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.
(Not answered)

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
(Not answered)

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
(Not answered)

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.
(Not answered)

(If you have problems watching the video, right click to open it in a new window)

3. Junior Maths Challenge Problem (UKMT)

3 marks

3.1 The kettle in Keith’s kitchen is 80% full. After 20% of the water in it has been poured out, there are 1152 ml of water left. What volume of water does Keith’s kitchen kettle hold when it is full?

1400 ml
1600 ml
1700 ml
1800 ml
2000 ml
(Not answered)

Show Hint (–1 mark)

–1 mark

20% of the water is the same as saying 20% of 80% of the full kettle. What is 20% of 80%.

20% of the 80% is 16% of the kettle's capacity. Therefore the volume of water left in the kettle after Keith has poured out 20% of the original amount is 64% of the kettle's capacity. So when full, the kettle holds 115264×100 ml, that is 1800 ml.

4. Powers of 4

2 marks

4.1 Calculate 43?

Correct Solution: 64

2 marks

4.2 Calculate −43?

Correct Solution: -64

2 marks

4.3 Calculate 0.43?

Correct Solution: 0.064

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.