Parallelogram 41 • Level 2 • 12 Jun 2025Aluminium Can Engineering
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. The smallest number...
1.1 What is the smallest number that is divisible by all the numbers from 1 to 6?
Correct Solution: 60
It is tempting to just multiply all the numbers, so 1 x 2 x 3 x 4 x 5 x 6 = 720, but this is not the smallest number that is divisible by the numbers 1 to 6.
If a number is divisible by 6, then it is also divisible by 1, 2 & 3, so the original problem is the same as asking: “What is the smallest number that is divisible by 4, 5 and 6?”
You could look at the factors of these numbers, or you could notice that every multiple of 12 is a multiple of 4 and 6, so the solution is 12 x 5 = 60.
2. Junior Maths Challenge Problem (UKMT)
2.1 My train left Southampton at 06:15 and arrived in Birmingham at 08:48 later that morning.
How many minutes did the journey take?
- 153
- 193
- 233
- 1463
- 1501
- (Not answered)
It is 2 hours from 06:15 to 08:15. Because 48 − 15 = 33, there are 33 minutes from 08:15 to 08:48. Hence, in total, the journey took 2 hours and 33 minutes.
There are 120 minutes in two hours. Therefore the journey took 120 + 33 minutes, that is, 153 minutes
2.2 Later the same day I left Birmingham by train at 19:04. My train arrived back at Southampton at 21:40. How many minutes did my return journey take?
Correct Solution: 156
3. Aluminium can engineering
I am a big fan of Bill Hammack (also known as “Engineer Guy”), whose YouTube channel looks at the clever engineering behind lots of machines and objects. In this video, he looks at the engineering behind an ordinary fizzy drinks can. This is only the opening of the video, but if you are really interested in engineering (or fizzy drinks) then make sure you watch the whole video later.
(If you have problems watching the video, right click to open it in a new window)3.1 Roughly how many aluminium cans are made each second?
- 15
- 150
- 1,500
- 15,000
- 150,000
- (Not answered)
3.2 A spherical tin can would be great because:
- It easily rolls around
- It looks the same from every angle
- It uses the least amount of aluminium for a given volume
- It would bounce back if you dropped it.
- (Not answered)
3.3 If you packed spherical cans, how much would be “void space”, unused space between cans?
- 0%
- 13%
- 26%
- 30%
- (Not answered)
3.4 A cylindrical can looks like...
- a sphere from the top and a cuboid from the side.
- a sphere from the top and the side.
- a cuboid from the top and the side.
- a cuboid from the top and a sphere from the side.
- (Not answered)
Try to watch the rest of this video about aluminium cans:
(If you have problems watching the video, right click to open it in a new window)4. Junior Maths Challenge Problem (UKMT)
4.1 The diagram shows a shape made from four 3 cm × 3 cm × 3 cm wooden cubes joined by their edges.
What, in cm2, is the surface area of the shape?
- 162
- 180
- 198
- 216
- 234
- (Not answered)
The surface of the shape is made up of the full surface of each of the four cubes that make it up.
Each face of the 3 cm × 3 cm × 3 cm cubes is a square with side length 3 cm. Hence the area of each of these faces is 3 cm × 3 cm = 9 cm2.
Each cube has 6 of these faces. Hence the surface area of each cube is 6 × 9 cm2 = 54 cm2.
Therefore the total surface area of the four cubes is 4 × 54 cm2 = 216 cm2.
Hence the surface area of the shape is 216 cm2.
4.2 The shape shown uses six 1 cm × 1 cm × 1 cm wooden cubes.
It has been made by removing two such cubes from the top layer of a 2 cm × 2 cm × 2 cm cube.
What, in cm2, is the surface area of the shape?
Correct Solution: 24
Work out the surface area of 6 separate cubes, then remove the area of the hidden surfaces.
Two cubes in the base layer have 3 hidden faces each.
Two cubes in the base layer have 2 hidden faces each.
Two cubes in the top layer have 1 hidden face each.
The 6 separate cubes have a total of 36 faces.
Two cubes in the base layer have 3 hidden faces each. Two cubes in the base layer have 2 hidden faces each. Two cubes in the top layer have 1 hidden face each. So, in total, there are 12 hidden faces.
24 faces are exposed, each with an area of 1 cm2, so the total surface area is 24 cm2.
5. An elegant proof
One of the most important aspects of mathematics is the PROOF. A proof is an argument or set of logical steps that demonstrates that a certain statement is correct.
The statement is not probably correct or almost certainly correct, but absolutely correct, because the proof says so.
Below is a screengrab of a tweet from @FermatsLibrary and it offers a proof of the statement that every odd integer (or every odd positive number) is the difference of two squares.
5.1 The odd number 11 is the difference of two square numbers. What is the smaller of these two square numbers?
Correct Solution: 25
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.