Parallelogram 45 • Level 1 • 10 Jul 2025The Coastline Paradox
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 Coastline Paradox
How long is the coastline of Australia (or any country)?
Seems simple enough - get your ruler our and set about taking your measurements.
But things turn out to be a bit more complicated than that, due to the jagged nature of these shapes.
Watch this video to learn all about the ‘coastline paradox’
(If you have problems watching the video, right click to open it in a new window)1.1 If you measure the coastline of Australia in lengths of 500km, then what measurement do you get?
- 1,000 km
- 5,000 km
- 12,500 km
- 25,000 km
- (Not answered)
1.2 Why are there different measures for the coastline?
- People made errors in their measurement
- Different-sized measuring instruments give different measures for the coastline
- The coastline is constantly changing in size
- The coastline of Australia is hard to track on a map
- (Not answered)
1.3 What happens to the measure of the coastline as the measuring instrument gets really small?
- It settles on a value of around 5,000km
- It settles on a value of around 25,000km
- It gets smaller and smaller, with the measure heading towards zero
- It gets larger and larger, with the measure heading towards infinity
- (Not answered)
1.4 The Koch snowflake shares similar properties to these weird coastlines.
How is it generated?
- Take a triangle and keep dividing it into thirds
- Take a triangle and keep adding smaller triangles in the middle portion of each edge
- Take a triangle and join the three midpoints of its edges together, then keep repeating this for all new triangles
- Take a triangle and draw a zigzag through it
- (Not answered)
Here is the shape after the first iteration
Here is the shape after several iterations
2. Tricky problems
2.1 In the diagram, the large equilateral triangle is divided into a number of smaller equilateral triangles.
What percentage of the larger triangle is shaded?
- 60%
- 65%
- 70%
- 75%
- 80%
- (Not answered)
Count the small triangles to work out how many are shaded and unshaded.
There are 12 shaded triangles and 4 unshaded triangles, so the percentage shaded is 75% (ie
2.2 Which one of the five numbers below is not a multiple of 7?
- 2345
- 2352
- 2359
- 2366
- 2374
- (Not answered)
Four of the options are separated by gaps of 7
Since 2345 and 2352 have a difference of 7, they are either both multiples of 7 or neither of them are. Likewise for 2352 and 2359, and 2359 and 2366.
So either all four are multiples of 7, or none of them are.
Since only one of the options is not a multiple of 7, it must be the case that only 2374 is not, and the other four are.
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...
- 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.
- 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.
- This was our last Parallelogram of the year, but be sure to come back in September for more puzzles and problems.
Cheerio, Simon.