Parallelogram 45 • Level 4 • 10 Jul 2025Catching Kendama

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.

2 marks

1.1 In 2014, Australian Suzy Walsham won the annual women’s race up the 1576 steps of the Empire State Building in New York for a record fifth time. Her winning time was 11 minutes 57 seconds.

Approximately how many steps did she climb per minute?

13
20
80
100
130
(Not answered)

We see that 11 minutes 53 seconds is just under 12 minutes, and 1576 is just under 1600. So the number of steps per minute is approximately

160012=4003∼133.

We conclude that, of the given options, 130 is the best approximation.

2.

3 marks

2.1. The equilateral triangle and regular hexagon shown have perimeters of the same length.

What is the ratio of the area of the triangle to the area of the hexagon?

5:6
4:5
3:4
2:3
1:1
(Not answered)

We can assume that we have chosen units so that each side of the regular hexagon has length 1. It follows that the length of the perimeter of this hexagon is 6. Therefore the length of the perimeter of the equilateral triangle is 6. Hence each of its sides has length 2.

It follows that we can divide the equilateral triangle into 4 smaller equilateral triangles each with side length 1 as shown. The hexagon may be divided into 6 equilateral triangles with side length 1 as shown.

It follows that the ratio of their areas is 4:6, that is, 2:3.

3.

3 marks

3.1. What is the remainder when 22×33×55×77 is divided by 8?

2
3
4
5
7
(Not answered)

Show Hint (–1 mark)

–1 mark

Notice that 33×55×77 is a positive odd integer greater than 1 and so is equal to 2n+1, for some positive integer n.

First, we note that 33×55×77 is a positive odd integer greater than 1 and so is equal to 2n+1, for some positive integer n.

It follows that

22×33×55×77=4×33×55×77
=42n+1
=8n+4

Therefore when 22×33×55×77 is divided by 8, the quotient is n and the remainder is 4.

4.

4 marks

4.1 You are asked to choose two positive integers, m and n with m>n, so that as many as possible of the expressions m+n, m−n, m×n and m÷n have values that are prime.

When you do this correctly, how many of these four expressions have values that are prime?

0
1
2
3
4
(Not answered)

Show Hint (–1 mark)

–1 mark

The answer is not equal to 4.

With m=2 and n=1, we have m+n=3, m−n=1, m×n=2 and m÷n=2. So it is possible for three of the expressions to have values that are prime. [Note that 1 is not a prime, so in this case m−n is not prime].

We now show that this is the best that can be achieved by showing that it is not possible for all four of the expressions to have values that are primes.

Since m and n are positive integers with m>n, we have m>1. Therefore m×n is prime if, and only if, m is prime and n=1. Then m+n=m+1 and m−n=m−1. Therefore, if all four expressions are primes, m−1, m and m+1 would be three consecutive primes. However a trio of consecutive positive integers that are all primes does not exist. We conclude that not all four of the expressions can have values that are primes.

5. Catching Kendama

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

3 marks

5.1 In order to make one of the toughest Kendama catches, Tadashi has to take advantage of a principle from physics known as…

The conservation of energy
The conservation of angular momentum
The conservation of mass
The conservation of atoms
The conservation of curiosity
(Not answered)

Finally, it’s up to you, but if you haven't already you might enjoy watching the rest of Tadashi’s video about paperclips

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

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.