Parallelogram 41 • Level 3 • 12 Jun 2025More Rules for Critical Thinking

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. Digital clock symmetry

2 marks

1.1 This time on a 24-hour clock has mirror symmetry:

Of the following options, how many minutes later will there also be a time with mirror symmetry?

404
414
424
434
444
(Not answered)

434 minutes is 7 hours and 14 minutes later, which 20.05, which has mirror symmetry.

2. Junior Maths Challenge Problem (UKMT)

3 marks

2.1 Billy has three times as many llamas as lambs. Milly has twice as many lambs as llamas.

They have 17 animals in total.

How many of the animals are llamas?

5
6
7
8
9
(Not answered)

Show Hint (–1 mark)

1 mark

Suppose that Billy has b lambs and hence 3b llamas. Then, in total, Billy has 4b animals.

Suppose that Milly has m llamas and hence 2m lambs. Then, in total, Milly has 3m animals.

Because the total number of animals is 17, we have 4b+3m=17.

Suppose that Billy has b lambs and hence 3b llamas. Then, in total, Billy has 4b animals.

Suppose that Milly has m llamas and hence 2m lambs. Then, in total, Milly has 3m animals.

Because the total number of animals is 17, we have 4b+3m=17.

We have two unknowns but just one equation. This equation has infinitely many different solutions. However, in this problem b and m are numbers of animals, and so each of them is a non-negative integer. This enables us to find their values.

Here b and m are non-negative integers. It follows that 4b≤17 and hence b≤4. So the only possible values for b are 0, 1, 2, 3 and 4.

It is easy to check that, of these values of b, only when b=2 is the corresponding value of m an integer.

When b=2, we have

4×2+3m=17,

that is,

8+3m=17,

and hence

3m=9,

from which it follows that

m=3.

Since b=2, Billy has 3×2=6 llamas. Since m=3, Milly has 3 llamas.

Therefore 6+3=9 of the animals are llamas.

3. More integer solutions

a and b are positive integers and 7a+5b=49. Find the values of a and b.

1 mark

3.1 What is a?

Correct Solution: 2

1 mark

3.2 What is b?

Correct Solution: 7

It is tempting to jump to the solution that a=7 and b=0, but 0 is not a positive solution.

Instead, you can remove 7s from 49 until you reach a number that is divisible by 5.

49 minus two 7s is 35.

So the answer is a=2 and b=7.

a and b are positive integers and 23a+17b=320. Find the values of a and b.

1 mark

3.3 What is a?

Correct Solution: 8

1 mark

3.4 What is b?

Correct Solution: 8

Simply try different values for a and check to see if there is a value for b that is a positive integer.

If a=1, then b=17.5ish, so that is not correct.

But if a=8, then b=8, and that is correct.

Alternatively, you might notice that 23+17=40, and that 320=40×8=23+17×8=23×8+17×8.

4. Bertrand Russell’s rules for critical thinking

As mentioned last week, Bertrand Russell (1872-1970) was a British philosopher and mathematician. He compiled 10 rules for critical thinking (which is the ability to think clearly and rationally, understanding the logical connection between ideas). In his words: “The Ten Commandments that, as a teacher, I should wish to promulgate (promote).”

The first five rules appeared last week… and here are the remaining five rules. Take a look at each rule, and think about why each one is important.

6. Do not use power to suppress opinions you think pernicious, for if you do the opinions will suppress you.
7. Do not fear to be eccentric in opinion, for every opinion now accepted was once eccentric.
8. Find more pleasure in intelligent dissent than in passive agreement, for, if you value intelligence as you should, the former implies a deeper agreement than the latter.
9. Be scrupulously truthful, even if the truth is inconvenient, for it is more inconvenient when you try to conceal it.
10. Do not feel envious of the happiness of those who live in a fool’s paradise, for only a fool will think that it is happiness.

5. More integer solutions

Take a look at this amazing video from the Mind Your Decisions YouTube channel.

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

3 marks

3.1 What is the explanation for the missing square?

Magic
Mathematics
Witchcraft
Voodoo
Brexit
(Not answered)

There is a full explanation of the problem at the Mind Your Decisions blog. Alternatively, this short video also gives a good explanation for the paradox.

In case you don’t know what a paradox is, it is a statement that appears to rely on solid reasoning and logic, but which results in nonsense or a contradiction.

(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.