Parallelogram 40 • Level 3 • 6 Jun 2024Rules 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 Clocks

3 marks

1.1 In a 24-hour period, how often does the number 5 appear in a digital clock?

Let me define the question in a bit more detail. Count how many 5's would be on the clock if you looked at it every minute. So, for example, at 05:53 there are two, then at 05:54 there are two, and at 05:55 there are three and so on. By the way, it is a 24-hour clock, so you will also have 15:55, which has three 5’s.

Correct Solution: 504

Show Hint (–0.5 mark)

–0.5 mark

The answer is between 500 and 600.

This question appeared on the NRICH website, which has lots of great maths resources.

In one hour, the 5 appears in the minutes 16 times.

In 24 hours, the 5 appears in the minutes 24×16=384 times.

The 5 appears 60 more times when the hour is 5.

The 5 appears 60 more times when the hour is 15.

In 24 hours, the 5 appears in the hours 2×60=120 times.

Therefore, the total is 384+120=504 times.

2. Junior Maths Challenge Problem (UKMT)

4 marks

2.1 The diagram shows a quadrilateral PQRS in which PQ and QR have the same length.

Also PR bisects ∠SPQ, the ratio of ∠SPR to ∠PRS is 2:3 and ∠PSR=110°.

How large is angle PQR?

124°
120°
110°
90°
28°
(Not answered)

Show Hint (–0.5 mark)

–0.5 mark

Because the angles of a triangle have sum 180° and ∠PSR=110°, it follows that ∠SPR+∠PRS=180°−110°=70°.

Show Hint (–0.5 mark)

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The ratio of ∠SPR to ∠PRS is 2:3. It follows that ∠SPR=25×70°=28°.

Because the angles of a triangle have sum 180° and ∠PSR=110°, it follows that ∠SPR+∠PRS=180°−110°=70°.

The ratio of ∠SPR to ∠PRS is 2:3. It follows that ∠SPR=25×70°=28°.

Since PR bisects ∠SPQ, we deduce that ∠RPQ=∠SPR=28°.

Since PQ=QR, the triangle PQR is isosceles, and therefore ∠PRQ=∠RPQ=28°.

Thus the triangle PQR has two angles which are each 28°. Therefore, because sum of the angles of a triangle is 180°, we deduce that:

∠PQR=180°−28°−28°=124°.

3. Richard Feynman explains science

Richard Feynman is a hero of science and a Nobel Prize winner in physics. In this video, he explains how being a scientist is like watching a game of chess when you don’t understand the rules.

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

3 marks

3.1 In science, what does Feynman say is the most interesting moment?

When you discover something that you expected.
When you measure something and the answer is what you expected.
When a chemical reaction behaves as expected.
When the mathematical equations correctly predicts a result in the laboratory.
When something unexpected happens, when something does not fit.
(Not answered)

4. Junior Maths Challenge Problem (UKMT)

4 marks

4.1 In the triangles PQR and STU, ∠RPQ=2×∠UST, ∠PRQ=2×∠SUT and ∠RQP=15×∠UTS.

How large is ∠UTS?

90°
100°
120°
150°
more information needed
(Not answered)

Show Hint (–0.5 mark)

–0.5 mark

Let the angles of the triangle STU be s°, t° and u° as shown in the figure.

Show Hint (–0.5 mark)

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The angles of a triangle have sum 180°. Therefore

s+t+u=180 (1) and
2s+15t+2u=180. (2)

Show Hint (–0.5 mark)

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By multiplying equation (1) by 2 we obtain

2s+2t+2u=360. (3)

Then, by subtracting equation (2) from equation (3), we deduce that

2−15t=180

Let the angles of the triangle STU be s°, t° and u° as shown in the figure.

It follows that the angles of the triangle PQR are 2s°, 15t° and 2u°, as shown.

The angles of a triangle have sum 180°. Therefore

s+t+u=180 (1) and
2s+15t+2u=180. (2)

By multiplying equation (1) by 2 we obtain

2s+2t+2u=360. (3)

Then, by subtracting equation (2) from equation (3), we deduce that

2−15t=180,

that is,

95t=180.

Therefore

t=59×180=100.

It follows that ∠UTS=100.

5. Bertrand Russell’s rules for critical thinking

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).”

I want you to think about each one in detail, so here are the first five … the next five will appear next week.

1. Do not feel absolutely certain of anything.
2. Do not think it worth while to proceed by concealing evidence, for the evidence is sure to come to light.
3. Never try to discourage thinking, for you are sure to succeed.
4. When you meet with opposition, even if it should be from your husband or your children, endeavor to overcome it by argument and not by authority, for a victory dependent upon authority is unreal and illusory.
5. Have no respect for the authority of others, for there are always contrary authorities to be found.

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.