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Parallelogram 21 • Level 3 • 25 Jan 2024Dripping Taps

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Noun: Parallelogram Pronunciation: /ˌparəˈlɛləɡram/

  1. 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. Are you as smart as an undergraduate?

Here is a mathematics question that appeared on the highbrow BBC quiz “University Challenge”.

2 marks

1.1. A tap leaks 1 ml of water every second. How many 10 L buckets would it fill in a day?

I don’t want to pressure you, but after one student got the wrong answer on the TV show, the other student answered it in 20 seconds without pencil or paper.

Correct Solution: 8

The number of seconds in a day is 60 × 60 × 24 = 86,400 seconds.

This means the tap drips 86,400 ml, or 86.4 litres, or enough to fill 8 buckets of 10 litres each.

The answer is 8 buckets.

2. An eclipse seen from space

I have been lucky enough to see five total solar eclipses: in America, the Faroes, Zambia, Libya and... Devon. But I have never seen one from space.

I spotted this image on a NASA website called Astronomy Picture of the Day. There is a different Astronomy Picture of the Day each day, so visit regularly if you are interested in the universe.

According to the NASA website:

“Here is what the Earth looks like during a solar eclipse. The shadow of the Moon can be seen darkening part of Earth. This shadow moved across the Earth at nearly 2,000 km/h. Only observers near the centre of the dark circle see a total solar eclipse - others see a partial eclipse where only part of the Sun appears blocked by the Moon. This spectacular picture of the 1999 August 11 solar eclipse was one of the last ever taken from the Mir space station. The two bright spots that appear on the upper left are thought to be Jupiter and Saturn. Mir was deorbited in a controlled re-entry in 2001.”

3. Junior Maths Challenge Problem (UKMT)

5 marks

3.1. The diagram shows an equilateral triangle, drawn inside of a rectangle. Angles x and y have been labelled.

What is the value of x+y?

  • 30
  • 45
  • 60
  • 75
  • 90
  • (Not answered)
Show Hint (–1 mark)
–1 mark

Notice that x and y are two of the internal angles of a pentagon.

We have highlighted that pentagon in this graphic.

Consider how this information can help you solve the problem.

Show Hint (–1 mark)
–1 mark

What do the internal angles of pentagon add up to? The formula for calculating the sum of the interior angles of a regular polygon is: n−2×180° where n is the number of sides of the polygon.

Show Hint (–1 mark)
–1 mark

Remember that the triangle is equilateral. What is the value of the biggest angle in the pentagon, above angle x and below angle y?

Show Hint (–1 mark)
–1 mark

You know that two of the angles in the pentagon are right angles.

This is tough question. Well done if you did it without hints. There are a couple of ways to find the value of x+y degrees, but here is the neatest method.

The marked angles are two angles of the pentagon PRSTQ.

The angles Q and T are both 90°. The interior angles of the equilateral triangle are all 60°. Hence the angle in the pentagon at R is 300°. The sum of the angles in a pentagon is 540°.

So x+y+300+90+90=540.

Therefore x+y=60°.

Alternatively, if you imagine a line drawn parallel to the top of and bottom of the rectangle, which intersects the triangle at point R, you can use alternate angles to find that angle PRS=x+y, and as it is an equilateral triangle, this must be 60°.

4. Junior Maths Challenge Problem (UKMT)

2 marks

4.1 What is the mean of 23 and 49?

  • 12
  • 29
  • 79
  • 34
  • 59
  • (Not answered)

The mean of two numbers, a and b, is the average: 12(a+b).

Hence the mean of 23 and 49 is:

5. Calculating Pi with Real Pies

A couple of weeks ago, we featured Matt Parker in a video talking about measuring π using a pendulum. To end this Parallelogram, here is a another π-culiar video by Matt, in which he measures pi using pies. Watch and enjoy.

(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.

Additional Stuff

  • Matt’s video is from a channel called Numberphile, which is full of fantastic mathematical stuff. Some videos are challenging, others are more straightforward, and a few are about the history or culture of mathematics. For example, this video asks who is right? Brits who says MATHS or Americans who say MATH? (If you have problems watching the video, right click to open it in a new window)

  • And this video, by Matt Parker, shows you how mathematicians tie their shoelaces. I don’t think it is part of the GCSE curriculum yet, but you will need to know about the maths of tying shoe laces for A Level, so it is worth learning the technique now.