Parallelogram 23 Level 3 8 Feb 2024Countdown and levitating water

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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.
  • 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.
  • As it is half-term, you have a bit of extra time to complete this Parallelogram. You should finish it by Sunday 21 February. And the next Parallelogram will appear on Thursday 25 February.

IMPORTANT – it does not really matter what score you get, because the main thing is that you think hard about the problems... and then examine the solution sheet to learn from your mistakes.

1. Crazy Coundown numbers

I am guessing that most people have played Countdown, and if you like maths then you have probably played along with the numbers game – you pick six numbers, the show generates a random number less than one thousand, and finally you have to create the big number from the smaller six numbers using the four basic operations of arithmetic.

Sometimes it’s easy, sometimes it’s hard, but rarely are the calculations as complex as the numbers game in the clip below. It is taken from the Australian version of Countdown, known as ‘Letters and Numbers’. What I particularly like about this clip is that there is a complex way and a simpler way to get the answer.

If you want to tackle the puzzle before you watch the clip, then this is the puzzle: make 821 from 25, 100, 75, 50, 6, 4.

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

2. Junior Maths Challenge Problem (UKMT)

Part 1 of this question is really sneaky. I got it wrong when I tried, then kicked myself afterwards.

3 marks

2.1. A list is made of every digit that is the units digit of at least one prime number. How many of the following numbers appear in the list?

  • 1
  • 2
  • 3
  • 4
  • 5
  • (Not answered)

The numbers 2, 3 and 5 are themselves prime numbers so they occur in this list. Also 1 is the units digit of the prime number 11 (and also 31, 41 and many more). However, a number with units digit 4 is an even number greater than 2 and so is not prime. Thus, there are 4 numbers in the list, namely 1, 2, 3 and 5 which are the units digits of at least one prime number. Therefore the answer is 4.

2 marks

2.2. How many of the other digits, 0, 6, 7, 8 and 9 are the units digits of at least one prime number?

  • 1
  • 2
  • 3
  • 4
  • 5
  • (Not answered)

The answe is 2, because primes can end in 7 and 9 (e.g., 7, 17, 19, 29, 37…), but not 0, 6 and 8. If the final digit is 0, 6 or 8, then the number is even and cannot be prime.

3. The Leidenfrost effect – levitating water

This video is by Alex Nickel from his Technicality channel (which is full of fascinating videos). It is quite long (9 minutes), and takes you through an explanation of a strange phenomenon known as the Leidenfrost effect. Can you understand the science? Can you follow his explanation? Does it stick in your brain afterward?

There is a question following the video, which you might want to look at first, so that you can look out for the answer during the video.

NOTE: If you are going to experiment with the Leidenfrost effect, check with your parents/teachers and be safe.

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

2 marks

3.1. Which planet could be powered by the Leidenfrost effect?

  • Mercury
  • Venus
  • Earth
  • Mars
  • 51 Pegasi b
  • (Not answered)

4. Junior Maths Challenge Problem (UKMT)

In case you have forgotten, or are new to Parallelograms, these Junior Maths Challenge questions come from an annual maths challenge that you might already have been part of or which you might be taking part in this year. The questions are sometimes tough, but that is the definition of a challenge. In fact, this one is so tough that I have added a hint.

3 marks

4.1. One cube has each of its faces covered by one face of an identical cube, making a solid as shown. The volume of the solid is 875cm3.

What, in cm2, is the surface area of the solid?

  • 750
  • 800
  • 875
  • 900
  • 1050
  • (Not answered)
Show Hint (–1 mark)
–1 mark

If the object is one cube surrounded by cubes then we have (1 + 6 = 7 cubes), so you can now work out the volume of each cube... and therefore the length of any side of any cube. When it comes to looking at the surface area, we have no surface area from the cube in the middle, and 5 faces from each of the other 6 cubes. So, just be careful when you work this out. Take it step by step, lay our your calculation clearly and check your answer before you submit it.

The solid is made up of 7 cubes. So each of them has volume 8757 = 125 = 53cm3. Hence the side length of each of the cubes is 5cm. Each of the 6 faces of the central cube is covered by one face of one of the other 6 cubes, each of which has 5 faces showing. So the surface of the figure is made up of 6 x 5 = 30 squares, each with side length 5 cm and hence area 25 cm2. Hence the total area of the figure is 30 x 25 = 750cm2.

5. The Indiana π Bill

Here is one of the oddest stories in the history of π – the day when politicians tried to change the value of π. Read this article from “Today I Found Out” and answer the question below.

1 mark

5.1. Who coached the politicians in the Senate so that they did not make a silly decision about π?

  • Professor Indiana
  • Edward Goodwin
  • Professor Waldo
  • The Temperance Committee
  • Senator Ellison
  • (Not answered)

6. Why don't country flags use the colour purple?

This is a short, curious and odd video about purple. No maths, just chemistry.

(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 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.
  • Finally, if you missed any earlier Parallelograms, don't forget to go back and complete them.
  • You earn reward points and badges from completing missed Parallelogams.

Cheerio, Simon.

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