Parallelogram 1 Year 10 4 Sep 2019All Dark Things

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Noun: Parallelogram Pronunciation: /ˌparəˈlɛləɡram/

  1. a portmanteaux word combining parallel and telegram. A message sent each week by the Parallel Project to bright young mathematicians.

Welcome to the first of our Parallelograms designed for Year 10 students, a collection of mathematical challenges designed to stretch your brain and make your neurons more squiggly.

These challenges are a random walk through the mysteries of mathematics, most of which will be nothing to do with what you are studying at the moment in your classroom. Be prepared to encounter all sorts of weird ideas, including a few questions that appear to have nothing to do with mathematics at all.

  • Tackle each Parallelogram in one go. Don’t get distracted.
  • When you finish, remember to hit the SUBMIT button.
  • Finish by Sunday night if your whole class is doing parallelograms.

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. The Rhind Papyrus

The Ancient Egyptian Rhind Papyrus dates back to around 1550 BC and is full of mathematical ideas and puzzles. Not surprisingly, it has a section about calculating the slopes of pyramids.

I particularly like that it opens with the statement: “Directions for Attaining the Knowledge of All Dark Things”. That’s quite a way to describe mathematics.

2 marks

1.1. The papyrus has a great deal about “Egyptian fractions”, which means that every fraction has to be described in terms of other fractions which have the numerator 1.

Or, instead of 215, you would write 110+130.

So, instead of 710, you would write 23+130.

If 23 is represented in Egyptian fractions, it could be written as 13+14+1x. What is the missing denominator x?

Correct Solution: 12

23=13+14+1x

Finding the lowest common denominator, this is:

812=412+312+1x

x=12.

3 marks

1.2. If 2101 is represented in Egyptian fractions, it equals 1101+1202+1303+1x.

What is the missing denominator x?

Correct Solution: 606

2101=1101+1202+1303+1x

Now subtracting 1101 from each side:

1101=1202+1303+1x

Using the lowest common denominator:

6606=3606+2606+1x

x=606.

3 marks

1.3. The Rhind Papyrus also has a riddle about 7 houses, each containing 7 cats, who each eat 7 mice, who each ate 7 stalks of barley, which each contained 7 grains of barley. The question is how many items do we have in total? To get you started, we have 7 houses and 49 cats, so we have 56 items in total.

(BTW, this is similar to the much more modern St Ives riddle. If you have never heard about that puzzle, then google it.)

Correct Solution: 19,607

The total number of items is 7+72+73+74+75=19,607 items.

2. A Beautiful Mind

“A Beautiful Mind”, an Oscar-winning film from 2001, tells the story of the real-life mathematical genius John Nash. After making an incredible discovery that would have a major impact on the world of economics, Nash began to suffer from mental illness, and the film is about his journey towards recovery and being awarded the Nobel Prize for Economics.

This journey is reflected in the clip below. His renewed interest in teaching young mathematicians is a sign that Nash is recovering from his period of illness.

4 marks

2.1 At the end of the clip, Nash mentions a problem about two bicycles squashing a fly, which is a famous problem in mathematics. Here is one version of that question:

“Two cyclists, 120 Km apart, approach each other, each pedalling at 20 Km per hour. A fly starts at one cyclist, flies towards the other one, then bounces off it and thus flies back and forth between the two cyclists at 25 Km per hour. When the cyclists meet head on and squash the fly between them, how far has the fly flown while going back and forth?”

CLUE: At first sight, this looks like a very complicated question, because it is very hard to work out where the fly meets the second bicycle, and where it then meets the original bicycle, and so on. However, there is a simple way to solve this problem. Think hard, but if you are stuck then try the hints below.

Correct Solution: 75 km

Show Hint (–1 mark)
–1 mark

Did you really think hard before asking for this hint?

Show Hint (–1 mark)
–1 mark

Let’s ignore the fly for a moment – how long will it take for the bikes to reach each other? Once you know this time, then you can easily work out how far the fly will have travelled during this time..

Let’s ignore the fly for a moment – how long will it take for the bikes to reach each other? They will meet in the middle, so it will take them 3 hours to cover the 60 Km (120/2 Km) at 20 Km/h.

We don’t need to worry about the details of the fly’s bouncing back and forth, but travelling at 25 Km for 3 hours, we know that it will cover 75 Km before it is squashed.

3. Intermediate Maths Challenge Problem (UKMT)

Every Parallelogram will contain two or more UKMT Intermediate Maths Challenge problems. This first one is fairly simple, but others will be much trickier... and for all of them, remember, NO CALCULATORS!

3 marks

3.1 Without using a calculator, what is the value of 6102 − 2016?

  • 3994
  • 4086
  • 4096
  • 4114
  • 4994

The direct method is to do the subtraction:

6,102 − 2,016 = 4,086

Note that it isn’t really necessary to do this subtraction. The digits of both 6102 and 2016 add up to 9. So both 6102 and 2016 are multiples of 9. It follows that their difference is also a multiple of 9. So the digits of the difference add up to a multiple of 9.

Of the given options, 4086 is the only one which satisfies this condition (4 + 0 + 8 + 6 = 18). So assuming, as you are entitled to, that one of the options is correct, you can deduce that B is the correct option.

Alternatively, if you do the subtraction, the fact that the sum of the digits of 4086 is a multiple of 9, serves as a useful check.

1 mark

3.2 Without using a calculator, find the remainder when 6,666,666 is divided by 9.

Correct Solution: 6

1 mark

3.3 Without using a calculator, find the remainder when 7,777,777 is divided by 9.

Correct Solution: 4

1 mark

3.4 Without using a calculator, find the remainder when 8,888,888 is divided by 9.

Correct Solution: 2

For each of these, sum the digits and then subtract the nearest multiple of 9.

8+8+8+8+8+8+8=56

9×6=54

5456=2.

4. Intermediate Maths Challenge Problem (UKMT)

4 marks

4.1 What is the value of 2.017 × 2016 − 10.16 × 201.7? Remember, NO CALCULATOR!

  • 2.016
  • 2.017
  • 20.16
  • 2016
  • 2017

To avoid a lot of complicated arithmetic, we exploit the facts that 2.017=201.7100 and 10.16=1016100.

Then we take out the common factor 201.7100. This gives:

2.017×201610.16×201.7=201.7100×20161016100×201.7
=201.7100×20161016
=201.7100×1000
=201.7×10
=2017.

I hope you enjoyed the first Parallelogram of the year. There will be more next week, and the week after, and the week after that. So check your email or return to the website on Thursday at 3pm.

In the meantime, you can find out your score, the answers and go through the answer sheet as soon as you hit the SUBMIT button below.

When you see your % score, this will also be your reward score. As you collect more and more points, you will collect more and more badges. Find out more by visiting the Rewards Page after you hit the SUBMIT button.

It is really important that you go through the solution sheet. Seriously important. What you got right is much less important than what you got wrong, because where you went wrong provides you with an opportunity to learn something new.

Cheerio, Simon.