Parallelogram 6 Year 11 15 Oct 2020Surprising Values

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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.

These challenges are a random walk through the mysteries of mathematics, most of which will be nothing to do with what you are doing 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. 0!

No need to shout it, I heard you the first time!

Do you know what zero factorial equals? If you do, do you know why? Watch this video from YouTube channel Numberphile.

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

1 mark

1.1 What is the value of 0!?

  • Undefined
  • 0
  • 1
  • Infinity
  • (Not answered)
1 mark

1.2 What is the value of -1!?

  • Undefined
  • -1
  • 1
  • 0
  • (Not answered)
2 marks

1.3 How many ways are there to arrange 5 objects in a line?

Correct Solution: 120

2. Intermediate Maths Challenge Problem (UKMT)

4 marks

2.1 The diagram shows a right-angled isosceles triangle XYZ which circumscribes a square PQRS.

The area of triangle XYZ is x.

What is the area of square PQRS?

  • 4x9
  • x2
  • 4x5
  • 2x5
  • 2x3
  • (Not answered)
Show Hint (–1 mark)
–1 mark

The diagram shows that triangle XYZ may be divided into 9 congruent triangles.

The square PQRS is made up of 4 of these 9 triangles.

3. 0.9999...

Do you know what 0.9 recurring equals? If so, do you know why? Watch this video from the Mathematical Association of America.

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

2 marks

3.1 If you choose to believe it has a value, what is the value of 1.999999...?

Correct Solution: 2

3 marks

3.2 If you chose to believe it has a value, what would be the value of ...33333 (the number that has infinitely many 3s before the decimal point)?

  • 13
  • 0
  • 1
  • 3
  • 13
  • (Not answered)
Show Hint (–1 mark)
–1 mark

S = ...33333

10S = ...33330

Compare S and 10S

Show Hint (–1 mark)
–1 mark

Looking at the values of S and 10S, we can see that 10S=S3.

Solve to find the value of S.

4. Intermediate Maths Challenge Problem (UKMT)

4 marks

4.1 The Pythagoras Patisserie sells triangular cakes at 39p each and square buns at 23p each. For her party, Helen spent exactly £5.12 on an assortment of these cakes and buns. How many items in total did she buy?

  • 15
  • 16
  • 17
  • 18
  • 19
  • (Not answered)
Show Hint (–1 mark)
–1 mark

One cake and one bun cost a total of 62p.

Note that 512÷62=8 remainder 16, that is 512=839+23+16.

Also note that 16=3923.

Show Hint (–2 mark)
–2 mark

512=839+23+3923. How many 39s and how many 23s is that?

One cake and one bun cost a total of 62p. Note that 512 ÷ 62 = 8 remainder 16 and note also that 16 = 39 - 23.

Hence512=839+23+16=839+23+3923=9×39+7×23.

As 39 and 23 do not have a common factor, other than 1, Helen must have bought 9 cakes and 7 buns.

5. Intermediate Maths Challenge Problem (UKMT)

5 marks

5.1 A regular tetrahedron with edges of length 6 cm has each corner cut off to produce the solid shown.

The triangular faces are all equilateral triangles, but not necessarily all the same size.

What is the total length of the edges of the resulting solid?

  • 28 cm
  • 30 cm
  • 36 cm
  • 48 cm
  • more information needed
  • (Not answered)
Show Hint (–2 mark)
–2 mark

For the triangular faces of the resulting solid to be equilateral, it is necessary for each of the solids removed at the corners to be a regular tetrahedron.

Removing a regular tetrahedron at a corner means that the perimeter of the equilateral triangle created is equal to the sum of the other three edges of the removed tetrahedron.

For the triangular faces of the resulting solid to be equilateral, it is necessary for each of the solids removed at the corners to be a regular tetrahedron. Removing a regular tetrahedron at a corner in this manner does not change the total length of the edges of the solid as the perimeter of the equilateral triangle created is equal to the sum of the other three edges of the removed tetrahedron.

The original tetrahedron had six edges, all of side 6 cm, and therefore the total length of the edges of the resulting solid is 36 cm.

I hope you enjoyed this Parallelogram. 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.