Parallelogram 31 Year 8 10 Jun 2021Floppy Pizza

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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. Junior Maths Challenge Problem (UKMT)

3 marks

1.1 Adding four of the five fractions 12, 13, 16, 19, and 118 gives a total of 1.

Which of the fractions is not used?

  • 12
  • 13
  • 16
  • 19
  • 118
  • (Not answered)

There are five different ways of selecting four of the five fractions given in the question. One method would be to try these different combinations in turn to find the four fractions that add up to 1.

However, it is much quicker to begin by adding all five of the fractions. This will give an answer greater than 1. We will then be able to see which fraction we need to remove to reduce the sum to 1.

By putting all of the fractions over the common denominator 18, we see that

12+13+16+19+118=9+6+3+2+118=2118=76=1+16.

We see from this calculation that to obtain a total of 1 it is the fraction 16 that is not used.

2. Don Bradman’s duck

Don Bradman had several pets, but his favourite pet was a duck called Duckminster Fuller...

Actually, that’s not true. Bradman was a heroic cricketer, and this video is all about his sad farewell to the game he loved. Even if you don’t like cricket, I challenge you not be moved by this tale of sporting genius and averages. By the way, a duck is the name given to the score when a batsman scores no runs.

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

3 marks

2.1 Prior to the last match, Bradman had scored 6,996 runs from 69 innings, and needed 4 more runs to have a career average of 100 from 70 innings.

How many runs would have given Bradman a career average of 99.99 (rounded to 4 significant figures)?

  • 0
  • 1
  • 2
  • 3
  • 4
  • 5
  • (Not answered)

6996+370=99.9857, which is 99.99 to 4 s.f.

3. Junior Maths Challenge Problem (UKMT)

3 marks

3.1 In Matt’s pocket there are 8 watermelon jellybeans, 4 vanilla jellybeans and 4 butter popcorn jellybeans.

What is the smallest number of jellybeans he must take out of his pocket to be certain that he takes at least one of each flavour?

  • 3
  • 4
  • 8
  • 9
  • 13
  • (Not answered)
Show Hint (–1 mark)
–1 mark

He could pick out a watermelon, then a vanilla, then a butter popcorn. He would have done it in 3 picks. Wouldn’t that be lucky! But what would be the most unlucky way to pick out the jellybeans, picking lots and lots but still not seeing all three flavours.

Show Hint (–1 mark)
–1 mark

It would be very unlucky indeed if he had picked out all the watermelon jellybeans first. He’d have picked out 8 jellybeans and only have one flavour. What if his luck is still down, and he picks out all of the vanilla next?

If Matt takes 12 jellybeans then he will have taken at least one of each flavour unless he takes all 8 watermelon jellybeans and either all 4 vanilla jellybeans or all 4 butter popcorn jellybeans.

In this case the 4 remaining jellybeans will all be of the flavour he has yet to take, so taking one more jellybean ensures that he will have taken at least one of each flavour.

4. Floppy pizza

When you are a mathematician, the world becomes amazing. Even eating pizza becomes amazing, as revealed in this video by Cliff Stoll. You might find some bits of the video confusing, but stick with it and see if you can understand why we eat pizza slices in a particular way.

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

1 mark

4.1 For shape A, what is the Gaussian curvature at the black dot?

  • Negative
  • Zero
  • Positive
  • (Not answered)

The horizontal line has (+) curvature and the vertical line has (-) curvature, so the Gaussian curvature is negative.

1 mark

4.2 For shape B, what is the Gaussian curvature at the black dot?

  • Negative
  • Zero
  • Positive
  • (Not answered)

The horizontal line has (+) curvature and the vertical line has zero curvature, so the Gaussian curvature is zero.

1 mark

4.3 For shape C, what is the Gaussian curvature at the black dot?

  • Negative
  • Zero
  • Positive
  • (Not answered)

The horizontal line has (+) curvature and the vertical line has (+) curvature, so the Gaussian curvature is positive.

5. Junior Maths Challenge Problem (UKMT)

4 marks

5.1 A large solid cube is cut into two pieces by a single plane cut.

How many of the following four shapes could be the shape of the cross-section formed by the cut?

  • 0
  • 1
  • 2
  • 3
  • 4
  • (Not answered)
Show Hint (–1 mark)
–1 mark

The third cross-section (with 4 sides) is definitely possible.

Show Hint (–1 mark)
–1 mark

The fourth cross-section (with 6 sides) is definitely possible.

We see from the figures below that all four shapes are possible.

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.