PG 34 27 Jun 2019Pet Puzzle

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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.
  • 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.

It’s the last normal Parallelogram of the school year... so it’s a toughie - and even better there is a super-giant Parallelogram next week to keep you busy over the summer holidays. Actually, it is not super-giant, just bigger than a normal Parallelogram. That’s because it is important that you do a bit of mathematics every week between the end of the summer term and the start of the autumn term to stop your brain from turning to sludge. Good luck!

1. Pet Puzzle

A cat and a dog weigh 10 Kg. A dog and a rat weigh 7 Kg. And a cat and a rat weigh 5 Kg.

3 marks

1.1 What is the combined weight of a dog, cat and rat?

(See if you can find a quick way to solve this problem.)

Correct Solution: 11 kg

SLOW APPROACH

(1) C + D = 10
(2) D + R = 7
(3) C + R = 5

Line (1) tells us that C = 10 – D, so we can swap the C in line (3) to give us…

[10 – D] + R = 5.

This means that R = D – 5, and we can swap this into line (2) to give us …

D + [D – 5] = 7
2D = 12
D = 6.

If D = 6, then line (1) tells us that C = 4.

Then line (3) tells us that R = 1.

So, the total weight is D + C + R = 6 + 4 + 1 = 11 Kg.

QUICK APPROACH

If we add (1) + (2) + (3), then we have two of each animal, which means.

2D + 2C + 2R = 10 + 7 + 5 = 22.

So, D + C + R = 11 Kg.

2. Three Simple Ways to Time Travel

1 mark

2.1 If you want time to slow down, which one of these will help?

  • Do nothing
  • Go for a walk
  • Climb to the top of a mountain
  • Watch paint dry
1 mark

2.2 Because of the stronger gravity, which part of your body ages most slowly? Which part of your body generally experiences time passing more slowly?

  • The top of your head
  • Your nose
  • Your heart
  • Your belly button
  • Your ankles
2 marks

2.3 How big is a femtosecond?

  • Thousandth of a second
  • Millionth of a second
  • Billionth of a second
  • Trillionth of a second
  • Quadrillionth of a second
2 marks

2.4 How big is a femtosecond?

  • One millionth of one millionth, of a second
  • One thousandth of one billionth, of a second
  • One hundredth of one billionth, of a second
  • One trillionth of one billionth, of a second
  • One millionth of one billionth, of a second

3. Junior Maths Challenge Problem (UKMT)

3 marks

3.1 The diagram shows a partially completed magic square, in which all rows, all columns and both main diagonals have the same total.

What is the value of x+y?

  • 10
  • 11
  • 12
  • 13
  • 14
Show Hint (–1 mark)
–1 mark

Label the number in the top right hand corner as “w”. This completes two lines, which are a diagonal and the right-hand column. We know that the totals in this diagonal and the right-hand column must be the same.

Show Hint (–1 mark)
–1 mark

The total of the diagonal is (6+7+w), and the total of the column is (x+y+w), so what must (x+y) be equal to?

Let w be the number in the top-right square, as shown in the figure.

Because we are dealing with a magic square, the total of the numbers in the right-hand column is the same as the total of the numbers in the diagonal from the bottom-left square to the top-right square.

This gives

x+y+w=6+7+w.

Hence, by subtracting w from both sides, we deduce that

x+y=6+7.

It follows that

x+y=13.

4. Junior Maths Challenge Problem (UKMT)

2 marks

4.1 How many integers are greater than 20+18 and also less than 20×18?

  • 320
  • 321
  • 322
  • 323
  • 324

We have 20+18=38 and 20×18=360.

The integers that are greater than 38 and less than 360 are the integers from 39 to 359, inclusive.

There are 35939+1, that is, 321 of these integers.

5. Junior Maths Challenge Problem (UKMT)

2 marks

5.1 Gill scored a goal halfway through the second quarter of a ‘teachers versus pupils’ netball match.

At that point, what fraction of the whole match remained to be played?

  • 14
  • 38
  • 12
  • 58
  • 34

Because 12×14=18, a half of a quarter is one eighth.

At the time when Gill scored the goal the first quarter and half the second quarter of the whole match had been completed. So the fraction of the game that had been completed was

14+18=28+18=38.

Therefore the fraction of the game that remained to be played was

138=8838=58

2 marks

5.2 Gill scored a second goal one-third of the way through the third quarter of the netball match. At that point, what fraction of the whole match remained to be played?

  • 712
  • 13
  • 512
  • 23
  • 14

The second goal was scored after:

14+14+13×14=14+14+112=312+312+112=712 of the game.

So, 512 of the game remained

6. Junior Maths Challenge Problem (UKMT)

3 marks

6.1 Beatrix places copies of the L-shape shown on a 4×4 board so that each L-shape covers exactly three cells of the board.

She is allowed to turn around or turn over an L-shape.

What is the largest number of L-shapes she can place on the board without overlaps?

  • 2
  • 3
  • 4
  • 5
  • 6
Show Hint (–1 mark)
–1 mark

The answer is not 4 shapes.

The figure shows one way of placing 5 L-shapes on the board without overlaps.

The 4 × 4 board is made up of 16 cells. The L-shape is made up of 3 cells.

Therefore, however 5 L-shapes are placed on the board without overlapping, there will be just one cell that remains uncovered. So there will not be space for one more L-shape.

Hence, the largest number of L-shapes that Beatrix can place on the board without overlaps is 5.

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.