Parallelogram 41 Level 1 13 Jun 2024I've Got The Power!

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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. Power shortcuts

Much of good maths actually avoids carrying out calculations.

Squaring numbers (also known as the power of 2) isn’t too bad, multiply it by itself:

42=4×4=16

But when the power gets bigger the problem becomes exponentially more difficult.

4854 is 515 digits long, we couldn’t possibly fit it in a standard size calculator!

However, by applying some clever maths we can still figure out some useful information about this number.

Watch this video to find out how:

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

1 mark

1.1 Calculate 41:

  • 1
  • 2
  • 4
  • 8
  • (Not answered)
1 mark

1.2 What is 43?:

  • 12
  • 16
  • 32
  • 64
  • (Not answered)
1 mark

1.3 How would you calculate 45?

  • 4 + 4 + 4 + 4 + 4
  • 4 × 4 × 4 × 4 × 4
  • 4 × 5
  • 5 × 5 × 5 × 5
  • (Not answered)
2 marks

1.4 What is the last digit of 4905?

  • 4
  • 6
  • (Not answered)
3 marks

1.5 Other numbers create last digit patterns when we raise them to different powers.

Using a calculator write down this first ten powers of 3:

  • 31 = ?
  • 32 = ?
  • 33 = ?
    ...
  • 310 = ?

What is the cycle created by the units digits?

  • 3, 9, 3, 9
  • 3, 9, 7, 1
  • 1, 2, 3, 4
  • 3, 6, 9, 0
  • (Not answered)
Show Hint (–1 mark)
–1 mark

Here are the first ten powers of three:

  • 31=3
  • 32=9
  • 33=27
  • 34=81
  • 35=243
  • 36=729
  • 37=2,187
  • 38=6,561
  • 39=19,683
  • 310=59,49

Look at the last digits, what is the repeating pattern?

2. Some more puzzles where you can use clever thinking to avoid doing any long calculations

2 marks

2.1 If today is Monday, what day of the week will it be in 698 days time?

  • Monday
  • Tuesday
  • Wednesday
  • Saturday
  • Sunday
  • (Not answered)

698 is only 2 less than 700, which is a multiple of 7.

Every multiple of 7 brings you back to the same day of the week!

So the 700th day will be a Monday again, counting back two the 698th day is a Saturday.

2 marks

2.2 In this figure, the triangles are all equilateral.

The dotted line is 12 cm long.

How long in total is the zigzag line?

  • 12 cm
  • 16 cm
  • 18 cm
  • 24 cm
  • 28 cm
  • (Not answered)
Show Hint (–1 mark)
–1 mark

The combined length of two bits of the zigzag line that make up the first triangle are double the length of the dotted base, because it is an equilateral triangle... and this will be true for every zigzag triangle.

The combined length of two bits of the zigzag line that make up the first triangle are double the length of the dotted base, because it is an equilateral triangle... and this will be true for every zigzag triangle.

So, the combined length of all the zigzag bits will be double the length of all the dotted base, i.e., 2 × 12 cm = 24 cm.

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.