Parallelogram 7 Year 11 22 Oct 2020Colliding Vortexes

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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 midnight on Sunday 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.

As most of you will have a half-term coming up, this Parallelogram is longer than usual and has a couple of extra questions. And your next parallelogram will be in two weeks time.

1. Intermediate Maths Challenge Problem (UKMT)

3 marks

1.1 A tourist attraction which opens every day needs 30,000 visitors per day on average to break even. Last week there were 120,000 visitors. What is the number of visitors needed this week to break even over the two-week period?

  • 43,000
  • 90,000
  • 180,000
  • 210,000
  • 300,000
  • (Not answered)

Over a two-week period, the required number of visitors is 14 × 30,000 = 420,000. Hence the number required this week is 420,000 - 120,000 = 300,000.

2. Intermediate Maths Challenge Problem (UKMT)

4 marks

2.1 Which of the following is the best estimate for the number of seconds in a year?

  • 3×104
  • 3×105
  • 3×106
  • 3×107
  • 3×108
  • (Not answered)
Show Hint (–2 mark)
–2 mark

There are 60 × 60 × 24 × 365 seconds in a non-leap year. This is roughly equal to 60 × 60 × 20 × 400.

There are 60 × 60 × 24 × 365 seconds in a non-leap year.

This is roughly equal to 60×60×20×400.

60×60×20×400=6×6×2×4×105=288×105=2.88×107.

So 3×107 is the best estimate.

3. Colliding vortexes

Science can be very beautiful, and it can also be hard work. This video from YouTube channel SmarterEveryDay, on colliding vortexes (or vortices, if you prefer), shows this brilliantly.

By the way, a vortex is a mass of gas or liquid with a whirling or circular motion, such as a smoke ring.

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

3 marks

3.1 When Destin completed his experiment, he realised that his exploration was not about fluid dynamics, but rather about…

  • Mechanics and dynamics
  • Vacuums and pressure
  • Persistence and patience
  • Accuracy and detail
  • Tom and Jerry
  • (Not answered)

4. Intermediate Maths Challenge Problem (UKMT)

4 marks

4.1 The diagram on the right shows that 1+3+5+7+5+3+1=32+42.

What is 1+3+5+...+1999+2001+1999+...+5+3+1?

  • 9992+10002
  • 10002+10012
  • 19992+20002
  • 20002+20012
  • none of these
  • (Not answered)
Show Hint (–2 mark)
–2 mark

The general form of the example given is 1+3+5+...+2n1+...+5+3+1=n12+n2.

Show Hint (–2 mark)
–2 mark

The largest number in the sum in question is 2001, so we have 2n1=2001, so n=1001.

The general form of the example given is 1+3+5+...+2n1+...+5+3+1=n12+n2.

The largest number in the sum in question is 2001, so we have 2n1=2001, so n=1001, which gives 1+3+5+ ... +1999+2001+1999+ ... +5+3+1=10002+10012.

If the general form was not obvious to you, you could have tried a 2×2 square or a 6×6 square to get some extra clues.

5. Prime numbers

Number theorists love to find formulas which generate prime numbers. You may have heard of Mersenne’s formula 2p1, where p is a prime number, which has generated the 18 largest known primes.

Here is a simpler procedure which seems to generate prime numbers. Write a list of 3s, and end with the digit 1. For example, 31 is prime, so is 331, and so is 3,331.

2 marks

5.1 What is the smallest number of this type (lots of 3s ending with a 1) which is not prime?

Correct Solution: 333333331

Show Hint (–2 mark)
–2 mark

You may have a ‘factorise’ function on your calculator which could help here. On the Casio fx-83GT X, it is the second function of the button for degrees/minutes/seconds (o ’ ”).

If not, you can use https://isthisprime.com/3333331 and replace the number at the end of the URL.

6. Intermediate Maths Challenge Problem (UKMT)

5 marks

6.1 A 4 by 4 'antimagic square' is an arrangement of the numbers 1 to 16 inclusive in a square, so that the totals of each of the four rows and four columns and two main diagonals are ten consecutive numbers in some order.

The diagram shows an incomplete antimagic square.

When it is completed, what number will replace the asterisk?

  • 1
  • 2
  • 8
  • 15
  • 16
  • (Not answered)
Show Hint (–1 mark)
–1 mark

The totals of the top row and the completed main diagonal are 30 and 39 respectively and therefore the ten consecutive numbers in question are 30 to 39.

Show Hint (–1 mark)
–1 mark

The number in the bottom right-hand corner must be one of 1, 2, 8, 15 or 16. Taking 1 or 2 makes the diagonal (from top left to bottom right) add up to less than 30.

Show Hint (–1 mark)
–1 mark

Making the number in the bottom right-hand corner 15 or 16 makes the diagonal (from top left to bottom right) add up to greater than 39. Hence, 8 must go in the bottom right-hand corner.

Show Hint (–1 mark)
–1 mark

If 15 is placed in the vacant square in the second column, we get a total of 45 for that column, which is too big. Try putting the 15 in another vacant square.

The totals of the top row and the completed main diagonal are 30 and 39 respectively, and therefore the ten consecutive numbers in question must be those from 30 to 39 inclusive.

The number in the bottom right-hand corner must be one of 1, 2, 8, 15 or 16. Taking 1 or 2 makes the diagonal (from top left to bottom right) add up to less than 30, while taking 15 or 16 produces a total greater than 39. Hence 8 must go in the bottom right-hand corner.

It now follows that * must be replaced by 15, since if 15 is placed in one of the other three vacant squares, we get a total of 45 (second column, too big), 34 (third column, same as diagonal) or 47 (third row, too big).

7. Intermediate Maths Challenge Problem (UKMT)

3 marks

7.1 A wire in the shape of an equilateral triangle with sides of length 9 cm is placed on a flat piece of paper.

A pencil is held in the hole at the centre of a disc of radius 1cm, and the disc is rolled all the way around the outside of the wire, and then all the way around the inside of the wire.

What shape is drawn by the pencil?

  • A
  • B
  • C
  • D
  • E
  • (Not answered)

On the outside of the wire, the pencil describes an arc of a circle as the disc rolls around each of the corners of the triangle, but this does not happen when the disc moves around the inside of the wire.

8. 1089 and all that

Here’s a trick you can use to impress your friends, as demonstrated by Aisling Bea on BBC’s QI.

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

1 mark

8.1 Follow the same procedure in the video, starting with the number 462. What is your answer?

Correct Solution: 1089

2 marks

8.2 Follow the same procedure in the video, starting with the number 464. The trick will not work because the number is a palindrome (i.e., the same backwards). What is your answer when you start with 464?

Correct Solution: 000

Once you have convinced yourself when it will and will not work, you may want to try to prove that it works algebraically. Watch this video of James Grime explaining why it works, and make sure you pause and check his algebra at each stage.

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

There will be more in TWO weeks, so check your email or return to the website on Thursday at 3pm on November 5th.

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.