PG 30 23 May 2019Amazing Grace

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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.
  • It’s half-term for most of you, so this Parallelogram is a bit longer and you have a bit more time to complete it. Importantly, double badge points!
  • Finish by midnight on Sunday 2 June 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. History of Mathematics

As mentioned last week, the terrific Mathigon website has loads of great material, including a timeline showing the great mathematicians of past centuries. Visit the timeline (click and it will open up in a new tab) and answer the three questions below. Just enter the name given in the plum box.

So, if the answer is John Napier, just enter Napier, because that is the name that appears in the plum-coloured box on the timeline:

2 marks

1.1 Looking at twentieth century mathematicians 1900 to 1999, what is the surname of the mathematician who is the only mathematician to have won the Abel Prize and the Nobel Prize for economics?

Correct Solution: NASH

2 marks

1.2 Looking at twentieth century mathematicians 1900 to 1999, what is the surname of the mathematician who won $1 million for proving the Poincaré conjecture, but refused to accept the money, stating: “I’m not interested in money or fame; I don’t want to be on display like an animal in a zoo.”?

Correct Solution: PERELMAN

2. Junior Maths Challenge Problem (UKMT)

2 marks

2.1 The diagram shows an equilateral triangle, a square, and one diagonal of the square.

What is the value of x?

  • 105
  • 110
  • 115
  • 120
  • 135

Let A, B, C and D be the vertices of the square, and A, D and E be the vertices of the equilateral triangle, as shown.

EDA=60° because it is the angle of an equilateral triangle.

ADC=90° because it is the angle of a square. The diagonal DB bisects the angle ADC, and therefore ADB=45°.

It follows that x°=EDA+ADB=60°+45°=105°.

Therefore x=105.

3. Junior Maths Challenge Problem (UKMT)

4 marks

3.1 Mathias is given a grid of twelve small squares. He is asked to shade grey exactly four of the small squares so that his grid has two lines of reflection symmetry.

How many different grids could he produce?

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

The answer is not 6.

Show Hint (–1 mark)
–1 mark

The answer is not 5.

Show Hint (–1 mark)
–1 mark

The answer is not 4.

The two lines of reflection symmetry of the grid are shown in the figure on the right as broken lines.

We see from this that if the grid with some of the small squares shaded grey has both these lines of reflection symmetry, then all the four squares labelled P must be the same colour. Similarly, all the four squares labelled Q must be the same colour, the two squares labelled R must be the same colour, and the two squares labelled S must be the same colour.

It follows that there are only three ways in which Mathias can shade exactly four of the small squares of the grid grey so that the result has two lines of reflection symmetry. These are

  • (i) shade grey all the squares labelled P, and no others,
  • (ii) shade grey all the squares labelled Q, and no others,
  • (iii) shade grey all the squares labelled R and all those labelled S, and no others.

The three different grids that Mathias could produce are shown in the figure below.

4. The Amazing Grace

Grace Hopper was a force of nature. A rear admiral in the American Navy, a pioneer in computing and a terrific mathematician. Find out more below and answer the question that follows.

2 marks

4.1 What caused Grace Hopper to invent the word bug, meaning a problem in a piece of computer hardware or software?

  • An ant
  • A cockroach
  • A ladybird
  • A moth
  • A spider

5. Junior Maths Challenge Problem (UKMT)

4 marks

5.1 What is the sum of the digits in the completed crossnumber?

Show Hint (–1 mark)
–1 mark

Which power of 11 has 5 digits? Try 114.

  • 25
  • 29
  • 32
  • 34
  • 35

The answer to 3 Across is a 5-digit power of 11. We have 112= 121, 113 = 1331, 114 = 14641.

Any higher power of 11 has more than 5 digits. We deduce that 3 Across is 14641.

It follows that 2 Down is a 2-digit square with units digit 4. Hence 2 Down is 64.

We now see that 1 Across is a 3-digit cube with units digit 6. The only such 3-digit cube is 216 (= 63).

Hence the completed crossnumber is as shown in the figure.

We see that the sum of the digits in the completed crossnumber is:

2 + 1 + 6 + 1 + 4 + 6 + 4 + 1 = 25.

6. Even more amazing Grace

Here is an interview between Grace Hopper (question 4) and Dave Letterman (who is a legendary chat show host). Take a look and answer the questions below.

As you might have guessed, Grace Hopper is one of my heroes.

2 marks

6.1 How long is a nanosecond?

  • 1/10 of a second
  • 1/100 of a second
  • 1/1,000 of a second
  • 1/1,000,000 of a second
  • 1/1,000,000,000 of a second
2 marks

6.2 How long is a nanosecond?

  • 3 mm
  • 3 cm
  • 30 cm
  • 3 metres
  • 3 Km

7. Junior Maths Challenge Problem (UKMT)

5 marks

7.1 The distance between Exeter and London is 175 miles. Sam left Exeter at 10:00 on Tuesday for London. Morgan left London for Exeter at 13:00 the same day. They travelled on the same road. Up to the time when they met, Sam’s average speed was 25 miles per hour, and Morgan’s average speed was 35 miles an hour.

At what time did Sam and Morgan meet?

  • 17:00
  • 15:55
  • 15:30
  • 15:00
  • 14:40
Show Hint (–1 mark)
–1 mark

Because the answer to this question depends only on the average speeds of Sam and Morgan up to the time when they meet, we may assume that Sam’s average speed was 25 miles an hour from 10:00 to 13:00 and also from 13:00 up to the time when they meet. If follows that by 13:00 Sam has gone 75 miles and is therefore at that time 100 miles away from London.

Because the answer to this question depends only on the average speeds of Sam and Morgan up to the time when they meet, we may assume that Sam’s average speed was 25 miles an hour from 10:00 to 13:00 and also from 13:00 up to the time when they meet. If follows that by 13:00 Sam has gone 75 miles and is therefore at that time 100 miles away from London.

As soon as Morgan sets off they are approaching each other at a combined average speed of 25 + 35 miles an hour, that is, at 60 miles an hour. Therefore it takes a further 10060 hours until they meet. Because:

10060 hours = 1 4060 hours = 1 hour and 40 minutes,

it follows that they meet 1 hour and 40 minutes after 13:00. Hence they meet at 14:40.

8. OK Go

OK Go is an American rock band, famous for their quirky one-take music videos. This one shows an elaborate machine that employs all sorts of mechanics. Watch and enjoy … and look out for the answers to these two questions.

1 mark

8.1 In the last minute of the video, there is an interesting version of Newton’s cradle. Normally a series of suspended metal ball bearings swing to and fro, with the one on the left passing its momentum to the right, via the ones in the middle... and then vice versa. What is OK Go’s cradle made from?

  • Water balloons
  • Sledge hammers
  • Baseball bats
  • Koala bears
1 mark

8.2 Such machines often involve domino toppling, and there is domino toppling within the first minute of this video. However, in the second minute of the video there is domino toppling, but there are no dominos. What four objects are toppled?

  • Chairs
  • Tables
  • Wardrobes
  • Fridges

Such machines have different names in the UK and the US, in honour of two eccentric inventors. Search on the internet to complete the names.

1 mark

8.3 USA:

Rube Correct Solution: Goldberg

1 mark

8.4 UK:

Heath Correct Solution: Robinson

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.
  • 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.
  • The next Parallelogram will be out on Thursday 6 June at 3pm.

Cheerio, Simon.