PG 29 16 May 2019Mathematical Timeline

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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.

1. History of Mathematics

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 mathematicians from the BC era (who lived more than 2,000 years ago), what is the first name of the mathematician who wrote ELEMENTS and was know as the father of geometry?

Correct Solution: EUCLID

2 marks

1.2 Looking at mathematicians from 0 to 1000, what is the name of the mathematician who lived in Alexandria and constructed astrolabes and hydrometers?

Correct Solution: HYPATIA

2 marks

1.3 Looking at mathematicians from 0 to 1000, who was called the father of algebra?

  • Hypatia
  • Aryabhata
  • Brahmagupta
  • Muhammad Al-Khwarizmi
  • Al-Karaji

2. An easy puzzle

I saw this in a newspaper – it was a puzzle for adults, fully grown humans, but I think you will find it easy.

All the digits from 1 to 9 are used in this grid, but only once. Can you work out their positions in the grid so that each of the six different sums work?

Two of the numbers have been inserted in order to get you started. Just identify the other seven numbers.

(NOTE: perform operations in the order they appear, e.g., 3+4×5=35 (not 23))

0.25 marks

2.1 a = Correct Solution: 6

0.25 marks

2.2 b = Correct Solution: 8

0.25 marks

2.3 c = Correct Solution: 7

0.25 marks

2.4 d = Correct Solution: 9

0.25 marks

2.5 e = Correct Solution: 3

0.25 marks

2.6 f = Correct Solution: 2

0.25 marks

2.7 g = Correct Solution: 4

The easiest number to pin down is d, which has to be 9.

5÷1×9=45.

You may have followed a different route, but I then worked out c and g. We know that 9c+g must equal 67, and c and g are single digits, so then c=7 and g=4.

Now we know c=7, then we also know that a×b=48.

This means that a = 6 and b = 8 or a = 8 and b = 6.

We know that b + 1 is factor of 18 from the middle column, so b=8.

We also know that a×5 is a factor of 90 from the 1st column, so a=6.

And the rest is easy.

3. Junior Maths Challenge Problem (UKMT)

4 marks

3.1 Which of the following integers is not a multiple of 45?

  • 765
  • 675
  • 585
  • 495
  • 305

Because 45=5×9, if an integer is divisible by 45, it is divisible both by 5 and by 9. Conversely, if an integer is divisible both by 5 and by 9, then, because 5 and 9 have no common factors other than 1, the integer is also divisible by 45.

Each of the integers 765, 675, 585, 495 and 305 has units digit 5, and therefore is divisible by 5.

The test for whether an integer is divisible by 9 is whether the sum of its digits is divisible by 9.

We see that 7+6+5=18, 6+7+5=18, 5+8+5=18 and 4+9+5=18. Because 18 is divisible by 9 we deduce that all of the first four integers given as options are divisible by 9. However 3+0+5=8 and so the sum of the digits of 305 is not divisible by 9.

We deduce that 305 is the only one of the given options that is not divisible by 45.

4. Square it!

The great maths website NRICH has a tricky game called “Square It”. Play against the computer in 1-player mode and see if you can win.

Play the game

If you do manage to win, then just take a photo or screengrab, which includes the date and time, and then email it to me at prizes@parallel.org.uk – we will pick one winner at random at midnight on Thursday (May 23rd) and send them a copy of my book “The Simpsons and Their Mathematical Secrets”.

5. Junior Maths Challenge Problem (UKMT)

4 marks

5.1 Seven squares are drawn on the sides of a heptagon so that they are outside the heptagon, as shown in the diagram.

What is the sum of the seven marked angles?

  • 315°
  • 360°
  • 420°
  • 450°
  • 630°

In the figure we have labelled some of the vertices so that we may refer to them.

Suppose that there is a flag whose pole is in the direction of GP and pointing as shown. Consider the effect of carrying out the following operations. First rotate the flag anti-clockwise about G through PGQ so that now its pole lies along GQ. Next slide the flag without rotation so that its pole lies along HR. Next rotate the pole about H through RHS so that it lies along HS, and so on, until the flag returns to its original position. The total angle that the flag has turned through is the sum of the seven marked angles. But in returning to its original position the flag has completed a full rotation of 360°.

Therefore the sum of the seven marked angles is 360°.

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.