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Parallelogram 37 • Level 2 • 16 May 2024Mathematical Timeline

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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. 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
  • (Not answered)

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
  • (Not answered)

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

5. Junior Maths Challenge Problem (UKMT)

4 marks

5.1 In a sequence of numbers, each term after the first three terms is the sum of the previous three terms. The first three terms are -3, 0, 2. Which is the first term to exceed 100?

  • 11th term
  • 12th term
  • 13th term
  • 14th term
  • 15th term
  • (Not answered)

The first thirteen terms of the sequence are −3, 0, 2, −1, 1, 2, 2, 5, 9, 16, 30, 55, 101, ... .

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.