Parallelogram 8 Level 5 26 Oct 2023Cibara numerals

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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.


3 marks

What is the value of 4002 - 2004?

  • 2004
  • 2002
  • 2000
  • 1998
  • 1996
  • (Not answered)

4004 - 2004 = 2000, so 4002 - 2004 = 2000 - 2 = 1998.


4 marks

In Niatirb they use Cibara numerals. These are the same shape as normal Arabic numerals, but with the meanings in the opposite order. So "0" means "nine", "1" means "eight" and so on.

But they write their numbers from left to right and use arithmetic symbols just as we do. So, for example, they use 62 for the number we write as 37.

How do the inhabitants of Niatirb write the answer to the sum which they write as 837 + 742?

  • 419
  • 580
  • 1579
  • 5317
  • 8420
  • (Not answered)

The sum is what we would write as 162 + 257 and this equals 419. However, in Niatirb it would be written 580.


4 marks

The diagram shows a square board in which strips of white squares alternate with strips of black and white squares. A larger board, constructed in the same way, has 49 black squares.

How many white squares are there on the larger board?

  • 176
  • 196
  • 245
  • 289
  • 392
  • (Not answered)

The 49 black squares will be in a 7 × 7 formation, so the board will measure 15 × 15 squares. Hence the number of white squares = 225 - 49 = 176.


4 marks

If the pattern shown is continued, what number will appear directly below 400?

  • 438
  • 439
  • 440
  • 441
  • 442
  • (Not answered)

Note that the number at the end of the nth row is n2, so 400 will lie at the end of the 20th row.

The row below will end in 212, ie 441, so the number directly below 400 will be 440.


5 marks

A square is divided into four congruent rectangles and a smaller square, as shown. The diagram is not to scale.

The area of the small square is 14 of the area of the whole square.

What is the ratio of the length of a short side of one of the rectangles to the length of a long side?

  • 1 : 2
  • 1 : 3
  • 1 : 2
  • 1 : 3
  • 1 : 4
  • (Not answered)

Let the length of a short side of a rectangle be x and the length of a long side be y. Then the whole square has side of length y+x, whilst the small square has side of length yx.

As the area of the whole square is four times the area of the small square, the length of the side of the whole square is twice the length of the side of the small square.

Therefore y+x=2yx i.e. y=3x so x:y=1:3.

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.