Parallelogram 39 • Level 5 • 30 May 2024A Trapped Knight

Noun: Parallelogram Pronunciation: /ˌparəˈlɛləɡram/

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.
When you finish, remember to hit the SUBMIT button.
Finish by Sunday night 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.

1. Intermediate Maths Challenge Problem (UKMT)

3 marks

1.1 How many of the following statements are true?

Any number which is divisible by 6 is even.
Any number which is divisible by 9 is odd.
The sum of any two odd numbers is even.
The sum of any two even numbers is odd.

0
1
2
3
4
(Not answered)

If a number is divisible by 6 then it must also be divisible by 2, so it is even. ‘The sum of any two odd numbers is even’ is also true. However, not all multiples of 9 are odd (e.g. 18) and the sum of any two even numbers is even rather than odd, so two of the four statements are true.

2. Intermediate Maths Challenge Problem (UKMT)

5 marks

2.1 Trinni is fascinated by triangle numbers (1, 3, 6, 10, 15, 21, etc.) and recently, coming across a clock, she found that she could rearrange the twelve numbers 1, 2, 3, … 12 around the face so that each adjacent pair added up to a triangle number.

She left the 12 in its usual place; what number did she put where the 6 would usually be?

1
4
5
10
11
(Not answered)

Show Hint (–2 mark)

–2 mark

It’s easy to work out which numbers are either side of the 12. It then does not matter which side the 3 and 9 go. It is then easy to work out which numbers go the other side of the 3 and 9. As you get to the end, it gets trickier, as there are some different options, but just remember that for every option clockwise and anticlockwise, the number has to be same at the bottom of the clock.

For ease of reference, label the points A, B, C,…, K as shown.

First note that A = 3 and K = 9 or vice versa. With no loss of generality, let A = 3. Then the only possible values for B, C, and D are 7, 8 and 2 respectively. This gives E = 1 or 4 and, as K = 9, J = 1 or 6.

If J = 1, then E = 4 and the only remaining possibilities for I, H and G are 5, 10 and 11 respectively.

This means that F = 6, but 11 + 6 is not a triangle number, so J is not 1 and must, therefore, be 6. This means that I = 4 and hence E = 1.

The remaining values may now be assigned: H = 11, G = 10 and F = 5.

3. Intermediate Maths Challenge Problem (UKMT)

3 marks

3.1 A shop advertises ‘Buy one, get one at half price’. For this offer, the average cost per item is the same as:

Two for the price of one
Three for the price of one
Three for the price of two
Four for the price of three
Five for the price of four
(Not answered)

The offer gives the purchaser two items for the price of one and a half items.

So the average cost per item is the same as four items for the price of three.

4. Intermediate Maths Challenge Problem (UKMT)

4 marks

4.1 Which of the following shaded regions has an area different from the other shaded regions?

(Not answered)

Show Hint (–1 mark)

–1 mark

It might be easier to look at the area outside the triangle in order to work out the area of the triangle, or to see which option has a different triangle area.

The area of triangle A=12×2×3=3;
The area of parallelogram B=1×3=3;
The area of triangle C=12×3×2=3;
The area of rectangle E=1×3=3.
However, the area of triangle D=3×3−3+3+12=212.

5. Intermediate Maths Challenge Problem (UKMT)

4 marks

5.1 Platinum is a very rare metal, even rarer than gold. Its density is 21.45 g/cm³.

Assuming that the world production has been about 110 tonnes for each of the past 50 years, and negligible before that, which of the following has a comparable volume to that of the total amount of platinum ever produced?

a shoe box
a cupboard
a house
Buckingham Palace
the Grand Canyon
(Not answered)

The volume of 1kg of platinum is (1000/21.45)cm³, that is approximately 50cm³.

So 1 tonne of platinum has a volume of approximately 50,000cm³, which is 120m³.

The volume of platinum produced per year is therefore about 5m³ and the total volume of platinum ever produced is approximately 250m³.

This is the volume of a cuboid measuring 10m × 5m × 5m, which is comparable to a house.

6. The Googol machine

This machine physically visualizes a googol (10100). Watch and be awe struck. Each gear has a reduction of 10:1, so when the first wheel makes 10 rotations, the second wheel makes only 1 rotation. Then when the first wheel makes 100 rotations, the second wheel makes only 10 rotations, and the third wheel rotates once. If the first, or when, the first wheel makes a googol rotations, the final wheel will have made one rotation.

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

The next Parallelogram will be out next Thursday at 3pm.

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.