Parallelogram 23 • Level 5 • 8 Feb 2024Tongue Twisters

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. Adding up 5 numbers

This question was offered up at my son’s school, as one of their monthly puzzles.

3 marks

1.1 Alice challenges Bob to an addition duel. She asks Bob for 5 six-digit numbers. Bob replies with these numbers: 275,381 ; 463,128 ; 978,425, but then Alice interrupts Bob and says, ‘You’re taking too long, I’m going to choose the last two numbers.’ Alice then chooses the numbers 724,618 and 536,871.

Alice says, ‘The first person to find the sum of these five numbers, without using a calculator, is the winner.’

They both line up the numbers in a column and then start adding at the same time. Alice gets the correct answer in 2 seconds – how did she do this?

She has superhuman powers
Her trick involved squaring and then square rooting
Her trick depended on the number 9
Her trick depended on the number 5
Her trick depended on using base 2
(Not answered)

Show Hint (–1 mark)

–1 mark

Alice chooses her numbers AFTER Bob, so she is picking numbers that somehow make it easier to add up all five numbers. Maybe each of Alice’s numbers pairs up with one of Bob’s numbers to make something easy to deal with?

Alice’s first number adds to Bob’s first number to make 999,999 and her second number adds to Bob’s second number to also make 999,999. This is easy to do, because you just need to make sure that each pair of digits adds to 9. If Bob picks 123,456, then you pick 876,543, because 1 + 8 = 9, 2 + 7 = 9, and so on.

Now the sum is easy. It is 999,999 + 999,999 + Bob’s final number, which is 978,425.

Or, to make it really easy, the sum is 1,000,000 + 1,000,000 + 978,425 – 2.

2. Sounds That Could Exist, But Don't

This video from Tom Scott is all about the language sounds that could exist, but don't. There is no solid connection with mathematics, but it is a fascinating video. It really made me think about the sounds we make with our mouths... and the sounds we don’t or can’t make.

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

2 marks

2.1 Why is the alveolar ridge (the hard bit of the roof of your mouth, just behind your teeth) known as the “pizza ridge”?

It is triangular, like a pizza slice.
It can burn when you eat pizza.
It is the colour of tomato sauce.
It has spots like mini pepperoni.
It has taste buds that respond to stuffed crusts.
(Not answered)

2 marks

2.2 In which year did linguists discover a sound that had been missing from their grid? It was the labiodental flap sound.

Correct Solution: 2005

3. Intermediate Maths Challenge Problem (UKMT)

4 marks

3.1
The diagram shows four equilateral triangles with sides of length 1, 2, 3 and 4.

The area of the shaded region is equal to n times the area of the unshaded triangle of side-length 1.

What is the value of n?

8
11
18
23
26
(Not answered)

Show Hint (–2 mark)

–2 mark

If one triangle is similar to and has twice the lengths of another triangle, then it has four times the area. This should allow you to quickly work out the areas of the different triangles, but be careful to take into account that 1 bit of each triangle is unshaded.

We suppose that the equilateral triangle with sides of length 1 has area a. Because the areas of similar figures are proportional to the squares of their side lengths, it follows that the equilateral triangles with side lengths 2, 3 and 4, have areas 4a, 9a and 16a, respectively.

It follows that the areas of the parts of these triangles that are shaded are 4a−a, 9a−a and 16a−a, that is, 3a, 8a and 15a, respectively.

Therefore, the total shaded area is 3a+8a+15a=26a. Hence n=26.

4. Intermediate Maths Challenge Problem (UKMT)

4 marks

4.1 The product of two positive integers is equal to twice their sum. This product is also equal to six times the difference between the two integers.

What is the sum of these two integers?

6
7
8
9
10
(Not answered)

Show Hint (–2 mark)

–2 mark

This is not really the way you should approach the problem, but if you are really stuck, then remember that it is multiple choice. Can you exclude any of the possible answers and take an educated guess?

Of course, it is much better to actually use algebra to get to the full solution.

We let the two positive integers be m and n, with m≥n.

From the information in the question we deduce that mn=2m+n and mn=6m−n.

It follows that 6m−n=2m+n.

Hence 6m−6n=2m+2n.

It follows that 4m=8n, and therefore m=2n.

Substituting 2n for m in the equation mn = 2m+n, gives 2n2=6n.

Because n≠0, we can divide both sides of this last equation by 2n to deduce that n=3.

Therefore m=6.

Hence m+n=6+3=9.

Therefore the sum of the two integers is 9.

5. Intermediate Maths Challenge Problem (UKMT)

5 marks

5.1 The diagram shows two rectangles and a regular pentagon.

One side of each rectangle has been extended to meet at X.

What is the value of x?

52
54
56
58
60
(Not answered)

Show Hint (–1 mark)

–1 mark

Regular or irregular, the internal angles of a polygon add up to (n – 2) x 180°, where n is the number of sides.

Show Hint (–2 mark)

–2 mark

If you are stuck, just try a brute force approach and try calculating a few values for increasing values of n. You will notice that many of the terms under the square root cancel out.

There are various approaches, and here is one of them.

We use the fact that the sum of the angles of a pentagon is 540°. Hence, each interior angle of a regular pentagon is 108°. We also know that the interior angles of the rectangles are each 90°.

We now consider the pentagon TUVWX, as shown in the figure. The interior angles of this polygon at T and W are each 90°. The interior angle at U is the interior angle of the regular polygon, namely 108°. The interior angle at V is the sum of an interior angle of the regular pentagon and a right angle, that is, 108° + 90°.

Therefore 90+108+108+90+90+x=540. Hence, x+486=540. Therefore x=540−486=54.

6. Intermediate Maths Challenge Problem (UKMT)

4 marks

6.1 A water tank is 56 full. When 30 litres of water are removed from the tank, the tank is 45 full.

How much water does the tank hold when full?

180 litres
360 litres
540 litres
720 litres
900 litres
(Not answered)

The 30 litres of water removed from the tank is the difference between 56 and 45 of the capacity of the tank. Now

56−45=5×5−4×65×6=25−2430=130.

We therefore see that 30 litres amounts to 130 of the capacity of the tank. It follows that when the tank is full the number of litres that it holds is 30×30=900.

7. Intermediate Maths Challenge Problem (UKMT)

5 marks

7.1 PQRS is a square. Point T lies on PQ so that PT : TQ=1 : 2. Point U lies on SR so that SU : UR=1 : 2. The perimeter of PTUS is 40 cm.

What is the area of PTUS?

40 cm²
45 cm²
48 cm²
60 cm²
75 cm²
(Not answered)

Show Hint (–1 mark)

–1 mark

Start by drawing a diagram.

Because PQRS is a square, SR=PQ and PS=PQ. Because the ratio PT : TQ is 1 : 2, we deduce that PT=13PQ. Similarly, we have SU=13SR=13PQ=PT.

The lines PT and SU are parts of opposite sides of the square PQRS. Therefore they are parallel. Because they are also of equal length, it follows that PTUS is a parallelogram. Therefore TU=PS=PQ.

It follows that the perimeter of PTUS is PT+TU+US+SP=13PQ+PQ+13PQ+PQ=83PQ. Therefore 83PQ=40 cm. Hence PQ=38×40 cm =15 cm. Therefore PT=13×15 cm =5 cm.

We also note that, because ∠SPT=∠PSU=90°, we can deduce that PTUS is a rectangle.

It follows that the area of PTUS is given by PT×TU=5 cm ×15 cm =75 cm².

There will be more next week, and the week after, and the week after that. So check your email or return to the website on 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.