Parallelogram 22 • Level 4 • 1 Feb 2024Mean Girls

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.
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. Mean Girls

Here’s a clip from “Mean Girls”, a 2004 film starring Lindsay Lohan as Cady, a student trying to settle into a new school. Cady tries to change herself in order to fit in, but in the end she returns to her original personality and joins the Mathletes in the state championship finals.

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

Cady answers a question about “limits”, which is something you probably haven't covered yet.

The concept of limits touches on areas such as finding the value of a function in unusual or extreme situations. For example, let’s explore the value of this function.

2z2−17z+88−z

When z=7, then fz=−13.

When z=9, then fz=−17.

But what about when z=8?

The denominator is 8−8, which is zero, and we cannot divide by zero. So what is the answer? Is there an answer?

We can factor the top half of the equation, so it can be re-written as:

2z2−17z+88−z=2z−1z−8−z−8=2z−1−1

Now it is clear that when z=8, then fz=−15.

Now, answer the question below, which relates to the function:

y2−4y−21y−7

1 mark

1.1. What is the value of fy when y=6?

Correct Solution: 9

1 mark

1.2. What is the value of fy when y=8?

Correct Solution: 11

3 marks

1.3. What is the value of fy when y=7?

Correct Solution: 10

y2−4y−21y−7=y−7y+3y−7=y+3=10

2. Demis Hassabis

Demis Hassabis is an extraordinary computer scientist, perhaps the greatest pioneer in artificial intelligence in the world. He grew up in North London, the son of a Chinese-Singaporean mother and Greek-Cypriot father, and went to his local comprehensive school.

In this short interview, he explains how chess played an important role in his development. Take a look and maybe think about starting to learn how to play chess. Or, if you play chess already, start taking it more seriously and join a chess club.

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

By the way, there are several great online websites, many of them offering free interactive lessons or the chance to play other people online (or play against a computer). A good place to start is Chess Kid.

3. Intermediate Maths Challenge Problem (UKMT)

This is one of the toughest questions from a UKMT Intermediate Maths Challenge exam, so be ready to stretch your brain.

4 marks

3.1 The diagram shows a square, a diagonal and a line joining a vertex to the midpoint of a side. What is the ratio of area P to area Q?

1:2
2:3
1:2
2:5
1:3
(Not answered)

Show Hint (–2 mark)

–2 mark

Let I, J, K and L be the vertices of the square. Let M be the midpoint of JK, let N be the point where the diagonal IK meets LM. Let the line through N parallel to LK meet IL at R and JK at S. Let T be the foot of the perpendicular from N to LK. Let the square have side length s.

In the triangles INL and KNM, the opposite angles ∠INL and ∠KNM are equal. Also, as IL is parallel to JK, the alternate angles ∠LIN and ∠MKN are equal. Therefore the triangles INL and KNM are similar. Hence INNK=ILMK=2.

Similarly, the triangles INR and KNS are similar. Therefore NRNS=INNK=2.

So NR=23s and NS=13s.

Similarly, the triangles INR and KNS are similar. Therefore NRNS=INNK=2.

So NR=23s and NS=13s.

Now, NTKS is square, because its angles are all right-angles, and ∠NKT=45°. Therefore NT=NS=13s.

It follows that the area of the triangles LNK, INL and MNK are 12s×13s=16s2, 12s×23s=13s2, and 1212s×13s=112s2, respectively.

The area of the region P, is that of the triangle LNK, that is, 16s2. The area of the region Q is obtained by subtracting the areas of the triangles LNK, INL and MNK from the area of the square. So region Q has area s2−16s2−13s2−112s2=512s2.

So the ratio of these areas is 16s2:512s2=16:512=2:5.

4. Intermediate Maths Challenge Problem (UKMT)

2 marks

4.1 Which of the following is divisible by 6?

one million minus one
one million minus two
one million minus three
one million minus four
one million minus five
(Not answered)

It helps to first write the options we are given in standard form using digits:

999 999
999 998
999 997
999 996
999 995

An integer is divisible by 6 if and only if it is divisible by 2 and by 3. It is easy to see that, of the given options, only 999 998 and 999 996 are divisible by 2, and, of these, only 999 996 is divisible by 3.

5. Intermediate Maths Challenge Problem (UKMT)

2 marks

5.1 A machine cracks open 180 000 eggs per hour. How many eggs is that per second?

5
50
500
5000
50000
(Not answered)

There are 60 seconds in a minute and 60 minutes in an hour. So there are 60 × 60 = 3600 seconds in an hour.

So 180 000 eggs per hour is the same as 1800003600=180036=50 eggs per second.

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.