Parallelogram 28 • Level 3 • 14 Mar 2024Laurels and Hardy

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. Laurels and Hardy

Take a look at this short video by YouTuber Vsauce.

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

Kevin explains this bit of maths is probably not going to help achieve world piece, and instead justifies it as a curiosity.

In other words, maths does not have to be useful – it can just be curious, interesting and/or beautiful.

2 marks

1.1 The great mathematician G. H. Hardy revelled in the idea that his mathematics was useless. He therefore suggested that this meant his mathematics could never do any harm.

I am not sure I agree with either of those conclusion.

In any case, what do we call mathematics that is not motivated by any application?

Clean
Clear
Refined
Pure
Sterling
(Not answered)

2 marks

1.2 Hardy once wrote: “A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ……..”

What is the missing word at the end of this quote?

Tip: take a guess if you think you know, but maybe Google the quote if you have no idea.

ideas
dreams
numbers
shapes
chalk
(Not answered)

1 mark

1.3 To justify the title of this Parallelogram (Laurels and Hardy), which prize (or laurel) did G. H. Hardy not win?

Smith's Prize
Royal Medal
De Morgan Medal
Chauvenet Prize
Nobel Prize
Copley Medal
(Not answered)

There is no Nobel Prize for mathematics, although many mathematicians have won the Nobel Prize for Economics (or, to give it is proper name, the Sveriges Riksbank Prize in Economic

2.

5 marks

2.1 What is the smallest four-digit positive integer which has four different digits?

1032
2012
1021
1234
1023
(Not answered)

Here it is easy just to check the options that are given. A, D and E are the only options in which all four digits are different. Of these, clearly, E is the smallest.

For a complete solution we need to give an argument to show that 1023 really is the smallest four digit positive integer with four different digits. It is easy to do this.

To get the smallest possible number we must use the four smallest digits, 0, 1, 2 and 3. A four digit number cannot begin with a 0. So we must put the next smallest digit, 1, in the thousands place, as a four-digit number beginning with 2 or 3 is larger than one beginning with a 1. For similar reason the hundreds digit must be the smallest remaining digit, 0. Similarly the tens digit must be 2 and the units digit must be 3.

So the required number is 1023.

3.

5 marks

3.1 Each square in the figure is 1 unit by 1 unit.

What is the area of triangle ABM (in square units)?

4
4.5
5
5.5
6
(Not answered)

Triangle ABM has base 3 units and height 3 units, so its area is 12×3×3 units², that is 412 units².

4.

5 marks

4.1 In the diagram shown, all the angles are right angles and all the sides are of length 1 unit, 2 units or 3 units.

What, in square units, is the area of the shaded region?

22
24
26
28
30
(Not answered)

Divide the whole figure into horizontal strips of height 1 unit: its area is (3 + 6 + 8 + 8 + 8 + 6 + 3) units² = 42 units². Similarly, the unshaded area is (1 + 4 + 6 + 4 + 1) units² = 16 units². So the shaded area is 26 units².

Alternative solution: notice that if the inner polygon is moved a little, the answer remains the same – because it is just the difference between the areas of the two polygons. So, although we are not told it, we may assume that the inner one is so positioned that the outer shaded area can be split neatly into 1 by 1 squares – and there are 26 of these.

5.

6 marks

5.1 The lengths, in cm, of the sides of the equilateral triangle PQR are as shown.

Which of the following could not be the values of x and y?

(18, 12)
(15, 10)
(12, 8)
(10, 6)
(3, 2)
(Not answered)

As triangle PQR is equilateral, x+2y=3x−y=5y−x. Equating any two of these expressions gives 2x=3y.

The only pair of given values which does not satisfy this equation is x=10, y=6.

6.

6 marks

6.1 Pablo’s teacher has given him 27 identical white cubes.

She asks him to paint some of the faces of these cubes grey and then stack the cubes so that they appear as shown.

What is the largest possible number of the individual white cubes which Pablo can leave with no faces painted grey?

8
12
14
15
16
(Not answered)

We can see 19 of the 27 cubes. Of these there are 12 which we can see have at least one grey face. The remaining cubes could have all their faces white. So the maximum number of cubes that could be all white is 27 − 12 = 15.

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.