Parallelogram 8 • Level 2 • 26 Oct 2023Maths Jokes

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. Maths Jokes

Can you match the mathematical joke to the correct punchline?

Punchline A: Henry the 18.
Punchline B: Remove the S.
Punchline C: Because it gives them square roots.

2 marks

1.1 Which king loved fractions?

Punchline A
Punchline B
Punchline C
(Not answered)

2 marks

1.2 Why do plants hate math?

Punchline A
Punchline B
Punchline C
(Not answered)

2 marks

1.3 How do you make seven an even number?

Punchline A
Punchline B
Punchline C
(Not answered)

2.

3 marks

The difference between 13 of a certain number and 14 of the same number is 3.

What is that number?

A) 24
B) 36
C) 48
D) 60
E) 72
(Not answered)

Since 13−14=4−312=112

it follows that 112 th of the number is 3. Therefore the number is 12×3=36.

3.

3 marks

In the expression 1 □ 2 □ 3 □ 4 each □ is to be replaced by either + or ×.

What is the largest value of all the expressions that can be obtained in this way?

A) 10
B) 14
C) 15
D) 24
E) 25
(Not answered)

In general, we obtain a larger number by multiplying two positive integers together than by adding them. The only exceptions are when one of the positive integers is 1, because 1+n>1×n, and when both are 2, because 2+2=2×2.

Therefore to obtain the largest possible value we need to replace the first □ with + but the other two with ×. This leads us to the answer 1+2×3×4=1+24=25.

4.

3 marks

Which of the following is divisible by all of the integers from 1 to 10 inclusive?

A) 23 × 34
B) 34 × 45
C) 45 × 56
D) 56 × 67
E) 67 × 78
(Not answered)

It is easy to rule out four of the options using the fact that a product of integers is divisible by a prime number p if, and only if, at least one of the integers making up the product is divisible by p.

Using this we see that, since neither 23 nor 34 is a multiple of 3, it follows that 23 × 34 is not a multiple of 3, since neither 34 nor 45 is a multiple of 7, 34 × 45 is not a multiple of 7, since neither 56 nor 67 is a multiple of 5, 56 × 67 is not a multiple of 5, and, similarly, 67 × 78 is not a multiple of 5.

This rules out the options A, B, D and E. In the context of the JMC this is enough for us to be able to conclude that option C is the correct answer.

However, for a full solution, we would need to check directly that 45 × 56 is divisible by all the integers from 1 to 10. This is straightforward.

Every integer is divisible by 1. Since 45 is divisible by 3, 5 and 9, so also is 45 × 56. Since 56 is divisible by 2, 4, 7 and 8, so also is 45 × 56. Since 45 × 56 is divisible both by 2 and by 3, which have no common factor, it is also divisible by 6. Since 45 × 56 is divisible by both 2 and by 5, which have no common factor, it is also divisible by 10.

Therefore 45 × 56 is divisible by all the integers from 1 to 10, inclusive.

5.

4 marks

The diagram shows a pyramid made up of 30 cubes, each measuring 1m × 1m × 1 m.

What is the total surface area of the whole pyramid (including its base)?

A) 30 m²
B) 62 m²
C) 72 m²
D) 152 m²
E) 180 m²
(Not answered)

The surface area that can be seen by looking up from below the pyramid is that of 4 × 4 = 16 squares each measuring 1m × 1 m. So the surface area of the base is 16 m².

The view looking down from above the pyramid is shown in the diagram. The surface area that can be seen is made up of a complete 1m × 1m square, some three-quarter squares and some half squares. Without adding these up, we can see that the surface area is also 16 m².

The view from each of the four sides is made up of a pyramid of 10 squares whose surface area is 10 m².

Therefore, the total surface area is 2 × 16 m² + 4 × 10 m² = 72 m².

6.

5 marks

The four straight lines in the diagram are such that VU=VW. The sizes of ∠UXZ, ∠VYZ and ∠VZX are x°,y°and z°.

Which of the following equations gives x in terms of y and z?

A) x=y−z
B) x=180−y−z
C) x=y−z2
D) x=y+z−90
E) x=y−z2
(Not answered)

Because VU=VW, the triangle VUW is isosceles and so ∠VUW and ∠VWU are equal.

Because they are vertically opposite ∠VWU and ∠YWX are equal.

Therefore ∠VUW, ∠VWU and ∠YWX are all equal. We let the size of these three angles be t°, as marked on the diagram.

We now apply the Exterior Angle Theorem in turn to the exterior angle, ∠VUW of triangle UZX and the exterior angle ∠ZYW of triangle WYX. This gives t=z+x and y=t+x.

Therefore, using the first equation to substitute z+x for t in the second equation, we have y=z+x+x=z+2x.

Hence, by rearranging the last equation, 2x=y−z.

By dividing both sides of the last equation by 2, we conclude that x=y−z2.

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.