Parallelogram 47 • Level 3 • 24 Jul 2025Pat's village visit

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.

1 mark

1.1 In this diagram, what is the value of x?

16
36
64
100
144
(Not answered)

The unmarked interior angle on the right of the triangle =360−324°=36°. So, by the exterior angle theorem, x=100−36=64.

2.

2 marks

2.1. In a triangle which has the angles x°, y°, z° the mean of y and z is x.

What is the value of x?

90
80
70
60
50
(Not answered)

If the mean of y and z is x, then y+z=2x. So the sum of the interior angles of the triangle is x+y+z°=3x°. So 3x=180, that is x=60.

3.

3 marks

3.1 The letters J, M, C represent three different non-zero digits, where JJ + MM + CC = JMC.

What is the value of J+M+C?

19
18
17
16
15
(Not answered)

The hundreds column shows us that J=1 or 2. [We can’t carry more than 2 from the units to the tens; and 2 plus the biggest feasible values 7, 8, 9 for the three letters is only 26.]

The units column shows that J+M is a multiple of 10 and it can’t be 0 (or else J+M=0); so J+M=10 and M=9 or 8 respectively.

Also, the sum of the units column is 10+C, so there is exactly 1 to carry to the tens column. The tens column now tells us that J+C+1=10J. So J=2 is not possible and therefore J=1, C=8 and M=9.

4.

4 marks

4.1 Pat needs to travel down every one of the roads shown at least once, starting and finishing at home.

What is the smallest number of the five villages that Pat will have to visit more than once?

1
2
3
4
5
(Not answered)

Villages which have more than two roads leading to them (or from them) must all be visited more than once as a single visit will involve at most two roads. So Bentonville, Pencaster and Wytham must all be visited more than once.

The route Home, Bentonville, Greendale, Wytham, Bentonville, Pencaster, Home, Wytham, Horndale, Pencaster, Home starts and finishes at Home and visits both Greendale and Horndale exactly once so the minimum number of villages is three.

5.

5 marks

5.1 Kiran writes down six different prime numbers, p, q, r, s, t, u, all less than 20, such that p+q=r+s=t+u. What is the value of p+q?

16
18
20
22
24
(Not answered)

Show Hint (–2 mark)

2 mark

The primes less than 20 are 2, 3, 5, 7, 11, 13, 17, 19. The number 2 is the only even prime, so you cannot use it, because otherwise one sum will be (even + odd) and the two others will be (odd + odd) (

The prime numbers less than 20 are 2, 3, 5, 7, 11, 13, 17, 19. It is not possible for 2 to be one of the six numbers Kiran wrote down, since that would give one of the pairs an odd sum, whereas both of the other pairs would add up to an even number.

The sum of the remaining 7 primes is 75 which is a multiple of 3. The sum of the six primes making up the three pairs must also be a multiple of 3 since each pair has the same total.

So the odd prime not used in the six pairs must be a multiple of 3 too. Therefore 3 is the odd prime not used. So each pair totals 72 ÷ 3, that is 24, and the pairs are 5 + 19, 7 + 17, and 11 + 13.

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.