Parallelogram 45 • Level 2 • 10 Jul 2025Trains and clocks

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.
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. My train was scheduled to leave at 17:40 and to arrive at 18:20. However, it started five minutes late and the journey then took 42 minutes. At what time did I arrive?

18:21
18:23
18:25
18:27
18:29
(Not answered)

The train arrived 5 + 42 = 47 minutes after 17:40, that is at 18:27.

2.

4 marks

2.1. A station clock shows each digit by illuminating up to seven bars in a display. For example, the displays for 1, 6, 4 and 9 are shown.

When all the digits from 0 to 9 are shown in turn, which bar is used least?

(Not answered)

The number of times each bar is used is: A 4; B 6; C 8; D 7; E 7.

3.

2 marks

3.1. Which of the following numbers is three less than a multiple of 5 and three more than a multiple of 6?

12
17
21
22
27
(Not answered)

Of the options given, only 27, which is three less than a multiple of 5, namely 30, and three more than a multiple of 6, namely 24, has both of the properties in the question.

4.

3 marks

4.1. The equilateral triangle XYZ is fixed in position. Two of the four small triangles are to be painted black and the other two are to be painted white. In how many different ways can this be done?

3
4
5
6
More than 6
(Not answered)

If the top triangle is painted black, then any one of the three remaining triangles may also be painted black. Similarly, if the top triangle is painted white, then any one of the three remaining triangles may also be painted white. So there are six different ways.

5.

5 marks

5.1 Points P and Q have coordinates (1, 4) and (1, −2) respectively. For which of the following possible coordinates of point R would triangle PQR not be isosceles?

(−5, 4)
(7, 1)
(−6, 1)
(−6, −2)
(7, −2)
(Not answered)

If R is (−5, 4) then PQ=PR=6.

If R is (7, 1) or if R is (−6, 1) then R lies on the perpendicular bisector of PQ (the line Y=1), so in both cases PR=QR.

If R is (7, −2), then QP=QR=6.

However if R is (−6, −2), then PQ=6, QR=7 and PR>7, so triangle PQR is scalene.

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.