Parallelogram 11 • Level 2 • 16 Nov 2023Mercury Trivia

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. More stuff about 1,729

Last week, I told you about the Indian genius Ramanujan, and his revelation concerning the number 1,729, which can be split into two cubic numbers:

1,729 = 1³ + 12³
1,729 = 9³ + 10³

1,729 is also a Harshad number, which means you can add up the digits (1 + 7 + 2 + 9 = 19) and their sum is a divisor of 1,729 (19 × 91 = 1,729).

Harshad means “giver of joy” in Sanskrit, an ancient Indian language.

2 marks

1.1. Which of these numbers is a Harshad number?

51
52
53
54
55
(Not answered)

The digits of 54 add up to 9, and 6 × 9 = 54.

3 marks

1.2. Which of these numbers is a Harshad number?

115
116
117
118
119
(Not answered)

The digits of 117 add up to 9, and 13 × 9 = 117.

3 marks

1.3. 1,729 is a special type of Harshad number because its digits add up to 19, and we can reverse these digits to get 91... and 19 × 91 = 1,729.

Only four other numbers have this peculiar property. The first number is 1, which is quite boring. The second number is 81, which is still a bit boring. But the third number 1,45X is a 4-digit number, where I have replaced the final digit with X. What is the value of X? Of course, it has to be between 0 and 9.

Correct Solution: 8

The digits of 1,458 add up to 18.

Reverse the digits and we get 81.

18 × 81 = 1,458.

2. Days on Mercury

Watch this short video about the rotation and orbit of Mercury and answer the questions below.

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

2 marks

2.1 How long is a day on Mercury?

About 8 seconds.
About 8 minutes.
About 8 hours.
About 8 Earth days.
About 8 Earth weeks.
(Not answered)

The video says that a day on Mercury is 59 days, which is roughly 8 weeks.

3. Junior Maths Challenge Problem (UKMT)

2 marks

3.1 In my row in the theatre the seats are numbered consecutively from T1 to T50. I am sitting in seat T17 and you are sitting in seat T39. How many seats are there between us?

23
22
21
20
19
(Not answered)

You could just count the seats on your fingers, or you could take a more mathematical approach.

The seats between us are numbered from T18 up to T38. So the seats between us are the 38 seats from T1 up to T38, not including the 17 seats from T1 up to T17.

So there are 38 − 17 = 21 seats between us.

2 marks

3.2 JMC follow-up question How many integers are there between 100 and 200 (not including either 100 or 200)?

Correct Solution: 99

2 marks

3.3 JMC follow-up question Suppose that k is an integer greater than 27, and that there are 50 integers between 27 and k, not including 27 and k. What is the value of k?

Correct Solution: 78

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.