Parallelogram 10 • Level 2 • 9 Nov 2023Ramanujan

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. Ramanujan and 1,729

Last week, I told you about the Indian genius Ramanujan. Here is one of the most famous stories about Ramanujan, told by his friend and fellow mathematician G. H. Hardy.

I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavourable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."

What did Ramanujan mean? He was explaining that 1,729 can be split into two cubic numbers:

1,729 = 1³ + 12³

That’s cool! But it can also be split into two other cubic numbers:

1,729 = 9³ + 10³

This is a very rare property, and 1,729 is the smallest number that can be split into two cubes in two different ways. And Ramanujan’s brain seemed to have an amazing ability to work out this sort of complex maths.

One of my favourite TV shows is Futurama. Many of the writers are mathematicians, and I met them a few years ago when I was writing “The Simpsons and Their Mathematical Secrets” (which has three chapters about Futurama). Because the writers admire Ramanujan, they featured 1,729 in three different episodes. Here are some stills from those episodes.

Another episode features the number 87,539,319 on the top of a taxi.

This is because 87,539,319 can also be split into two cubes, but this time it can be split into cubes in THREE different ways. For example, here is one split:

87,539,319 = 167³ + 436³

2 marks

1.1. Here is a second way to split 87,539,319 into two cubes. What is the missing number?

87,539,319 = 228³ + _ _ _ ³

Correct Solution: 423

2 marks

1.2. Here is a third way to split 87,539,319 into two cubes. What is the missing number?

87,539,319 = 255³ + _ _ _ ³

Correct Solution: 414

2. Negative Cubes

Earlier, I said that 1,729 was the smallest number that you can split into two cubes in two different ways, but that is only true if we assume that the cubes have to be positive.

If we allow negative numbers, there is a smaller number, 91, which can be split into two cubes in two different ways.

91 = 4³ + 3³
91 = 6³ + x³

2 marks

2.1 What is the value of x?

-1
-2
-3
-4
-5
(Not answered)

91 = 63 + (-5)³ = 216 – 125

3. More Futurama Maths

2 marks

3.1 In the Futurama episode “Rebirth”, there is a nightclub called “Studio 1².2¹.3³”.

The dots mean multiplication, so you could write this as “Studio 1² × 2¹ × 3³”.

It is named after a real New York nightclub from the 1970s and 1980s. What was that nightclub called?

Studio 6
Studio 36
Studio 54
Studio 99
Studio 216
(Not answered)

2 marks

3.2 James Cook, the British explorer, was born in the year 1³ × 3³ × 4³. What year was that?

Correct Solution: 1728

1³ × 3³ × 4³ = 1 × 27 × 64 = 1,728.

So Cook was born in 1728.

4. Junior Maths Challenge Problem (UKMT)

2 marks

4.1 How many minutes are there in 112 of a day?

240
120
60
30
15
(Not answered)

There are 24 hours in a day. So in 112 of a day the number of hours equals 112×24=2.

In 1 hour there are 60 minutes. Hence the number of minutes in 2 hours is 2×60=120.

So there are 120 minutes in 112 of a day.

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.