Parallelogram 35 Level 2 2 May 2024Hidden Figures

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Noun: Parallelogram Pronunciation: /ˌparəˈlɛləɡram/

  1. 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. Hidden Figures – another maths movie

Last week we looked at the film “Good Will Hunting” and explored one of the mathematical problems that appears in it. I also posed a question about another film: “What is the title of the film that celebrates the role of the pioneering women mathematicians at NASA in the 1960s?”

Most of you correctly identified the film as Hidden Figures. Let’s start by watching the trailer.

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

1 mark

1.1 Just to check that you were concentrating on that trailer, for how many hours did the children say their mother had been gone?

  • 3 hours
  • 30 hours
  • 300 hours
  • 3,000 hours
  • 30,000 hours
  • (Not answered)

The answer is 300 hours, which the mother immediately converts into 12½ days.

The film tells the story of three brilliant black women who helped send some of the first American astronauts into space. These are just a few of the many women who played a crucial role in the space race. It is a terrific film, which received three Oscar nominations and which is both inspiring and exciting. Well worth watching if your parents or teacher can get hold of the DVD.

2. Moore's Law

The women were known as “computers”, because the original term computer referred to a person who did calculations. Only later did the term computer become more associated with machines. Increasingly, machine computers at NASA took over from human computers, and many of the women became some of the first computer programmers and operators. Those early computers were incredibly difficult to operate and relatively slow, certainly compared to today’s computers, as discussed in a blogpost on the ZME Science website:

…back on Earth at the Goddard Space Flight Center thousands of flight technicians and computer experts employed the IBM System/360 Model 75s mainframe computer to make independent computations and maintain communication between Earth and lunar landers. These computers cost $3.5 million a piece and were the size of a car. Each could perform several hundred thousand addition operations per second, and their total memory capacity was in the megabyte range. Programs were developed for the 75s that monitored the spacecraft’s environmental data and astronauts’ health, which were at the time the most complex software ever developed. Today, however, even a simple USB stick or WiFi router is more powerful, let alone an iPhone. The iPhone 6 uses an Apple-designed 64 bit Cortex A8 ARM architecture composed of approximately 1.6 billion transistors. It operates at 1.4 GHZ and can process instructions at a rate of approximately 1.2 instructions every cycle in each of its 2 cores. That’s 3.36 billion instructions per second. Put simply, the iPhone 6’s clock is 32,600 times faster than the best Apollo era computers and could perform instructions 120,000,000 times faster. You wouldn’t be wrong in saying an iPhone could be used to guide 120,000,000 Apollo era spacecraft to the moon, all at the same time.

Transistors are important components of computer circuitry, and the more transistors a computer circuit has, the more calculations it can do at one time.

This remarkable improvement in computing is known as Moore’s Law. Wikipedia says:

Moore's law is the observation that the number of transistors in a dense integrated circuit doubles approximately every two years. The observation is named after Gordon Moore, the co-founder of Fairchild Semiconductor and Intel, whose 1965 paper described a doubling every year in the number of components per integrated circuit, and projected this rate of growth would continue for at least another decade. In 1975, looking forward to the next decade, he revised the forecast to doubling every two years

You might not find this doubling of the number of transistors on a computer circuit particularly impressive, but it is actually mind-blowing. It has literally changed civilisation.

When something doubles, and doubles again, and again and again … then the results are phenomenal. It is a type of increase known as exponential growth. The graph below shows Moore’s Law between 1971 and 2011. Take note of the funny scale on the y-axis, which increases by a factor of 10 at each step. So, in 40 years, the number of transistors on a microprocessor increased from 2,300 to 2,600,000,000, which is an increase by a factor of more than a million.

1 mark

2.1 Look at the number of transistors on the 80486 microprocessor (~1990) and the number of transistors on the AMD K8 microprocessor (~2003). By what factor did the number of transistors increase in just 12 years?

  • 2
  • 10
  • 50
  • 100
  • 500
  • (Not answered)

The 80486 had roughly 1,000,000 transistors, and the AMD K8 had about 100,000,000 transistors, so there was an increase by a factor of 100.

3. The Chess Problem

There is a very old story that demonstrates the power of exponential growth and Moore’s Law, but it relates to chess and rice, not computers and transistors.

In India, or Persia or somewhere east of the English Channel, the King or Maharajah or Sultan wanted to reward a mathematician, because he had just invented chess. The King offered a chest of gold coins or a barrel of diamonds, but instead the mathematician asked for a grain of rice on the first square of a chessboard, 2 grains on the second square, 4 grains on the third square and so on, doubling the number of grains of rice on each square until the 64th square.

The King laughed at what seemed like a trivial reward, but he did not laugh for long.

Grab a piece of paper and work out the number of grains of rice on each of the first 32 squares. You will almost certainly need a calculator, and make sure you check your answers. I have given you a headstart by listing the answers for the first 10 squares.

1 mark

3.1 How many grains were on the 15th square?

Correct Solution: 16,384

If the tenth square has 512 grains (from the table), then you should have worked out (11th = 1,024), (12th, 2,048), (13th, 4,096), (14th, 8,192) and (15th, 16,384). The answer is 16,384.

1 mark

3.2 How many grains were on the 20th square?

Correct Solution: 524,288

The answer is 524,288.

1 mark

3.3. How many grains were on the 32nd square?

Correct Solution: 2,147,483,648

The answer is 2,147,483,648.

You may have worked out that the number of grains on the nth square is 2n1. So, instead of doubling over and over again, you could use the xy button on your calculator.

If you carried on the calculation for the full 64 squares, then you should arrive at 9,223,372,036,854,775,808 grains on the 64th square.

If you want to add up all the grains on all the squares, then the total is double the number of grains on the final square minus 1. (You can check this by looking at the table above. The total number of grains for the first, say, five squares is 1 + 2 + 4 + 8 + 16 = 31, which is the same as [2 × 16] – 1 = 31.)

This means that the total number of grains on all 64 squares is: (2 × 9,223,372,036,854,775,808) – 1 = 18,446,744,073,709,551,615.

That is about 100 billion tonnes of rice, which is about hundred times more rice than the whole Earth grows each year. That is the power of exponential growth.

2 marks

3.4 In another version of the story, the King gets revenge by saying that the mathematician can only have the grains of rice if he can count them. If the mathematician counts at the rate of 1 grain per second, how long will it take him to collect his reward of 18,446,744,073,709,551,615 grains? Which of these answers is closest?

  • 400 years
  • 500 million years
  • 600 billion years
  • 700 trillion years
  • 800 quadrillion years
  • (Not answered)

To count 18,446,744,073,709,551,615 grains will take 18,446,744,073,709,551,615 seconds, which is roughly 18×1018 sec = 3×1017 mins = 5×1015 hours = 2.1×1014 days = 5.8×1011 years = 580 billion years, so the answer is roughly 600 billion years.

4. Summer salaries

You get a summer job that lasts for 6 weeks. You have two salary options:

Option A – £100/week.
Option B – a salary that doubles each week, starting at £10/week.

1 mark

4.1 How much would option A deliver in total after 6 weeks? (Note: no need to include £ in your answer, as we have done that for you!)

£ Correct Solution: 600

Obviously option A delivers 6 x £100 = £600.

1 mark

4.2 How much would option B deliver in total after 6 weeks? (Note: no need to include £ in your answer, as we have done that for you!)

£ Correct Solution: 630

Option B delivers £10 + £20 + £40 + £80 + £160 + £320 = £630.

So option B is preferable.

5. Parallel emails

If you use Parallel on an email address from your school, some school IT systems prevent you from receiving the emails we send out to remind you when a new Parallelogram is released, or when we have another exciting and nerdy maths thing to tell you about. To avoid this, you could give us a different email address we can contact you on - either a personal email address, or one for you parents or guardian. If you'd like to do that, you can put it in this form, but please ask a parent first if you are not yet 13 years old.

1 mark

5.1 If you do want to give us an alternate email address, be sure to click the link above (if you haven’t already). Either way, here's a free mark, just for being a fan of Parallel.

  • Thank you!
  • (Not answered)

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.

Additional Stuff

If you want to find out more about the topics in this week’s Parallel Challenge, then I recommend:

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


Moore's Law graph taken from this Wikipedia page