Parallelogram 1 • Level 4 • 22 Aug 2024Sing-A-Long-A Googol

Noun: Parallelogram Pronunciation: /ˌparəˈlɛləɡram/

a portmanteau word combining parallel and telegram. A message sent each week by the Parallel Project to bright young mathematicians.

Welcome to the first of our Parallelograms, a collection of mathematical challenges designed to stretch your brain and make your neurons more squiggly.

These challenges are a random walk through the mysteries of mathematics. Be prepared to encounter all sorts of weird ideas, including some questions that have nothing to do with mathematics.

Tackle each Parallelogram in one go. Don’t get distracted.
When you finish, remember to hit the SUBMIT button.
You can new earn reward points and badges – find out more here.
Finish by midnight on Sunday if your whole class is doing parallelograms.

IMPORTANT – it does not really matter what score you get, because the main thing is that you think hard about the problems... and then examine the solution sheet to learn from your mistakes.

1. Googol

The mathematical term “googol” was invented in 1920 by 9-year-old Milton Sirotta, nephew of American mathematician Edward Kasner. A googol is 10¹⁰⁰, which means it is written as 1 followed by one hundred zeroes.

The fact that it can be written in such a compact form, 10¹⁰⁰, is deceiving, because it represents a phenomenonally gigantic number of mind-blowing proportions. A googol is about one hundred billion billion times bigger than the number of particles in the visible universe (10⁸⁰).

Although the company and search engine Google is spelt differently, it based its name on the huge number googol, because its ambition was to provide users with huge amounts of information.

2 marks

1.1. What is googol squared?

10¹⁰⁰
10²⁰⁰
10^10,000
10^{100 × 100}
(Not answered)

If you are squaring a number with an exponent, then you double the exponent. For example:

(3³)² = 3⁶ = 729
27² = 729

2 marks

1.2. What is √(googol)?

10¹⁰
10⁵⁰
3.16¹⁰⁰
3.16¹⁰
(Not answered)

If you are square rooting a number with an exponent, then you halve the exponent. For example:

√(5⁶) = 5³ = 125
√15,625 = 125

2. Googol song

This is Helen Arney, a physicist and geek songstress, singing about a googol… and trying her best to compose a verse about a googolplex. A googolplex = 10^googol, and it is a vast number, even compared with a googol.

Helen is accompanied by Matt Parker, who will reappear in a future Paralellograms.

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

2 marks

2.1 What is googolplex squared?

10^googol
20^googol
100^googol
10^{(2 × googol)}
(Not answered)

If you are squaring a number with an exponent, then you double the exponent.

2 marks

2.2 What is √(googolplex)?

10^googol
5^googol
10^{(0.5 × googol)}
3.16^googol
(Not answered)

If you are square rooting a number with an exponent, then you halve the exponent.

3. Intermediate Maths Challenge Problem (UKMT)

If you are a Year 9 student, then it is likely that you will be taking part in the United Kingdom Maths Trust (UKMT) competition known as the Intermediate Maths Challenge (IMC). If you do particularly well, you might earn yourself a gold, silver or bronze certificate, but you will have to work hard as you will be competing against students in Years 9, 10 and 11 from across the country.

Your teachers will help you prepare for this national maths competition, but in each week's Parallelogram we will always include one UKMT Intermediate Maths Challenge question.

(If you have already taken the JUNIOR Maths Challenge, aimed at younger students, then you will notice that these INTERMEDIATE Maths Challenge questions are generally a bit more tricky.)

2 marks

3.1 Without using a calculator, what is the value of 4.5 × 5.5 + 4.5 × 4.5?

36.5
45
50
90
100
(Not answered)

Since you are not allowed to use a calculator, it is a good idea to look for a way to avoid having to do multiplication sums. The presence of the factor 4.5 in both products provides a clue to an efficient method. If we take out this common factor, we obtain:

4.5 × 5.5 + 4.5 × 4.5 = 4.5 × (5.5 + 4.5) = 4.5 × 10 = 45.

3 marks

3.2 Without using a calculator, find the value of 123.4 × 123.4 + 123.4 × 876.6

Correct Solution: 123,400

123.4 × 123.4 + 123.4 × 876.6 = 123.4 × (123.4 + 876.6) = 123.4 × 1,000 = 123,400.

4. History of Mathematics

The terrific Mathigon website has loads of great material, including a timeline showing the great mathematicians of past centuries. Visit the timeline (click and it will open up in a new tab) and answer the three questions below. Just enter the name given in the plum box.

So, if the answer is John Napier, just enter Napier, because that is the name that appears in the plum-coloured box on the timeline:

2 marks

4.1 Which mathematician from the 19th Century has a surname which is an anagram of “Hanging tile”? Please give their surname only. (Clue: initials FN)

Correct Solution: Nightingale

2 marks

4.2 The mathematician from 4.1 was the first to use which type of chart?

Gantt Chart
Pie Chart
Bar chart
Stem and leaf diagram
(Not answered)

2 marks

4.3 Which mathematician from the 20th Century has a surname which is an anagram of “Not marbled”? Please give their surname only. (Clue: initials BM)

Correct Solution: Mandelbrot

2 marks

4.4 Which of the following is not stated as an interest of the mathematician from question 4.3?

Roughness
Chaos
Clouds
Trigonometry
(Not answered)

I hope you enjoyed the first Parallelogram of the year. There will be more on Thursday September 14th, and the week after, and the week after that. So check your email or return to the website on Thursday at 3pm.

In the meantime, you can find out your score, the answers and go through the answer sheet as soon as you hit the SUBMIT button below.

When you see your % score, this will also be your reward score. As you collect more and more points, you will collect more and more badges. Find out more by visiting the Rewards Page after you hit the SUBMIT button.

It is really important that you go through the solution sheet. Seriously important. What you got right is much less important than what you got wrong, because where you went wrong provides you with an opportunity to learn something new.

Cheerio, Simon.