Parallelogram 33 • Level 2 • 18 Apr 2024A Matter Of Factorial!
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. Musical Challenge
1.1. This clip is from University Challenge on BBC2 – listen carefully and add up all the numbers in the song, then write the total in the box below.
(If you have problems watching the video, right click to open it in a new window)Correct Solution: 75
5, 4, 3, 2, 1 repeated five times, i.e.:
1.2. What is the total of all the numbers in the song at the beginning of this clip?
(If you have problems watching the video, right click to open it in a new window)Correct Solution: 44
“
Note: The questioner, Jeremy Paxman, gives a different and larger answer, because he played a longer clip with more numbers.
2. Junior Maths Challenge Problem (UKMT)
2.1 This is an extension problem based on an old Junior Maths Challenge question. What is the smallest ten-digit positive integer which has ten different digits?
Correct Solution: 1023456789
The obvious strategy is to put the smallest number on the left, as it represents billions, then the next smallest number and so on until the biggest number is placed in the units column. However, if we put zero in the billions column, then we have a 9-digit number instead of a 10-digit number, so we have to swap the ideal order of the zero and the one.
3. Junior Maths Challenge Problem (UKMT)
3.1 Find all the positive integers greater than 1 and less than 40 which have exactly one factor other than 1 and itself. What is the sum of these integers?
Correct Solution: 38
The numbers are 4, 9 and 25. So the sum is 38. These numbers are the squares of prime numbers. If a number only has one factor other than one and itself then it must be a repeated factor (as all factors come in pairs)... so the number must be square.
4. Margaret Hamilton shoots for the moon
A couple of weeks ago, we showed you a picture of the amazing software engineer Margaret Hamilton, which showed a print out of the computer code she wrote for the Apollo moon mission. The stack of paper was taller than her. Here is a video in which she talks about her incredible career – as you watch it, think about the following question:
4.1 Which of the following job titles did Margaret Hamilton invent?
- Computer programmer
- Computer scientist
- Software engineer
- Software developer
- Coder
- (Not answered)
By the way, if you want to learn a bit more about exactly what Margaret did to save the Apollo moon project, and make us all safer, then watch the video in the additional stuff section.
5. Elemental States
Following on from last week’s Venn diagram, here is a new one.
The elements of the Periodic Table are represented by 1, 2 or 3 letters, and the states of the USA are represented by 2 letters. Sometimes they coincide, so they sit in the intersection of the Venn diagram. The diagram above is missing six letter pairs. Where should they sit? You might need to google the Periodic Table and a list of abbreviations for US states.
5.1. Which of these states should sit in the intersection?
- Connecticut (CT)
- Florida (FL)
- Ohio (OH)
- (Not answered)
Florida is shortened to FL, which is also short for Flerovium.
5.2. Which of these elements should sit in the intersection?
- Lithium
- Cadmium
- Neon
- (Not answered)
Neon is shortened to Ne, which is also short for Nebraska.
6. Even more about factorials
A short while ago, we asked: “How many ways can we re-arrange the English alphabet of 26 letters?” Take a look back at Parallelogram (#7) if you need to remind yourself what we did.
In short, there are 26! ways to re-arrange 26 letters, where 26! means 26 × 25 × 24 …. × 1.
The (!) is called "factorial", and can be applied to any number, e.g., 4! = 4 × 3 × 2 × 1 = 24.
6.1. 403,291,461,126,605,635,584,000,000 = 26!
It ends in six zeroes because 26! involves multiplying by all the numbers from 26 down to 1, which includes multiplying by numbers like:
- 15 & 2 which adds 1 zero
- 20 & 5 which adds 2 zeroes
- 25 & 4 which adds 2 zeroes
Which number provides the last zero?
Correct Solution: 10
Multiplying by 10 provides the final zero.
6.2. What is the answer to (10! ÷ 9!)?
Correct Solution: 10
Imagine all the numbers you multiply together to make 10!, and then imagine all the numbers you multiply together to get 9!, and then cancel common numbers. What are you left with?
If you imagine all the numbers you multiply together to make 10!, and then imagine all the numbers you multiply together to get 9!, and then cancel common numbers, you are left with just 10.
6.3. What is the answer to (8! ÷ 5!)?
Correct Solution: 336
So,
Another way to think about it is to say:
So,
6.4. How long is 10! seconds?
Even if you get the answer right, make sure you come back and check the answer, as there is a very neat way of doing this.
- 6 hours
- 6 days
- 6 weeks
- 6 months
- 6 years
- (Not answered)
10! seconds = 3,628,800 seconds.
If you ÷3,600 then you get 1,008 hours, and if you ÷24 then you get 42 days, and if you ÷7 then you get 6 weeks.
Another way to think about this is to take the elements of 10! and observe how they can be paired up conveniently to turn seconds into weeks.
10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1
- Multiplying by the 10 and the 6 gives you 60, which turns seconds into minutes.
- Multiplying by 5, then 4, then 3 also gives you 60, which turns those minutes into hours.
- If we split 9 into 3 × 3, and then take one of those 3s and multiply it by the 8, we get 24, which turns those hours into days.
- Multiplying by the 7 turns those days into weeks.
- And then we are only left with the 2 and the other 3 (that we got from the 9), which gives us 6 weeks.
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
- This “Great Minds” video explains a bit more about Margaret Hamilton's software breakthrough that saved the Apollo moon project.