Parallelogram 26 • Level 3 • 1 Mar 2024Rattlebacks
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.
(If you have problems watching the video, right click to open it in a new window)Correct Solution: 176
M62 + 39 miles + 45 minutes + 30 miles = 176.
2. Junior Maths Challenge Problem (UKMT)
2.1 A pattern that repeats every six symbols starts as shown below:
♥ ♣ ♥ ♦ ♥ ♠ ♥ ♣ ♥ ♦ ♥ ♠ ...
Which is the 1001st symbol?
- ♥
- ♠
- ♦
- ♣
- (Not answered)
Our pattern repeats after six symbols, so to find the 1,001st symbol we need to work out: 1,001 ÷ 6 = 166 remainder 5.
So, the 6-symbol pattern repeats itself 166 times, and the 5th symbol will sit in the 1,001st position. So the answer is the heart symbol, ♥.
3. Junior Maths Challenge Problem (UKMT)
3.1. One day Beth, Carolyn and George read a book of 240 pages. They take it in turns to read a page, always in the order Beth, Carolyn and George.
The next day their cousin Sam comes to stay and they read the book again, taking it in turns to read a page, always in the order Sam, George, Carolyn and Beth.
How many pages are read by the same person on both days?
Correct Solution: 60
Beth used to read every 3rd page
Now Beth reads every 4th page, ie 4, 8, 12, 16, 20, 24, 28,...
Beth read 1 in 3 of the original pages, and then Beth reads 1 in 4 of them again.
So, if she read
The same will be true for Carolyn and George, so 3 × 20 pages = 60 pages will be read by the same person when the book is re-read with the help of Sam.
4. How does the sun shine?
Derek Muller from YouTube’s Veritasium channel asks people to explain the mechanism that is responsible for the sun shining. Have a think yourself before you watch Derek explain the physics behind sunshine.
(If you have problems watching the video, right click to open it in a new window)5. Junior Maths Challenge UKMT bonus
As the Junior Maths Challenge is now not far away, here is a bonus (and particularly tough) JMC question to stretch your brain.
5.1. The year 2010 belongs to a special sequence of twenty-five consecutive years: each number from 1988 to 2012 contains a repeated digit.
Each of the following belongs to a sequence of consecutive years, where each number in the sequence contains at least one repeated digit.
Which of them belongs to the next such sequence of at least twenty years?
- 2099
- 2120
- 2199
- 2989
- 3299
- (Not answered)
The only way to have a decade of consecutive years with a repeated digit is if the tens digit and the hundreds or thousands digit are the same (as in the decade 1990s).
In the 21st Century, to obtain a sequence of two years or more then either a 2 or a 0 must be repeated in each year, or the sequence include years such as 2011, 2033, 2044 etc.
So the only sequence after that mentioned in the question will be from 2020 to 2030, but this is too short.
In the 22nd Century, either a 2 or a 1 must be repeated. The first such sequence is 2110 to 2129 which does include 20 years, one of which is 2120.
6. Rattleback
Watch this video of one of the most mysterious objects in the universe. It’s called a rattleback.
(If you have problems watching the video, right click to open it in a new window)7. Decimal operations
7.1 What is 8 – 4.563?
Correct Solution: 3.437
7.2 What is half of 1.01?
Correct Solution: 0.505
7.3 What is 0.8 × 20?
Correct Solution: 16
7.4 What is 0.1 + 0.2 + 0.3 + 0.4 + 0.5 + 0.6 + 0.7 + 0.8 + 0.9?
Correct Solution: 4.5
7.5 What is 0.9 – 0.8 – 0.7 – 0.6 – 0.5 – 0.4 – 0.3 – 0.2 – 0.1?
Correct Solution: -2.7
7.6 What is 1 – 0.9 – 0.08 – 0.007?
Correct Solution: 0.013
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.