Parallelogram 27 • Level 3 • 7 Mar 2024Colour Code Challenge
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. The Colour Code Challenge
Below you can see the numbers from 1 to 20 represented by coloured circles, but what is the relationship between each number and its colours? If you can work out the relationship, then you will notice that one of the numbers is linked to the wrong colours.
1.1. Which number is linked to the wrong colours? There are some hints below if you need them.
Look for patterns. Which colours are linked to which numbers?
What colour is 2, and which numbers share that colour?
Which numbers only have a single colour?
2 is orange, 4 is doubly orange, 8 is triply orange. Can you figure out why?
2 is orange, 3 is green and 6 is orange and green. Why?
Correct Solution: 19
The prime numbers - 1, 2, 3, 5, 7, 11, 13 and 17 are single colours. All of the other numbers are coloured according to their prime factors - so 4 is made up of two oranges, because the 2 is coloured orange, and 4 = 2 × 2; 18 is made up of two greens (3s) and one orange (2), because 18 = 3 × 3 × 2.
The exception is number 19, which is made up of two oranges (2s) and one green (3). However, 2 × 2 × 3 ≠ 19, and as 19 is a prime number it should be a single colour.
2. Junior Maths Challenge Problem (UKMT)
2.1 The diagram shows a pattern of 16 circles inside a square.
The central circle passes through the points where the other circles touch.
The circles divide the square into regions. How many regions are there?
- 17
- 26
- 30
- 32
- 38
- (Not answered)
Each of the five outer circles is divided into six regions, giving 30 regions in total. In addition, there is one region in the centre of the diagram and one region between the circles and the sides of the square. So, in all, there are 32 regions.
3. Junior Maths Challenge Problem (UKMT)
3.1. The diagram shows a weaver’s design for a rihlèlò, a winnowing tray
from Mozambique.
How many lines of symmetry does the design have?
- 0
- 1
- 2
- 4
- 8
- (Not answered)
The four lines of symmetry are shown in the diagram.
4. April Fools’ Day Prank
April Fools’ Day is just a few days away, so I wanted to feature this video with Professor Matthew Weathers, lecturer in the Department of Mathematics and Computer Science at Biola University in California. If you study maths or computing at university, I cannot promise that all your lectures will be as fun as this one.
(If you have problems watching the video, right click to open it in a new window)5. Felix Klein’s birthday
Felix Klein is one of the most famous mathematicians in history. You can find out more about Felix Klein in the “Additional Stuff” section, but for this question you only need to know his date of birth.
5.1. He was born on 25 April 1849, or 25/4/1849. Why is this this a remarkable birthday for a mathematician?
- Each number shares a particular property linked to addition.
- Each number shares a particular property linked to subtraction.
- Each number shares a particular property linked to multiplication.
- Each number shares a particular property linked to factorials.
- Each number shares a particular property linked to fractions.
- (Not answered)
Each of the date of birth numbers is linked to multiplication because 25/4/1849 could be written as 52/22/432 (and squaring is an operation based on multiplication).
6. The Möbius loop
Felix Klein is best known for the Klein bottle, a weird bottle (below left) that has no inside and no outside. If you are a fan of the sci-fi comedy “Futurama”, then you might remember a shelf of Klein bottles appearing in the episode “The Route of All Evil” (below right).
Although in many senses it is impossible to create a Klein bottle in our normal 3-D world, it can in theory be constructed by joining two Möbius loops. You might now be wondering, “what is a Möbius loop?”. In short, it is an object that only has one side. Every piece of paper, of course, has two sides – a top and a bottom, or maybe a back and a front. However, you can turn a piece or strip of paper into an object that has only one side! (By the way, a Möbius loop is sometimes called a Möbius strip or a Möbius band.)
Making a Möbius loop involves taking a strip of paper, forming a loop, then making half a twist in one end, and finally joining the two ends with sticky tape.
The video below shows how you can make a Möbius loop for yourself. Once you have created a couple of Möbius loops, have a go at answering the questions below.
(If you have problems watching the video, right click to open it in a new window)6.1. What happens when you cut a Möbius loop in half lengthways?
- You create zero loops.
- You create a half a loop.
- You create one loop.
- You create two loops.
- You create a mess.
- (Not answered)
6.2. What happens when you cut the Möbius loop along its length, but starting one third of the way from the edge (NOT in the middle)? Keep cutting until you get back to where you started and the cut is complete. What have you created?
- No loops, just one long strip.
- One big loop.
- Two loops of equal size.
- One small loop and one big loop.
- Three loops.
- (Not answered)
Hopefully you will have got the right answers and been shocked by them. Right or wrong, take a look at this video that examines what happens if you cut a Möbius loop a third of the way from the edge. It also looks at a couple of other cutting mysteries, all in just a couple of minutes.
(If you have problems watching the video, right click to open it in a new window)For an easy to read guide to Möbius loops and the solution to the central cutting question (6.1) visit this website on Möbius loops.
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
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This week’s Klein birthday mystery was featured in Alex Bellos’s terrific puzzle column in the Guardian. Always worth a visit if you fancy a challenge.
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This animation gives further insights into what happens when you cut a Möbius loop.