Parallelogram 13 • Level 3 • 30 Nov 2023Prime Time Level 3
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. Primes against the clock
Below is a simple website that tests how well you know your prime numbers and how quickly you can apply some prime tests. It will show you a series of numbers and you have to decide whether or not they are prime. If you do not know straight away, then there are some easy checks that you can apply to filter out the non-primes:
- If it is even, then it is divisible by 2.
- If the digits add to 3, 6, or 9, then it is divisible by 3.
- If it ends with a 0 or 5, then it is divisible by 5.
- Remember, 1 is NOT a prime.
Just visit the Is this a prime? website and see how many numbers you can identify as primes or non-primes.
1.1. When you get a score bigger than 15, what does “Game over” page say?
- You correctly sorted x numbers.
- Well done – you correctly sorted x numbers!
- You’re a prime superstar for sorting x numbers!
- Bronze level – you correctly sorted x numbers!
- Sorting x numbers is great, but now aim for 20 numbers.
- (Not answered)
2. Don’t celebrate too early
Watch and learn a lesson from this video.
Don’t celebrate too early. Check your work. Don’t leave a job half done.
(If you have problems watching the video, right click to open it in a new window)3. Junior Maths Challenge Problem (UKMT)
3.1 The diagram shows a square divided into strips of equal width.
Three strips are black and two are grey.
What fraction of the perimeter of the square is grey?
1 5 1 4 4 25 1 3 2 5 - (Not answered)
Two sides of the square are wholly black, and
So the length of the perimeter that is grey is equal to 2 ×
The length of the perimeter is 4 times the length of one side.
So the fraction of the perimeter that is grey is
3.2 A square is divided into seven strips of equal width. Four of the strips are black and three are grey.
What fraction of the perimeter of the square is grey?
1 28 3 14 1 7 3 7 11 14 - (Not answered)
Two sides of the square are wholly black, and
So the length of the perimeter that is grey is equal to
The length of the perimeter is 4 times the length of one side.
So the fraction of the perimeter that is grey is
3.3 A square is divided into an odd number of strips of equal width.
The strips are alternately black and grey, with one more black strip than grey strips. The fraction of the perimeter that is grey is
How many strips are there?
Correct Solution: 25
We don’t know the number of strips, but we could call it
We know that 2 sides of the square have no grey, because there is no grey strip that touches any corner or which stretches along any side.
We know that along the other two sides, the fraction of grey on the perimeter is
So, if we can say the square has unit length 1, then the fraction of grey along the perimeter is:
If you rearrange and simplify this equation, then NS = 25.
3.4 Imagine that the square has trillion-and-one black strips and a trillion grey strips What fraction (give or take a bit) of the perimeter of the square is grey?
0 1 16 1 5 1 4 1 2 - (Not answered)
Two sides of the square will be black, as previously. The other two sides will effectively be half black and half grey, because the difference between a trillion and a trillion-and-one is insignificant.
Two sides of the square will be black, as previously. The other two sides will effectively be half black and half grey, because the difference between a trillion and a trillion-and-one is insignificant. So, the fraction of the perimeter that is grey is ( ¼ + ¼ ) ÷ 4 = ¼.
4. Junior Maths Challenge Problem (UKMT)
4.1 What is 2014 − 4102?
- -2012
- -2088
- -2092
- -2098
- -2112
- (Not answered)
It is easier to subtract the smaller number, 2014, from the larger number, 4102. Now
4102 − 2014 = 2088
and so
2014 − 4102 = −2088.
5. 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:
5.1 What is the last name of the mathematician from the 1st century CE created detailed maps of the Earth?
Correct Solution: Ptolemy
5.2 What is the last name of the mathematician from the 13th century CE (the 1200s) first developed a method which is now named after a British mathematician, William George Horner?
Correct Solution: Jiushao
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.