Parallelogram 6 • Level 3 • 10 Oct 2024Black Holes
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. Chicken mathematics
1.1. A farmer has 3 hens, which lay 3 eggs in total in 3 days.
If the farmer buys 9 more hens, how many eggs will the 12 chickens lay in 12 days?
Correct Solution: 48
If 3 chickens lay 3 eggs in 3 days, then it is tempting to say that 12 chickens lay 12 eggs in 12 days, but we can’t just swap 3 for 12 throughout. There is no justification for that.
Instead, this tells us that 1 chicken lays 1 egg in 3 days.
Therefore, 1 chicken lays 4 eggs in 12 days.
Therefore, 12 chickens lay 48 eggs in 12 days.
(Another way to think about this is to say that we are multiplying the number chickens by 4 and the number of days by 4, so we should have (4 × 4) or 16 times as many eggs. And 3 × 16 = 48 eggs.
2. Theory of Everything
“Theory of Everything” is a brilliant movie about the life of the cosmologist Professor Stephen Hawking, with the lead male actor Eddie Redmayne winning an Oscar for his performance. Here is the film’s trailer:
(If you have problems watching the video, right click to open it in a new window)2.1 Professor Hawking once said: “We are just an advanced breed of monkeys on a minor planet of a very average star. But we can ____ _____ ____. That makes us something very special.” What are the missing words? Have a guess, but also use Google to find the right answer.
- create new inventions
- think about God
- understand the Universe
- dream and imagine
- do clever mathematics
- (Not answered)
Many of Professor Hawking’s greatest ideas concerned the nature of black holes. This clip tells you a bit more about black holes and the astrophysicist who first suggested that they might exist.
(If you have problems watching the video, right click to open it in a new window)2.2 After suggesting the idea of a black hole, how long did Chandrasekhar have to wait before he received a Nobel Prize.
- Days
- Weeks
- Months
- Years
- Decades
- (Not answered)
The video mentioned that he suggested the idea in 1931, but Chandrasekhar did not receive a Nobel Prize until 1983. This is often the case with radical ideas, because it takes time to find the evidence that proves that the idea is correct.
3. Junior Maths Challenge Problem (UKMT)
3.1 For Beatrix's latest art installation, she has fixed a 2 × 2 square sheet of steel to a wall. She has two 1 × 2 magnetic tiles, both of which she attaches to the steel sheet, in any orientation, so that none of the sheet is visible and the line separating the two tiles cannot be seen. As shown alongside, one tile has one black cell and one grey cell; the other tile has one black cell and one spotted cell.
How many different looking 2 × 2 installations can Beatrix obtain?
N.B. If one arrangement can be transformed into another by rotating the installation, then these count as two separate installations.
- 4
- 8
- 12
- 14
- 24
- (Not answered)
One approach is to write down all the different ways that you could arrange the tiles, but you have to be careful to (a) draw them systematically, so that you don’t miss out any arrangements, and (b) cross out any duplicates. I did this myself and found 12 distinct installations.
Here is another approach to achieving the same answer.
In the 2 × 2 square sheet, there are 4 positions in which Beatrix could place the spotted cell. For each position of the spotted cell, there remain 3 positions where she could place the grey cell. Once she has placed these, she has no choice for the two black cells. Hence there are 4 × 3 = 12 possible installations.
It can be checked that Beatrix can make each of these installations by positioning the two tiles appropriately. [She can create each of the installations in which the black squares occupy the diagonal positions in two ways, and all the other installations in just one way.]
So Beatrix can create 12 different looking installations.
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.