Parallelogram 18 • Level 4 • 4 Jan 2024Teapot Time
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. Disney recycling
I was fascinated when I heard that in many of Disney’s classic early animated films, the studio recycled and rehashed earlier sequences to save time on drawing new scenes. The clip below shows some of the most remarkable examples. Watch carefully and answer the question below.
(If you have problems watching the video, right click to open it in a new window)1.1. One of my favourite films as a kid – even now – is Disney’s “Robin Hood”. The dance scene is largely built by stealing sequences from other films.
Which film is NOT recycled in a dance scene from “Robin Hood”?
- The Jungle Book
- Snow White
- The Aristocats
- Lady and the Tramp
- (Not answered)
Nowadays most animated films rely on mathematics to draw and animate scenes, which means we have amazing films like “Toy Story” and “Shrek”. In fact, mathematics (and computer science) is at the heart of the animation film industry.
Below is a short video clip about the Utah teapot, which played an important role in the history of computer animation. Take a look at the video and find out how and why mathematics was used to model the teapot, and then answer the question below.
(If you have problems watching the video, right click to open it in a new window)1.2. Why was the Utah teapot such a good object for testing how to mathematically model 3D objects?
- It has concave surfaces
- It has convex surfaces
- It can cast shadows on itself
- It is not too complicated
- All of the above
- (Not answered)
2. Intermediate Maths Challenge Problem (UKMT)
2.1. The three blind mice stole a piece of cheese. In the night, the first mouse ate
Later, the second mouse ate
Between them, what fraction of the cheese did they eat?
16 27 17 27 2 3 19 27 20 27 - (Not answered)
Our first method is a direct calculation: The first mouse ate
The second mouse ate
The third mouse ate
So between them, the amount of the cheese that the three mice ate was
Our second method is to first calculate the amount of cheese that remains. Each mouse eats
So the fraction which is eaten is
3. Mouse extension problem (UKMT)
3.1. Suppose there are
Find a formula for the fraction of cheese that they eat between them.
n n n n n 1 − n − 1 n n 1 − 1 n − 1 n n − 1 n n - (Not answered)
For this question, it is easier to work out how much of the cheese is left after each stage, and then subtract the total amount eaten from 1 to find out how much has been eaten. This is already a big hint, because it probably means that the answer is (1 – something).
A crude way to do this problem is to test out the formulae given as with a simple example, or the example you have already looked at in question 2.
If
For this question, it is easier to work out how much of the cheese is NOT eaten after each stage, and then subtract the total amount NOT eaten from 1 to find out how much has been eaten. If each mouse eats
A crude way to do this problem is to test out the formulae given with a simple example, or the example you have already looked at in question 2. You will find out which formula works, without really understanding why, but at least you will probably get the right answer.
4. Find the letter
This is a terrific puzzle from puzzle master Scott Kim. You’ll be able to find out more about him and how he constructed this puzzle when you click on the solution sheet.
4.1 The figure below is a letter of the alphabet that has been cut out of paper and folded just once. It is NOT the letter L. What letter is it?
In case it helps, the letter is taken from this alphabet, with this particular font (or type-face).
If you unfold the folded bit of paper, perhaps there is another bit of paper hidden underneath?
Correct Solution: F
The obvious approach is to unfold the paper and then we have an L-shape, but rotated, so that it is almost upside-down. But we have been told that the answer is NOT L. However, it is possible that there is a piece of paper under the folded arm of paper, so when the arm is unfolded we would see another arm, revealing something that looks like a flipped F-shape. Of course, because it is just a bit of paper, we can turn it over and make a real F.
You can find out more about Scott Kim’s thoughts on this puzzle and more generally by reading this article by him, or you can visit his website.
5. Intermediate Maths Challenge Problem (UKMT)
5.1 The number 6 lies exactly halfway between
Which of the following is not halfway between a positive integer and its square?
- 3
- 10
- 15
- 21
- 30
- (Not answered)
The first approach might be to check each answer A to E. For example, D (21) is halfway between
The second approach is to realise that to fit the rule, then a number,
Rearranging this equation gives
6. A New Year reminder
As it is the start of the year, I thought it would be good just to remind you that Parallelograms are really tough, they cover topics that you might not have seen before and they are designed to confuse you. If too many people get 100%, then I will just have to make the Parallelograms tougher, weirder and more confusing. In short, if you only get 50%, don’t worry. If you get less than 50%, don’t worry. Just make sure you go through the solution sheet and try to find out why you got a question wrong. I really do mean it when I say that I am overjoyed if a student gets 25%, as long as they did their best, looked at the solution sheet, and then learned something. That is far better than getting 100% and learning nothing... although well done if you do get 100%.
Does that make sense? I just wanted to remind students, teachers and parents that Parallelograms are a major challenge, and to a large extent the scores don’t matter.
By the way, it does say at the start of every Parallelogram: “Don’t worry if you score less than 50%, because it means you will learn something new when you check the solutions.”
6.1 Are you cool about Parallelograms being hard and therefore not necessarily getting a high score every week?
- Yes, I am very cool about it. Positively chilled. Or maybe negatively chilled.
- (Not answered)
6.2 Are you going to promise to go through the solution sheet every week and learn from your mistakes?
- Yes, I promise to go through the solution sheet every week. Really. Seriously. I mean it. If not, may the ghosts of dead mathematicians haunt me during maths lessons.
- (Not answered)
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.