Parallelogram 34 • Level 2 • 25 Apr 2024Good Will Hunting

• 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 Good Will Hunting problem

“Good Will Hunting” is a 1997 film that won two Oscars and made Matt Damon and Ben Affleck famous, as they starred in and wrote the film. It tells the story of a janitor called Will Hunting, who keeps the floors clean and the lecture theatres tidy at Massachusetts Institute of Technology (MIT), one of world’s best universities. However, unknown to the professors, Will is a mathematical genius, and the film tells the story of his battle to come to terms with his remarkable talent for numbers and his struggle to understand his relationship with his best friend, his girlfriend and himself. Along the way, he is helped by a psychologist played by Robin Williams, who appears in the film’s poster next to Matt Damon.

Take a look at this clip, which starts with a maths professor confronting his students, because someone has solved one of his problems and he doesn’t know who – of course, we know who solved the problem.

Although the problem that Will is working on is said to be incredibly difficult, that is just a bit of Hollywood exaggeration, because this is a problem that you – yes, you! – might be able to solve.

This is the problem: draw all homeomorphically irreducible trees of size n = 10.

First, don’t panic! Don’t be scared by the scary words. It is not as terrifying as it sounds. Let me translate what it means into terms that you will understand.

In maths, trees are dots connected by lines, but in this case the lines cannot form loops. Also, the question requires that a tree is not allowed to have a dot connected to only two lines, because the dot could be removed and it would not look very different.

We can show the rules as they apply to drawing trees with 5 dots.

Tree (a) is not allowed, because the red dot has only two lines, one going in and one going out. That is just waste of a dot.

Tree (b) is not allowed because it has a loop. Remember, no loops.

In fact, the only tree that you can make with 5 dots is this one. No loops. No dots with just two lines.

The only other thing you need to know about trees is that all of the 5-dot trees below are considered to be the same. They are copycat trees because they can be bent, stretched, twisted and flipped, so that they look like the 5-dot tree above. That is why we say there is only one 5-dot tree. One way to see that they are all the same tree is to note that each tree consists of one dot with four lines and four dots with only one line.

Now it’s time to see if you have understood the maths of trees.

Hopefully, you are now getting the idea, but before we get to the full Good Will Hunting problem, here are three warm ups.

You will need to get a piece of paper and start drawing some trees. The key things are: don’t create forbidden trees, and don’t create copycat trees. Copycat trees are sometimes hard to spot, so really twist and turn your trees to check that one of them cannot be transformed into another.

Now you are ready for the full Will Hunting problem.

The Good Will Hunting problem is not incredibly difficult, despite what the film implies. However, you have just been learning some maths that you would not normally do until A level. So, well done. Don’t worry if you did not understand everything about trees, but if you did get most of the answers right then double well done.

2. NASA’s massive blackboard photograph

This week I came across two terrific photographs of NASA mathematicians writing down some mathematics on their giant blackboards. The boards were used for calculating rocket orbits and for getting astronauts to the Moon, and I suppose they had to be large so that everyone in the department could watch, learn and contribute.

3. The Councillor Problem

A local councillor is knocking on doors. He is carrying out a survey, because he is wondering whether the community needs a new school.

When he knocks on one door, a lady answers and he asks her two questions: “How many children do you have, and what are the ages of your children?”

The lady is a mathematician, so she decides to make the councillor do some maths to work out the answers. She tells him that she has 3 children, but she does not give their ages. Instead, she gives him a clue.

CLUE 1: The product of their ages is 36.

The councillor is confused. The children could be 2, 3 and 6 (because 2 x 3 x 6 = 36), but there are lots of other possibilities. The mathematical mum decides to help him with another clue.

CLUE 2: The ages add up to the number of my house.

The councillor is happy for a moment, but then realises that he still can't tell what their ages are so he asks for a final clue, and the mum agrees.

CLUE 3: The lady then explains that her eldest child plays chess.

Straightaway he knows their ages, says thank you and moves on to the next house (hoping that the next parent will be more straightforward).

4. When not knowing mathematics can cost you $15,000

This clip from the TV quiz show Who Wants to be a Millionaire? shows just one reason why it’s a good idea to be a confident mathematician.

5. Junior Maths Challenge Problem (UKMT)

6. Junior Maths Challenge Problem (UKMT)

7. Maths Jokes

Can you match the mathematical joke to the correct punchline?

• Punchline A: It improved di-vision.
• Punchline B: “What’s wrong?” The father asked. “I really don’t like long division,” the son answered, “I always feel bad for the remainders.”
• Punchline C: Dive-ision!

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

Here's a little more about the NASA blackboard photos.

If you want to find out more about Good Will Hunting, then just Google it. There are tons of articles about one of the greatest films to come out of Hollywood. You should certainly watch it, but it is aimed at adults, so perhaps wait until you are a bit older.

Number nerd James Grime has made an interesting video asking “Who was the real Good Will Hunting?”