Posts Tagged ‘maths’

I like Futurama. I like it a lot. In fact, I would even say I prefer it to The Simpsons (the newer episodes at least). For those of who who have been living under a rock, Futurama is by the same chap that created The Simpsons (Matt Groening) and it is a cartoon about a delivery boy who is cryogenically frozen until he wakes up in the year 3000. Then come the wacky shenanigans with a cyclops, a robot, an old professor, and whatever-the-heck Dr Zoidberg is.

What may be of interest to you is that some of the showrunners and writers of both Futurama and The Simpsons have a background in mathematics. There have been many many mathematical references or jokes embedded into The Simpsons, which might be the topic of another blog post – for more info, I recommend Simon Singh’s book ‘The Simpsons and their Mathematical Secrets’. No, today I’m going to be talking about one of the greatest pieces of mathematics embedded into a television programme that I have ever seen: The Futurama Theorem.

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The Futurama Theorem was invented by Ken Keeler one of the writers of the show who also happens to have a PhD in applied mathematics. It was created for the sole purpose of explaining a concept behind an episode of Futurama – think of it like the ultimate ‘deus ex machina’. Let me explain: in the Season 6 episode entitled ‘The Prisoner of Benda’, Professor Farnsworth and Amy Wong invent a mind-switching machine (this allows two people to switch minds). However, once two people have switched minds, they cannot switch back directly. So for most of the episode, both characters are trying to get back to their original bodies by switching minds with a whole host of other characters. Just when all hope is lost, the Professor comes up with a mathematical solution to their situation:

It’s so…beautiful…

What this proves, in not so many words, is that no matter how many mind switches between two bodies have been made, they can still all be restored to their original bodies using only two extra people, provided these two people have not had any mind switches already. Pretty cool, huh?

Now the truly amazing thing about this theorem is that it was made just so the writers could continue the story in a resolute manner: they didn’t want the classic cut to ’20 hours later’ and everything would be sorted. No, they got one of the writers to come up with an actual valid mathematical theorem (that had never been published before) as a way of concluding an episode. Ken Keeler, you fantastic S.O.B.!

For those of who who cannot ‘read’ maths and are interested in what the heck the formula states, here it is in English:

  • Step 1: Have everybody who’s messed up arrange themselves in circles, each facing the body their mind should land in (e.g., if Fry’s mind is in Zoidberg’s body, then the Zoidberg body should face the Fry body).
  • Step 2: Go get two “fresh” (as of yet never mind-swapped) people. Let’s call them Helper A and Helper B.
  • Step 3: Fix the circles one by one as follows:
  • 3.a) Start each time with Helper A and Helper B’s minds in either their own or each other’s bodies
  • 3.b) Pick any circle of messed-up people you like and unwrap it into a line with whoever you like at the front
  • 3.c) Swap the mind at the front of the line into Helper A’s body
  • 3.d) From back to front, have everybody in the line swap minds with Helper B’s body in turn. (This moves each mind in the line, apart from the front one, forward into the right body)
  • 3.e) Swap the mind in Helper A’s body back where it belongs, into the body at the back of the line. Now the circle/line has been completely fixed. The one side effect is that for each time a circle is fixed, the Helpers’ minds will switch places, but that’s OK, see below
  • Step 4: At the very end, after all the circles have been fixed, mind-swap the two Helpers if necessary (i.e., in case there was originally an odd number of messed-up circles)

So there you have it: if you ever find yourself in a mind-swapped mishap then you may have need of The Futurama Theorem. If not, then at least you can marvel at how freakin’ awesome it is!

Tom

N.B. I have been told that I need to make my posts more ‘accessible’ to the ‘hoi polloi’ of the Internet world. Therefore, starting from now, if I post something maths-y then I will make sure to include something that non-mathematicians can appreciate and enjoy.

So here’s a picture of a dog dressed as Spiderman:

At first glance, you might be curious as to what this post is about. In that case, I would recommend learning the difference between ‘fallacies’ and ‘phalluses’. Dirty buggers. No, today won’t be spent talking about the male appendage – rather, some maths! Don’t sound too excited…

Yes, this post will be all about mathematical fallacies: these are ‘mistakes’ in mathematical proofs. However, the difference between a genuine mistake in a proof and a fallacy is that a fallacy will often lead to absurd results and conclusions that may seem flawless. Only when we examine the proof itself do we find the fallacy. So, today I’m going to show you some famous-ish fallacies that I find particularly interesting.

1. Fun With Fractions

Fractions is probably the part of maths that most people hate. Personally, I don’t mind them, but to some people they are the spawn of Satan. After all, there are so many rules to follow: “when can I divide?”; “Which fraction do I flip to divide?”; “Why am I even doing this?!” Many many questions to ponder. Below is a little example of some of the misunderstandings that go on when people deal with fractions. Now be warned, this is NOT how you simplify fractions. It just so happens that it works for this one example. Try it: type into your calculator ’16/64′ and it’ll simplify to 1/4. To all you non-mathematicians, sit back and marvel at how delightfully simple this fallacy is; to all you mathematicians, I challenge you to find another example where this kind of error works.

“But…erm…it’s right…”

2. Imaginary Numbers

To many people, the thought of imaginary numbers is ridiculous. In fact, I would wager that a lot of people wish that all numbers were imaginary and thus irrelevant. But I’m afraid that these imaginary numbers are far from irrelevant. The main thing to be aware of in terms of imaginary numbers is that the square root of -1 is called ‘i‘. Now, using that knowledge, behold as I prove that 1 = -1.

I feel a disturbance in the Mathematical Force…

At first glance this proof seems completely logical, however, there is an ever-so-tiny thing wrong with it. Now I could explain why this proof makes no sense, but it’s far more fun to let you work it out for yourselves. So, if you think you know the answer, please leave a comment!

3. MORE Imaginary Numbers

Wow, people just cannot get enough imaginary numbers, eh? This little fallacy is in the same style as the previous one. This time, the proof is showing how the square root of -1 is just 1 (which we know is impossible – remember I said it was that funny little number called ‘i‘). So, have a bash at this one!

Handy Hint: consider the fourth roots of 1…

“This is madness!”

4. Tricky trigonometry

‘SOHCAHTOA!’ – No, that’s not a made-up Japanese word, but a way of remembering the 3 trigonometric formulas! (‘Sine, Opposite, Hypotenuse, Cosine, Adjacent, Hypotenuse, Tangent, Opposite, Adjacent’) Ah, such fun! Trigonometry is the study of triangles and the relationships between their angles and sides. You can do lots with good ol’ trig: calculate angles, calculate sides, erm…calculate…other things too. Anyway, this fallacy proves that 0 = 2! Doesn’t that sound dramatic? Well, you’re probably aware of the fallacious proof that 1 = 2 (done so by dividing by zero – a huuuuuge no-no in maths) but this proof uses trigonometry! Have a look-see at this bad boy!

Handy Hint: it’s very close to number 3 in the list…

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Screw the maths, just look how neat my handwriting looks!

So, I’m afraid our mathematical journey must come to an end! We’ve had some ups, some downs, but most importantly – we’ve learnt some maths. I appreciate that this entire post may have gone over some people’s heads, but hey, sometimes I like to do maths.

Till next time,

Tom.

Is it time to hang up the pistols, Jack?

I loved the first Pirates of the Carribean film: it was a fun cinema experience. The second one was alright, but started to get a little complex in terms of plot and then the third one just didn’t seem to end. Now, On Stranger Tides was where the franchise changed: the executives at Disney realised that Johhny Depp’s character Jack Sparrow practically IS the POTC franchise and so they focused on him for this film. They scrapped the Elizabeth/Will stroyline and concentrated on Captain Sparrow’s shenanigans on the high seas. In my opinion, they ‘over-Jack‘ed Depp’s character, but that’s another story.

A report done by NerdWallet has looked into the decline in ratings and box office takings for film sequels. We all have those films we can say ‘they weren’t as good as the first one‘, well now there is some data to try and prove this hypothesis. N.B. Being a mathematician I feel obliged to point out that whilst this data seems to prove the fact that movie sequels decline in quality and profitability, there were limitations to the test (not every film ever was involved in the test, for example). Plus, there are certain films that prove exceptions to the rule, which I will touch upon later.

The reason I (and the report by Mr. Anderson – wow, total Matrix moment) concentrate on the POTC franchise is because the decline of the series has been harsh: the first film received an average rating of 79% from critics; Dead Man’s Chest only attained 54%; then a measly 44% for At World’s End and, finally, a disappointing 33% for On Stranger Tides. That’s a drop of 58% from the first film (before you start questioning my maths, the drop in rating score is measured by percentage change, not percentage difference – the difference between the first and fourth’s average reviews is 46%). The same principle, it was found, applies to the box office ratings.

Next, using data from 130 film series (comprising of 475 films), Anderson looked at reviews from popular film review website Rotten Tomatoes and revenue figures from Box Office Mojo and he came up with the following graph:

So, if we are to go by the data (remember though: there are lies, damned lies, and statistics) and using linear regression (don’t even ask) we get an average rating for POTC:5 of 31% – which is a 61% drop than the original. Whilst the box office revenues are not much help themselves (some films don’t make anywhere near $208 million) but the percentage change in revenue can be calculated and analysed (feel free to do this at home).

N.B. POTC:5 hasn’t come out yet (it’s set to be released in 2015) and who knows – they may make a fantastic film that lives up to the excitement and adventure of the original: we can’t judge a film before it has come out – that would be silly. I am NOT saying ‘Don’t watch the fifth film – it’s going to be rubbish!‘ I am simply commenting on the interesting decline in ratings and revenue that occur for film sequels. 

Dammit, Bond!

As I have mentioned earlier, there are many many examples of film series in which the sequels haven’t declined in ratings or revenue: Fast and Furious 5/6, Saw II/III/3D, The Matrix Reloaded, the list goes on. There is an important collection of films that are huge exceptions to the rule: James Bond. The 007 franchise is a tricksy one, in terms of ratings and revenue because it doesn’t technically have sequels. As each film is a standalone story (OK, there’s a bit of continuation between the Daniel Craig films) its ratings aren’t affected by those of the previous film (just look at Skyfall – an average rating of 92% comapred to Quantum of Solace‘s 64%). That’s why this particular franchise wasn’t involved in Anderson’s report – as it doesn’t follow the sequential nature that other series do.

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So, a more serious blog post today – but I hope that you found it interesting. All statistics and facts were provided by Mike Anderson from NerdWallet – the full article is available here. Next up, I’ll have to post about the scary world of Hollywood financing – in which accountants try and prove that nothing has made any money.

Till next time,

Tom

As you may or may not be aware I am studying Mathematics at University. Yes, I’m one of those ‘strange’ people that can grasp maths and I’ve decided, of my own free will, to pretty much devote the rest of my life to the study of it. Wow, when I write it like that it DOES seem kind of depressing. No matter, too late to change now I guess. Anyway, below you’ll find a list of reactions I have encountered when telling people that I study maths and how I personally have dealt with them. I suppose you could apply the following situations to any subject, but I find that maths is the one that gets the most…diverse…reactions. I hope you enjoy them.

1) The Question.

Ok, so let’s pretend that you’re on a first date. Everything’s going swimmingly when suddenly the other person asks you what you study, expecting a response of the more dociel subjects like Geography, English or even *snigger* Theatre Studies. And so, not wanting to lie this early on in what could be a fully-formed relationship, you reply ‘I’m doing maths’. To which the other person responds with one of my favourite responses:

‘Why?’

Now this is one of my favourites because you have no idea how to respond to that, and I don’t think the other person does either. I mean, it’s not like I’m going to say all of a sudden ‘Oh my God! Your questioning of my university studies has suddenly made me rethink my entire life! I’m going to give up my prosperous career as a mathematician and do Media Studies instead!’ And so you are forced instead to say cringe-inducing justifications for your studies, such as ‘I like it’ or ‘I was good at it in school’, or my personal favourite ‘Because it’s the same in every language’ (that is, in fact, a quote from Mean Girls and I advise you not to use it…). However, thereis a light at the end of the tunnel. After being asked this mundane question so many times, I have come up with a sentence that should halt any further probing into my study of maths:

‘Someone has to.’

See, this is hopefully funny enough to make the other person laugh and move on to talking about the weather or whatever, or it is so blunt and direct that the other person has no choice but to quickly change topic so that I don’t inevitably bore them to death with more blunt responses.

2) The Barrage of Compliments.

Right, so let’s go back to the dating scenario mentioned before. You’ve just told the other person that ‘I study maths’, but THIS time they will give a response that is not very common, but can catch you off-guard if you are not prepared:

‘Wow, you must be so clever! I wish I was as clever as you to do maths!’

At first glance this may sound sarcastic (this might just be because I write in an extremely sarcastic manner and so you have no choice but to interpret everything I write as being sarcastic…) but it is actually intended as a compliment from the other person. This puts you in a HUGELY awkward position as, well I don’t know about you but I cannot take compliments to save my life, you have to either downright accept these compliments or try and turn them into some sort of self-criticism. For example, there is the blunt answer of:

‘Yes, I am clever. And yes, I too wish that you were as clever as me.’

which, even I am willing to admit, is just plain rude. The other option is to reply in a sort-of awkward self-criticism, such as:

‘Well, I’m not GREAT at maths. I only chose it because I’m worse at everything else. I don’t even like the subject.’

when inside you are in fact thinking, ‘Why are you saying these things, Tom?! You LOVE maths!’ which may be closer to the truth. I must admitI still find this sudden ‘attack’ of compliments still a little awkward and I am forced to use the cringey ‘I-pretend-to-hate-maths-even-though-I-actually-love-it’ technique.

3) The Look.

Read the very first sentence of this post again, the one where I say I study maths. Now think back to the facial expression you were pulling when you first read that (or what you were thinking inside). It probably looked like a grimace, kind of like Moe from The Simpsons. Now, if you pulled this face in your head (you know what I mean) that’s perfectly fine: you are not displaying your disgust/questioning/anger/hatred openly and therefore I personally cannot tell your reaction to my studing of maths.
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When ‘The Look’ decides to manifest itself out in the open, that’s when things get personal. If I had a penny for every time someone has given me The Look when I told them I did maths, I would have about £2.30. The point is, that The Look is usually people’s default expression: a sort of ‘Ouch’ face. As in, ‘Ouch, you do maths.’ Well, I CHOSE to do maths so surely it doesn’t ‘hurt’ me? Why on Earth would I choose to study it when, in fact, I am ‘pained’ by it? Thus discrediting The Look.

N.B. I apologise if the previous paragraph was more of a stream-of-consciousness rather than coherent text. My writing style is quite…messy.

I can think of many, many more ‘reactions’ that people have shown me upon them finding out I study maths, and perhaps they’ll appear in another post. If you are a mathematician, I hope that you have at least gotten a few chuckles out of this and maybe you have witnessed first-hand some of the things mentioned; if you are not a mathematician (a ‘muggle’, as we like to call them) then I hope that the message is clear: next time someone says they study maths, think about how you’re going to react to it.

Later bitches,

Tom