So I'm a mathematician, and I'm a Futurama fan, and I've got this one episode which I particularly like. It's called "Prisoner of Benda." It's got a mathematical theorem in it and proof of a mathematical theorem. Well, let's have a look at this clip here, and then talk a little bit about it. "Wait a second... if we switched bodies, I could live life to the extreme, and you could indulge your demonic food lust!" *gasp* "We're just the people this mind's switcher was made for by us!" "Wait, can I still change my mind?" "Let's find out!" "It didn't work. You doddering old- oh!" (pause) Well, they're really happy at this point in time. They go off and, you know, enjoy their lives for a little bit. But then they find out, well, wasn't that good after all, so they want to switch back into their own bodies and, well, let's just see what happens then. *beeps indicating it didn't work* "Professor, what's not happening?" "Bad news, me. I failed to take into account the cerebral immune response." "You idiot!" "Once two bodies have switched minds, they can never switch back... "However, perhaps they can. Maybe we can swap back using a third body for temporary storage space." "I'm 60% storage space!" "Be warned, I-" *sci-fi noises* "Well, I'm off to the royal yap for a quick seduce and snatch." "See ya..." "Now then, Amy we'll simply switch bodies, and "then we'll- no, I'd be back in MY body But then you and Bender would be switched. "And the Amy and Bender bodies can't trade minds again since they just did." "Oh, no! Is it possible to get everyone [back] to normal using four or more bodies?" "I'm not sure I'm afraid we need to use... *DUN* "MATH." *dramatic sting* Math. Heh, well, let's use math, but first, let's figure out what's actually going on here. It's a little bit confusing. So, there's some sort of restriction in place here. The restriction is that once two bodies set on a chair they can't sit there again. Well, they can sit there again, but nothing is going to happen when you push the button. Okay, got nothing to do with the minds. Once two bodies have sat on the chair, that's it, right? Next time, we put them all, nothing is going to happen. So keep that in mind, so let's just kind of try and figure out what's happened so far, okay? So you've got our - well, three heroes here, three heroes and the way I've done this is... Always put down two diagrams. One diagram sense for the body, that's right here, and then one stands for the mind. That's the one with the triangle in it, okay? So we've got the Amy body and Amy mind, Professor body the Professor mind and a Bender body in the Bender mind. Then to start with... The Professor and Amy switch minds, so that goes like this. And now I've indicated that these two bodies have sat on the mind switcher machine by this Green Arc much just keep track of what's going on all right then well, they want to switch back but they can't right because the bodies have just sat down so they have to bring in Bender, so bender comes in and now the amy body and the Bender body sit on the chair and Minds get switched and then really when you have to look at the diagram. That's really only one thing left You can do you can sit those two bodies on the on the chair? That's betty what the professor suggests, so let's put them on and then he says well, then I'm in my own body again But you other guys you'll never be able to sort yourselves out because everything's been used up So we've already put all combinations of those three bodies or on the chairs Okay, and now the idea is to kind of bring in maybe even more bodies and you know try this idea again with temporary storage space and eventually sort everybody out Well, they bring a lot more bodies actually in the end there are nine people involved and most of them are you know familiar faces? except maybe the wash bucket Even wash bucket has a mind in this episode, and there's also robot emperor in this episode. So the robot emperor also plays a role apart from that, it's just the usual crowd or professor and leela and hermes all these guys and all right now They really give this a good go But in the end everybody's kind of all over the place mind wise and let's just see what what they end up with? It's this right so in the end. You know that we've got the wash buckets mind in amy's body for example and Fry's mind in zoidberg's body, and you know it's a big big huge mess All right, but then they really do the maths and well, they bring in some help from outside Some mathematicians. It's actually the Harlem globetrotters of the future they turn out to be excellent mathematicians apart from being excellent basketball players and they sort it out for them and so With this mess here they come up with a theorem Which allows them to well which allows them to sort out any mess not just this one here and even better They don't really have to know how this mess came about right I'm usually what you have to do is you kind of have to keep track who separates who I didn't really indicate this so far you don't have to notice if you apply their special Superduper Harlem Globetrotters Theorem, so let's just have have a look at what happens they bring in two of those harlem globetrotters as Temporary storage people and they sort out everything you've got a bit of a clip here and at the end We'll see the theorem, so [let's] [all] look Okay, so there's the theorem and Something happened obviously nobody can follow what's going on here because you don't really know What what the minds are doing? You don't see them as you only see the body Okay ah well, let's have a really really close look at the blackboard there, the blackboard has the theorem on it and I've Highlighted a few things, so that's that's really maths talk, and you'll only be able to really understand this if you're a mathematician like myself We can try you know we can try I've underlined a few words here So there's an arbitrary underlined here which basically stands for an arbitrary mess they talk about permutation But it's really an arbitrary mess, so we can deal with an arbitrary mess We do this by getting in two new guys for temporary storage and Then they're talking about cycles somewhere up there and the cycles whatever mess you're facing There's actually always a little bit of structure in a in a mess and these are the cycles of the mess Anyway, so I'll just try to make sense of this theorem now for you, and let's just see what we get, okay? So I'll give this my best try Okay, so here's the mess again. Here's the mess again. We'll Focus on one of the guys here, so there's the professor's body in Benders The professor's mind and Benders body. So we've got a professor here. We're looking for a second professor Picture well, it's over there. There's the professor's body, so we all make a connection like that again So we're going from mind to body now. There's a mind in here. So that mind Where's the corresponding body? It's there so we'll make another connection and Then Hermes' mind is in there. Where's his body it's down there and Amy's mind is in there. So was in his body [so] [over] there, and we can keep on going like this And we can do the same sort of thing for any sort of mess. Let's say We've got like a thousand people, and they're going wild on the mind switcher You know we just have them here in our room And we'll pick the first person and we'll do this, we kind of chase things around so Always mind to body, mind to body, mind to body, mind to body and eventually well what's going to happen well? We've got 2,000 people eventually we have to kind of get back to the beginning And we'll get back to the beginning like in this case if you just keep on going we get back to the beginning like there but you see we haven't picked up everybody here, so You know we've picked up some people in this cycle. It's a seven cycle. You involve seven bodies so minds But there's some people left over so now what we do is We'll just pick one of those left over guys and start the whole process again, and it will give a second cycle And if you've got a really really big group and a really really a big mess What you usually get is like lots and lots of these cycles but eventually Everything's kind of split up into this cycle. So in this case. We [actually] get one more cycle [a] mini cycle So they are we going zoidberg mind zoidberg's body? Fry's mind Fry's body, so it's kind of a mini cycle like this okay, and now that theorem tells us that we're supposed to sort out these cycles one at a time and I've got a slightly streamlined version of what's going on if you actually? Remember what happened in the clip you'll need see that like in every single one of this mind switcher that happens there one of the basketball players is involved in fact they kind of shared a job, so it seems like they both do the [same] number of Mind switching in the version. I'm going to show you which is also based on the theorem I think I think it's a little bit more transparent so let let this go for this essentially the same thing But I think it's a bit more transparent. What's happening okay? So here. We've got our seven cycle and here We've got our two cycle. Okay two cycle, and so you can see now in the seven cycle the way I've Put it here on the screen You always have the mind next to the body, mind next to the body, mind next to the body, mind next to the body and it Kind of closed in like this all right now we want to bring in these two new guys to Harlem globetrotters of the future and Use them to sort out these Cycles one at a time, so that's what the theorem tells us okay, so let's start with the big one bring in tune you guys and there's the one with the big hair and I also call him the the red guy okay, so the red guy, and there's the other harlem globetrotter I call him the blue guy because I've underlined in blue okay now. We have to start somehow This is all completely symmetrical you might as well start with the first one. We see here with your mind switch Okay, and we note down that this body has set with the red guy, okay, okay, so dial on Do a mind switch and that sorts out the professor, and then we dial on to a mind switch And that sorts out Lela And we're going to keep on going like this and as we do we always note on this body has set with red guy this body has set gets the red guy this body has set with the right guy we just Keep on going okay, so just keep on going and we are sorting out these so you can really see what's happening here It's really quite pretty Now I put an exclamation mark here, what's happening? Well? We're nearing the end right. We're nearing the end We're nearing the place where we started, okay? So let's just continue. You know. What would be next where we dial. Okay? We'll dial We switch minds and when we dial and then we're back to the beginning right, okay? But now if you have a look these two guys they already set on the mind switcher so we can't Put them on a mind switcher again So we really have to use the second guy now, okay? So we have to put the two basketball players on the chair and they switch minds okay? that's fine, and now we can actually sort out like the cycle, but just kind of putting those two guys on the mind switcher and You know Bender is Sorted But there's a bit of problem here the problem is Well that we put the basketball players on the chair and so basically their switch has been used you can't use it again yeah, but now there's others other cycles coming up, and we want to process them in exactly the same way right, so You know we might have to do a second cycle, and we can process it exactly the same way But at some point in time. We would have to put the two two guys on the seat again and We've probably used that right so we somehow have to avoid this and actually quite easy to avoid so let's just go back in time A little bit okay, let's go back in time. So we're now in the dialing face like dial Herman is a sorted dial err a me sort of dial the Washbucket is sorted and now we're In the critical phase, okay now. We're not dialing on what we'll do is We'll bring the second guy into play okay, so the second guy is going to switch with the emperor here, okay? and now we basically keep on going as before so we just kind of dial in and Presort it out and now we just set those two down and those two are sorted out and what's important about this maneuver is That the two basketball players have not sat on the chair So we can just bring in the next cycle sort it out in exactly the same way next cycle sort it out in exactly the same way Eventually we have finished and then either the basket players are already sorted out they'll have the right minds associated with the right bodies or we'll just put them down on a chair and sort them out then and That's it. That's basically What the theorem does okay now just to sort out the second cycle here so a second cycle That's very easy So there's zoidberg and fry and swap them around so we just go like this we go like that we go like this, and we go like that and now everybody's sorted okay? And you can use the same as I said same method to just sort out any mess whatsoever so any mess Split into cycles bring in two new guys. It doesn't matter how the mess came about you can sort it out like this. Okay, now it actually turns out that Mind switching is not a new thing. It's not something that Futurama invented It's also present in some other well Movies TV episodes there's one in particular. Which is quite interesting which is Stargate, so one of the stargate episodes has a mind switching machine It is the episode Holiday Okay, let's have a look at that one too So we've got the usual four guys who go out and explore new worlds So they go through the stargate and end up in distant places and explore so there's Samantha, Jack, Teal'c and Daniel and And now they go to this this planet, and that's really only one guy really old guy Marchello waiting for them and well, let's just see what happens okay, so they're just met Marchello and he's going to say you Quickly it fetches to fight the ghouls Daniel I'm fine I think Hmm. So what's happened? Well? I mean you've just watched the futurama minds switch so you know what happened? What happened Is that these two guys switched minds but actually Marchello that was his plan all along. He actually wanted a body a new body He was about to die, really all is about to die so he wants a new body He's got a new body now and he pretends to Be Daniel, so he pretends to be Daniel. Nobody's really picks up on it Until a little bit later when these two guys Jack and Teal'c try to pick up the mind switcher and bring it back to Earth So that's what happens then so I've got all the special gear on and they're ready to go on three one two three What? Hmm, so at that point in time They know that in a bit of trouble and they start well Then we have to chase down Marchello in Daniel's body having fun somewhere on Earth Okay, so we've switched minds here right and we've got a mess like this And now the person who always sort out things [really] really smart person in this in this series is Samantha So Samantha figures out how to sort out this mess and actually she doesn't bring in any additional guys She just uses the people that have been involved in the mind switches those four, okay? So those four so she she says well I've got a got a method of doing this just pay attention Follow what I say and then you know let's just have a look at this okay? So how if you actually knew that this works? What would you Do hmm? Well the first thing here is this set up is completely symmetrical so it shouldn't really matter How we start okay, so let's just do one mind switch here. It's possible this one here, okay, all right What should come next right? So let's say it somehow works. What should come next well? There's really two essentially different moves. You can make either you can make this or that that should if it works either should work or that one here right, so Which one works well let's try this one first okay, so let's try this one. What happens. Well Marchello is sorted out But this body has used up all its moves right and well. It's not sorted out here So we're stuck right, so this is a dead end. So let's just trace back here So what we should really do is do this move here, so we do this move here, and then well again It shouldn't matter which one. We do so let's just do one let's do the other and People are sorted out and the theorem Well this case Amantha takes a bow previously the theorem takes bow, and it's it's done Is that the earliest occurrence of of the mind switching in action no even even not? a stargate invented this one here even earlier I know something so 1996 there was an episode The Tick an animated series for kids really where that it was a mind switching You've got different machines. There's a mind switch in The Tick as a mind switcher in stargate, and there's the mind switch in futurama Well Different levels of sophistication. He is just a complete mess You don't know really how to sort it all out in the end this one here. You know pretty nice You know the way she does it and this one here is just amazing right? great Mathematics like in many other futurama episodes you know completely correct and You know something that every mathematician will like anyway, so that's about mind switching mathematics for today
Haha wow never thought id see my crazy Maths lecturer being shared around on Reddit. If you're interested in Professor Burkard and his crazy antics, he also has a book called "A mathematical guide to the best and worse ways to lace your shoes" http://monash.edu/research/people/profiles/profile.html?sid=2576&pid=3239
I still remember trying (and failing) to understand this when the episode first aired. Now I finally get it, great explanation. In the Stargate mind switching episode that is discussed at the end of the video no people external to the mind mess are employed as temporary storage to sort out things. While watching the video it occurred to me that the same is possible for the mind mess in the Futurama episode--just using Fry and Zoidberg instead of the two additional players should work. The theorem and this video are about sorting out an arbitrary mind mess and so it strikes me as a little bit odd that the Futurama producers opted for using a "tame" mind mess that does not strictly require the two external players to illustrate the theorem.
A really great Youtube channel. Also check out his video on proof for the irrationality of the square root of two.
If we think of it as 4 tiles in a square and number then as so: 1 2
3 4
We can represent a swap occurring with a line between two tiles. Begin with the first two to create the following and then work back to the beginning.
2 - 1
3 4
2 - 3 / 1 4
4 - 3 X 1 2
1 - 3 | X 4 2
1 - 2 | X | 4 3
And finally 1 - 2 | X | 3 - 4
Burkhard! He's a lecturer at my university. Never actually had him as a lecturer, but I hear only good things.
Mirror?
Does anyone else think the opening chords sounds like the Metroid theme, or am I taking crazy pills?
Holy shit that's my maths lecturer at Monash. He's a nutter, but an amazing lecturer.