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  • So we're gonna do chemistry today.

  • Just kidding.

  • I can't do any chemistry, but this one is for the chemist because they're gonna talk about something called the centrifuge Problem.

  • Okay, so this centrifuge is this thing that chemists used to take test tubes and they spend them to separate the contents of the test tube because it's something that's spinning at a very high speed.

  • It needs to be balanced in orderto work effectively.

  • And so here's what the centrifuge problem is.

  • If you have a centrifuge with, say, and holes, let's say six and you have K test tubes.

  • Is it possible to balance the centrifuge?

  • That's the question.

  • Let's assume they're evenly spaced on on the edge.

  • Let's assume the test tubes all have the same way.

  • It's that kind of thing just to sort of idealized the problem.

  • It's hard enough as it is.

  • So that's the problem.

  • So if you look at this example, so I've drawn this example with an equal six.

  • If Kay was equal to one, we have no chance.

  • If you only have one test two and you try and balance this thing, you're never gonna make it work, right, because one side will be waited because it's got the only test tube in there, so we can't balance it with one right.

  • But we can balance it with two.

  • We just put them opposite each other.

  • Okay, so cables to weaken.

  • D'oh!

  • And similarly, for three, you could imagine putting one here one here and one there.

  • And then the centrifuge will be balanced because they're evenly spaced along the border.

  • S o K.

  • Equals three.

  • We can do it.

  • And for K equals four, you might not see right away that K equals four can be done.

  • But here's the thing K equals four.

  • Trying to balance that is the same problem as trying to balance cables to right.

  • Because if we have six spots total than missing, two spots has the same weight balance is filling two spots.

  • All right, so if K is equal to four, we can do it.

  • And maybe you either see that directly, or you can think of it as being sort of the opposite of doing to test tubes.

  • Right?

  • So rather than feeling too, I'm leaving to empty.

  • So if one of them balances it, then so will the other.

  • Okay, So yes, we can do K equals four.

  • And for K equals five, though we again have no chance for the same reason is K equals one, which is that if we leave one empty, then there's no chance of balancing this thing.

  • And of course, with six.

  • Then we just feel every spot so right away.

  • If Enos six we see that one in five or the bad configurations, everything else is okay.

  • And if you're me and you like prime numbers, the first thing you think to yourself is well, I see exactly when it can be done.

  • It's when K and N have some common factor, right, Because one and six, they don't have a common prime factor.

  • Five and six don't have a common prime factor.

  • But every other number on this list does have a common factor with six.

  • That was my guess when I first saw this problem, and before I tell you how well my guest does, let me point out two things.

  • So one is that we've already discussed, which is doing this for K tubes is the same as doing it for n minus K tubes.

  • All right, so if I can do it for safe and a six.

  • If I could do it for one, I could do it for five.

  • If I can't do it for when I can't do it for five and so on.

  • So that's one thing to notice.

  • Maybe a more interesting thing to notice is that we can add up configurations, right?

  • So if I can balance some number of tubes, Okay, put him in a centrifuge.

  • It's balanced now.

  • If I have additional tubes and I can balance those, then just add them in and I again stay balanced, right, because it's like adding 0 to 0.

  • If you were balanced and then you add another balance configuration on top of it, you stay balanced.

  • If we want to think about the cake wills, for example.

  • Another way to think about it is two plus two.

  • I put one configuration here that's balanced, and then I could choose any other configuration that's balanced for the other two, and this will always work.

  • This adding configurations so long as there's room for it.

  • In this example, the numbers are small and it's kind of okay.

  • There's a lot of room in the centrifuge, but It's not always gonna be true.

  • Okay, okay.

  • All right.

  • So here's why.

  • I'm wrong.

  • Let's do an example with 12 spots, let's pretend these air all evenly spaced like before.

  • And so now we have an equals 12 and I claim that we can actually balance this with seven tubes.

  • No common factors with 12 in fact, is a prime number.

  • And it's not any of the primes that divide 12.

  • And it looks kind of weird, right?

  • Like I can't really visualize putting seven dots in here to balance this thing.

  • But think about what I said a minute ago about adding configurations.

  • If I can do three in a balanced way and I can do four in a balanced way, as long as I can put them in here so they don't overlap, then I could do it.

  • And that's exactly how we'll do this, because three divides 12.

  • I can just put them every four spots, so there's three that are balanced and same thing with four.

  • I could just put them every three spots and get something balanced, so we'll put say one here.

  • 123120 that's not gonna work.

  • It's already filled right?

  • So let's start again.

  • So can I do it by skipping every three?

  • Well, if I start here, I run into a field spot.

  • We already tried starting here, and I ran into a filled spot.

  • If I start here, I'm going to run into a field spot over here so that configuration isn't gonna work.

  • But it turns out there's actually a different four configuration that will work.

  • And the reason we know that is cause for is two plus two, right?

  • So let's not do four.

  • Let's do two pairs of opposite tubes.

  • Here is the one opposite pair, and here's another opposite pair.

  • And so, even though this looks kind of strange, I guess we've got three here and then two spaces and then 12 and then a space.

  • Even though this configuration looks kind of strange, we know for sure it's balanced because it was the sum of three balanced configurations.

  • Wow, that's that's yeah, it's not obvious, even after you do it.

  • Looking at it, you're like, I don't know how do I tell, but because we built it up from balance configurations, we know that it's balanced, so when you say balance.

  • You're saying I could You know I could I could balance that on a pin.

  • Yes, that's a that's a I mean, if you want to talk about centres of Mass and that kind of thing, you should probably talk to a physicist for me.

  • I'll tell you the mathematical interpretation of this.

  • We've talked about complex numbers before, and we've talked about the complex plane before.

  • So remember, a complex number is a real number, plus another real number times this eye and the way we visualize it on the plane is just using the co ordinates for the first and second reel number, which I've written is A and B.

  • So here's my like A and B.

  • So if I draw a circle on this plane that centered at zero and goes through the 00.1, then I can draw this centrifuge as a bunch of points on this circle.

  • Let me do the 61 It's a little easier, and it turns out that you can always write these complex numbers in terms of the first ones.

  • Let's say this is E then this is the squared.

  • This is a cube, which is minus one Actually, this is E to the fourth, and this is eat of the fifth.

  • And so the mathematical translation of this problem is can I add together powers of that one number z and get zero our very first configuration.

  • This was when we had four tubes in six spots.

  • Right?

  • This is the same as saying so I have to rotate this picture a little bit.

  • Let's say this is my Z.

  • This is the same as saying Z plus C squared plus And then I skipped a spot and then see to the fourth plus Z to the fifth is equal to zero for that particular complex number.

  • And so the way that you actually prove this is by showing that certain combinations like this of powers of some complex numbers can add up to zero.

  • But let's actually go back to the original thing.

  • So we thought maybe at first that the answer was having a factor in common, a prime factor in common.

  • Now we know that that's not true.

  • But it turns out that your next gas is true.

  • Which is if you think about this process, what did we D'oh!

  • We took the number of tubes we had.

  • We broke it down into prime factors, and those we knew we could balance right just by spacing them evenly as long as there was no overlap.

  • And it turns out that's exactly when this will work is that you can balance a centrifuge if and only if both the number of tested you have, which I've called Kay.

  • And the number of empty spots, which is an minus K, can be written as sums of prime factors of end.

  • So let's look at this an equal six.

  • So one is not possible to write as a sum of prime factors of end, because the prime factors are two and three.

  • Remember, we need both K and and minus K to be written that way.

  • So too can be, and four is two plus two.

  • Three can be and minus K is three, which is also the prime factor of and K equals four.

  • We've already checked on and minus forest to we've already checked that 1 to 5 can't be done, even though five is two plus three and minus five is one which can't be written as a sum of prime factors, event and so this condition of whether or not the number of test tubes and the number of missing spots can be written as a sum of prime factors of the number of holes is exactly when you can balance a centrifuge, which is something we understand really well, right?

  • So this is always useful in sort of geometry of that kind of study.

  • If you can change something just a little bit and get back to something, you understand really well, so there's a natural way to view this thing.

So we're gonna do chemistry today.

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A2 初級

離心機問題 - Numberphile (The Centrifuge Problem - Numberphile)

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    林宜悉 發佈於 2021 年 01 月 14 日
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