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  • You can try many many other variations

  • depending on where to put the rubber bands, and also using more paperclips.

  • Let me show you one or two examples.

  • We have tried the position No. 1 or position No. 2,

  • but one position that we have never tried-

  • This is quite interesting, rubber band in the middle,

  • and it has to stay on the paper

  • and you can put the paperclips and see what happens.

  • Rubber band now in the middle here and paperclips at the top.

  • So, now what's going to happen when I pull...

  • This is what happens. So rubber band is hanging from the paper but it's ok.

  • but those 2 paperclips got linked against each other,

  • but each one of them is linked with the rubber band.

  • So, they're all 3, if you like 3 circles meeting in the middle and any 2 of them being linked.

  • - [Not Borromean?] - It's not Borromean, because...

  • even if I make, for example, the rubber band disappear

  • the 2 paperclips are still linked, and so on...

  • You can probably guess at this stage, what's going to happen

  • if I put the rubber band in the same position

  • that one paperclip at the top, but the other paperclip at the bottom,

  • so paperclips are on the wrong side of each other.

  • So, now what's going to happen when I pull...

  • Do you have a guess? Interesting, let's try this.

  • That's interesting, it looks like the previous one.

  • But this time,

  • each paperclips is linked to the rubber band, but not between themselves.

  • Let's finish with something that is 'work in progress'.

  • So far, we have been linking paperclips together,

  • and sometimes we refer this as "addition".

  • I mean those 2 paperclips coming together, so it's like adding.

  • What about subtracting?

  • By which I mean, can we start from 2 paperclips that are linked already

  • and slotting them in some configuration on the bent paper,

  • Can you take them apart?

  • Now, I've tried many many different possibilities,

  • and it turns out that this time the orientation matters.

  • You know, you can link paperclips in many many different ways.

  • For example, you can link them here,

  • or you can link them, for example, here,

  • or you can link them here, and so on.

  • So the position matters, and also,

  • you know, you can link them like this,

  • or you can turn the paperclip around and you can link them like this.

  • It's very very confusing.

  • So, let's try to link them here,

  • and see how we fare.

  • It turns out that this kind of link will end up tearing the paper,

  • and it just distorting the paperclips and it ends up in disaster,

  • or so I think, because, you see, I haven't done many times so...

  • I have to be very careful each time.

  • But if I change the orientation of the paperclips,

  • and put them in this configuration, that's the right configuration.

  • You may not see much of a difference, but the key is how those are linked at the top.

  • Let's do it very very carefully.

  • Those 2 paperclips are already linked.

  • I bend the paper in that standard position,

  • but I slot the paperclip here and here,

  • by allowing this linked pair to straddle over the paper like so, here

  • and here like this, okay.

  • And then I slot them very carefully.

  • And here is the preparation; paperclips were linked.

  • Now I pull the ends, and...

  • hopefully this works, if it doesn't, we'll be very sad, and something... some disaster happens.

  • OK let's see, will it be able to do it?

  • They came apart, so we can now subtract.

  • - [You like that, don't you?] - Yeah, yeah, yeah that's very nice.

  • Because you can make them come apart, yeah, yeah...

  • [from Part One]: —computation, or calculation in general.

  • We always think naturally, because that's how we learn these things in school

  • and about numbers - calculating with numbers - or, perhaps, at a more advanced level

  • calculating with formulas.

  • And, you know, you do some algorithm, you do some recipe, and then the expected results come out.

  • But here, your brain, although it hasn't formalized anything,

  • - you know, it hasn't written any numbers or formulas - has, effectively, computed.

You can try many many other variations


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A2 初級

ペーパークリップの引き算 - Numberphile (Subtracting Paperclips - Numberphile)

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    林宜悉 に公開 2021 年 01 月 14 日