Name: Ham Sandwich Problem - Numberphile
Uploaded: 2020-03-27T18:08:16.000Z
Duration: 5 min 54 s
Description: Thousands of YouTube videos with English-Chinese subtitles! Now you can learn to understand native speakers, expand your vocabulary, and improve your pronunciation...

To-infinitive

[Hannah] Do you want to share this sandwich with me, Brady?

[Brady] Uh, do you know what? I kinda do.

Gerund

[H] It looks really appetizing. [B] It's nice.

[H] This bread's been squashed in my bag a little bit.

[B] Yeah, I love that you went for white bread. Like, you didn't, you know, go all trendy and go...

[H] No. None of that, none of that. I want full-on carbs, that's what I want.

Okay, if we're gonna do this fairly then, Brady, I think we need to cut the sandwich in half...

But that does leave me with a bit of a quandary, because how do I know exactly

Because look, you know, you've got this bread's sort of lopping around there,

If this sandwich is a bit badly made, as I am prone to doing,

[B] Well, first of all, how are we defining half a sandwich?

[B] Does the cut have to be straight? [H] Oh yeah, and only one cut.

And it doesn't matter where the sandwich is.

But thankfully, the Ham Sandwich Theorem can help us!

when you have three objects in three-dimensional space,

you will be able to cut each of them exactly in half using only one cut.

And.. The best way to explain this is by building up from just one slice.

Let's say that you want to cut this slice of bread in half.

But... Let's imagine someone's already come past and taken a bit of a nibble off of it.

Just so it's a bit harder. 'Cus, you know.

[H] Hmm, that's a bit better. Okay. Uh, now-

[B] So it's less obvious where to cut. [H] Less obvious where to cut.

Let's muck it up a little bit more, here we go.

Less obvious where to cut. Right, we want to get this exactly in half,

all of the bread is on this side of the knife.

now, all of the bread is on this side of the knife.

So what that means is that there must be some point in the middle

where exactly half of the bread is on this side,

[B] The crossover point. [H] Exactly, exactly.

But that was me holding the knife at this angle

It would also be true if I heard the knife that way,

or... [Cracks up] Slightly more difficult to get the angles, but that way.

So any angle at which I hold the knife, I can guarantee that there's gonna be some point

where half of the bread is on either side.

Regardless of the fact that the bread is all, kind of, messed up and been eaten by mice.

Now, let's imagine that we add in our ham.

Let's make it a little bit more difficult, let's kinda fold the ham up in a slightly awkward way...

[H] Yum. [B] Nice. I'm going off the sandwich a bit.

[B] I hadn't thought about that! Now I'm definitely off it.

[H] We already know that we can cut this slice of bread here at any angle, okay?

So what we can do is we can pick a point on this bread,

where we know that 50% of the bread is on this side,

But, what you'll notice is that all of our ham, now, in this example, is on this side of the knife.

But because we know that we can cut this bread at any angle,

You can rotate this knife so that now all of the ham is on this side.

Which means that there must be a point in the middle,

Where exactly half of the ham is on this side of the knife and half on the other side?

[B] There's one magic line. [H] At least one magic line.

Now, the third component: this extra slice of bread. Yum! Brady, look at this delicious sandwich, I've made!

It's slightly trickier to explain this one, but the idea is the same.

Except in this case, you now have the angle of the knife to play with as well.

So you're sorta sweeping over, turning it around and rotating it that way, and that way.

There should be a way where you can make a cut through all three perfectly in half.

One small thing; we know for a fact that this line exists.

The theory doesn't actually help you find it very much.  So, there's that.

[B]: Who came up with this? [H]: It was originally suggested by Steinhaus and Banach.

That's Banach, from a Banach Tarski thing,

where you can rearrange a sphere to make two spheres that look basically the same as the original one did.

And, but then it was proved in the N dimensional case, by Stone and Tukey.

[B]: There seem to be a lot of mathematical problems that center around things that happen in lunch rooms and tea breaks, isn't there?

Yeah, you'd always think that maybe that was the most productive part of a mathematicians day.

Do you think you can draw a line through them, that will divide them into two equal parts with the same area and perimeter?

Apparently the answer is yes, and it uses the Ham Sandwich problem that you just watched a video about.

If you'd like to find out more about this, go to brilliant.org/numberphile

Brilliant is a website full of science and maths, quizzes and puzzles, and things like that.

That will help you not just see mathematics and science in action, but really understand it all better.

Now if you go to brilliant.org/numberphile

The first 314 viewers of this video who do so it could get 20% off Brilliant's Premium Package.

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Ham Sandwich Problem - Numberphile

awkward

weird

episode

obvious