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

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

[H] Don't you think? [B] Yeah.

[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

what a half a sandwich is?

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

the ham isn't exactly even...

You know, how do you- how do you...?

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

how can you work out where halfway is?

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

What is half a sandwich?

[H] I want exactly half of this slice,

I want exactly half of the ham,

and I want exactly half of that slice.

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

And it doesn't matter where the sandwich is.

I want it even if it's like that.

But thankfully, the Ham Sandwich Theorem can help us!

So, the Ham Sandwich Theorem says that

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.

[B] Okay.

[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,

this slice of bread.

Now if I hold the knife here,

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

And if I move the knife over there,

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,

and half of the bread is on that side.

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

But that was me holding the knife at this angle

and moving through that way.

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.

So! That's fine.

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] [Laughs] [B] Cool.

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

[Both laughing]

[H] Is it 'cus I keep touching it?

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

[Both continue laughing]

[H] "Welcome to cooking with Hannah!"

[B] Right, yeah.

[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,

let's say like that,

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

and 50% is on that 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!

[B] Nice, nice. [H] Okay.

You can apply the same principles.

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?

[H]: Weird, huh? I know. [B]: Right.

[H]: Exactly.

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

See these five circles on the screen?

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

***Spoiler Alert***

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

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