So I'd like you to imagine a cake, are you imagining a cake?
Good, uh better still let's use a cake shall we? - (Brady: Look and I've even got a sign for you.)
Two candles and one cake - so that is that is the new puzzle. The puzzle was originally set without a
cake, it was a stick. I've deliberately got a linear cake; we have two candles, I like your choice of
candles, these are fresh candles, unlit and they're both blue. That might be important later on that we
can't tell the difference, although a spoiler alert it's not gonna be that important. And the puzzle
is very simple, Brady, I would like you to imagine that the candles are placed randomly on the cake.
There should be a klaxon going off, like what do- what do I mean by random? But I
will tell you, I mean uniformly randomly on a line along this cake. I don't care about
the sideways movement, like imagine there's a line along the centre. The candles go in somewhere.
If I wanted to do it- simulate it, maybe I could generate a random number from zero to one
in some uniform way and use that to mark how far. So maybe that one goes there and this one
goes somewhere else randomly there. The puzzle now comes when you pick up a knife and you say,
the cut of this cake, we're going to split it into two portions, happens also randomly. If I cut it
here then- I'm not too worried about the unequal pieces, we'll just deal with the aftermath of that
another day. The point I want to talk about this time is that there is not a candle on each piece.
So the question becomes now a probability question; if the candles go on randomly and
the cut is done randomly what are the chances that each piece has a
candle? I'm interested to ask your intuition Brady, what is the probability
of each piece having a candle? - (Feels like it's going to be 50/50.)
Why? I know I'm asking you wh-
while you're operating your camera and that's always tough but- - (It feels like the cut should be 50/50.)
What about the cut? - (It feels like when you average out where all the cuts happened after a zillion)
(cuts the average is in the middle.) - The average is in the middle. I agree, and actually that's true of
the candles too but that makes it slightly less intuitive because they're all averaging to
the same place now so- The thing I like about this puzzle is that at first glance to me - and
I'm a mathematician but I'm not you know I'm not an A-grade mathematician, I'm a maths
teacher really - it feels to me at first glance I don't know how to start. I don't know how to
start putting numbers on this other than my intuition. A lot of people say 50% to me,
they think that's- so your intuition is like other people's- - (Can I ask a question? Is it possible)
(to cut the candle? I'm assuming it's impossible to cut the candle.) - I think that's a good assumption and
I love working with you Brady, you're an excellent mathematical modeller, you are aware of
the complications that could get involved and you decide to make things simple. Now we- once we solve
this one we could start having non infinitely small point candles and do it, but I suggest we
don't bother yet. This is what mathematical modelling is all- right let's keep it simple
until we've sorted the simple one out. That's the puzzle, if you haven't already paused the video
and you want to think about it now's a good time. What I'd like to do for this one is solve it in
more than one way and that's kind of the point I wanted to talk about here, is that mathematicians
solve things in more than one way, and that's almost always what makes some profound insight
occur. Because if you say it from one perspective and another perspective you've got like a
joining up at the back and that's a nice moment so we're going to try that in this one. Other piece of
problem-solving advice, and for you as well, that if you're stuck on something try a simpler puzzle.
So if I took one of the candles away, my question has now no meaning right? What the chances we get
a candle on each piece? Obviously can't happen. So candle goes on and I cut somewhere randomly, what
are the chances it's on the left? The candle- - (50/50!)
That- it's got to be 50/50, it feels like there's
symmetry all over the place. They're both random and actually the easy way to answer that I think
is an ordering idea; either the candle comes first or the cut comes first and there's no reason why
one should be more than the other so you've got to go 50/50. That gives us a clue for our first
way of solving the problem: hey, if we want to split this into three ways to solve this problem, here's
part one, ordering. So ordering solved the simpler version, it was either the candle first or the
cut first, so why don't we try some ordering argument? And this is where I probably have
to write some things down. C for candle? Do I- do I differentiate between the two candles?
This is the other problem mathematicians have is like, do I need to model them as different?
And we've got two candles that look identical so I'm just going to say they're indistinguishable.
So let's say the candle could come first and then the other candle and then the
cut, that would be one way of ordering the outcome. - (What are you going to call the cut?)
We could use like a V shape for a cut or a K? - (K for knife.)
Yeah I mean it's annoying- it's an annoying
uh initial letter to use frankly. - (I like K for knife because a K is what a strike in baseball as well so)
That is one possibility. I think this is where your instincts from the simpler one kick in, you realise
well there are other possibilities and they don't feel like they should be any more likely or less
likely, because this is these are all random. Candle, candle, cut - cut with a K is now going to be in my
head forever more. What else could happen Brady? - (You could do the knife and)
(then the two candles?) - Yeah, and that doesn't feel like less likely or more likely. Is there another option?
(Yeah you could do a candle, a knife, and a candle.) - And that's the one that sort of answers
the question I started with, is that's the only time we're going to get a candle on each piece.
And since all of these are equally likely - although I haven't done a strong sort of careful argument
for that - but it feels like it's believable that one should be not more likely than the others,
which means that we get a third for that, a third for that, and a third for that. I now think this is
one way of claiming that the answer to, what are the chances we get a candle on each piece, is a
third. Which isn't a million miles from your 50%, but what I like- - (It's quite a way!)
Well I'm trying to be kind.
The intuition for the single candle one, however stupid a question it is - it feels like a dumb
question - but it's much more strong intuition for me that that's 50%. And then when you can upgrade to 2
you feel like, well maybe it's less likely and so if I'd given you the chance I'm sure you'd have
come down. Anyway, maybe that's right and maybe it's not; there's another way to solve this which uh
maybe feels a bit more mathematical but the key to this one is that each of those options
was equally likely, otherwise you can't make an argument. I guess we should mention that there's
a danger in every probability question about this, like what's the chance you win the lottery?
One argument says you either win or you don't, so it's 50/50 right? And obviously the misconception
there is they're not equally likely. So whenever you're making an argument about probability from
studying cases, you've got to check they're equally likely which I think we have done. - (And that would have been the)
(same if we numbered the candles, if we had candle 1 and candle 2?) - I'd forgotten about that, and you
should- you're right to pick me up on that, I'm glad you asked. So let's say it's candle 1, candle 2,
cut - or knife whatever you call it - now we've gotta we've got more options. So maybe if I mark these,
right that's a 1 and that's- these are the best numbers you've ever seen, almost as good as my mice
from a long time ago. Now there's another option right? With the cut over there I could have candle
2, and counter 1 and these would switch round and that would be a different option. But by a
similar argument you might see where this is going, there's no reason to say that one's more likely
than the other, and so actually the other option you said: cut, candle, candle could be these two
and you can see how this is going to pan out. And now we've got six options, each of which is equally
likely, and two of them satisfy our conditions so I still think we're in a third and it's nice to know
that if you're worried about whether the candles matter - I know some of you are worried - the uh it
doesn't. The argument works both ways around. Hey that's already two ways and that was only part one.
Method two, let's do another way of solving it, because if it comes up with the same answer we're
golden. If it doesn't we've got to ask questions about at least one of the methods. So method two
uh I want to go mathematical on this, I said that these are three decisions; there's a candle, candle,
and a cut - let's represent each one of those random things by a number. So candle 1 is a number
between zero and 1, uniform random distribution, candle 2 is a number zero and 1, and the cut
is a random number between zero and 1 which tells me where it will land. Now if you have two
numbers you could draw a two dimensional picture of what could- like you could have one dimension
and the other dimension for the other number. So if you have three numbers, like in our case, we need to
draw a three-dimensional picture of the outcome. So actually what I'm talking about is a possibility
space but it's going to be a three-dimensional picture. If I start drawing it on here- being able
to draw in 3D is a surprisingly useful skill for a mathematician, and I'm not doing a geometry problem
here, but I'm drawing a probability space. So that's x,y and z - 'zed' if you're in England. And let's say x
is the is candle 1, y is candle 2 and let's say z be the knife? Somewhere between zero and 1
on this axis, somewhere between zero and 1 on this axis, and some between zero and 1 on that axis
will describe the situation we end up. So what we've actually got is a cube. And any arrangement
we've got here is a point within the cube. I mean I can hold one right? So a point in this cube
represents the outcome of our randomness, and we can simulate this nicely on GeoGebra. - (Of course!)
Of course I really like GeoGebra as a sort of tool for tinkering, and what I did here was I drew
my cake and I've got a line that I can cut. So I can choose where the cut goes, I can choose
where candle 1 goes, and I choose where candle 2 goes. And you can see that over on the right
it's moving things around, so actually let's let's make this spin a bit so we can see what's going on.
The x-axis for candle 1, the y-axis for candle 2 and the vertical axis for the knife, and all
three measurements give you where the point is in space. If I put the cut somewhere in the middle
and the candle over on the left here, can you see that this point I'm representing in the 3D space
is it's gone bold and I've got a little announcement here saying 'cut is between
candles'. You can see that from the cake, but if I put the other candle over here it hasn't gone
bold. So basically there's a whole region of the cube where we've got a success, and
there's a region of the cube where we haven't got a success. So I'm just going to colour in in the cube
the points where the cut is between the candles. And I think you might be able to see where this
is going in terms of getting a probability out of it because the entire space is all the possibles,
and those two regions - it's necessary to spin this to see what those regions look like -
you can imagine that those are the regions where the knife is in between and it's really
narrow in the middle if the candles are close together, there's not a big region, and it's wide
if the candles are far apart. So the only question now is what volume of the cube is
shaded? What proportion? It's not trivial but it's also not that difficult, particularly if I
uh spin one of these regions; without turning, I promise you I'm not transforming, I'm just going
to spin it around. And we get a square base pyramid in fact, that's occupying one side
of the cube and going to a point. In fact, here's a cube right? And if I pull one of these off
this is the shape we've just made. Now the way I've configured it it didn't automatically turn
into this shape but it is that volume. And I don't know if you remember the volume of a pyramid Brady,
there's no reason why you should. - (Is it base times height?) - Base times height gives you the
uh volume of a cube- - (And then what? Do you third it?) - It is a third yeah. And this cube was volume 1
so this volume is a third. Which is a relief because that's the number we got earlier and what
we've done is we've proved this in two different ways, we've done it by an ordering argument
from our paper earlier - and ordering in space not time - and we've done it by a geometry argument
which is the volume of a cube. Now the one catch is, what if you didn't know the volume of a pyramid?
And so here's my point about solving things in two ways, what I like about this by accident is
that we're solving a probability problem, we did it with ordering, we did with geometry, and if
I don't know what the volume of a square based pyramid is I can prove it to you by probability.
I'm saying the answer is a third so the volume of this shape is in fact the third
of a cube; and hey a proof of the volume of a square based pyramid via a cake cutting exercise.
This is a serious point, if you can solve things in more than one way you have
two perspectives and there's quite often a moment of profound insight; like that's when
mathematicians get excited, you have two different ways of looking at a problem. Worth mentioning
there's another way of proving these fit together to make a full thing, and there's three of them
if I lock them together - these are 3D printed - then they obviously make a cube. Now that's
not a mathematical proof but it's pretty hard to argue that those three shapes do make a cube and
therefore if they're identical they are third each. So we've done the two different ways but I promise
there was deep water here; maybe this isn't so deep, it's a profound moment to do things from
two ways but there's at least another way of doing this and a glimpse of why questions like this are
maybe more dangerous than you want them to be. I'm gonna refer to something which you talked about
with Grant in a recent video about Bertrand's paradox and how thinking about cake led me on to
thinking a lot about what you and Grant talked about. So the third way of solving this,
and this is a nice thing for all mathematical problems, is instead of theorising just do it. So
uh you're gonna record a thousand takes of me just- no no we're not gonna do it. The trouble is this is
where you start noticing how would you actually pick a random point for the candle without like
a random number generator, it's a pain. So without cutting the cake and spoiling our tea break later
we'll simulate it, uh what mathematicians might call a Monte Carlo method. But we'll simulate it
because if it doesn't give us a third we start to doubt all the theory which we've carefully proven
twice so let's simulate it. So apologies for the crudity of this model, I feel like I'm quitting Doc
Brown there, but this one's not a scale model of Hill Valley, this is a scale model of a rubbishy
rectangular cake. I've got two tiny candles on here and a cut and all it's going to do if I
click on one of the buttons here, it's just going to pick random positions for each of them and you
can see this one there's a cut just in between the candles - success - and it's going to count them.
And I'm not going to keep clicking, I'm just going to let it go. This isn't rigged, but it's done
50 odd trials now, it's not doing super quickly, but we've got a success rate ticking away up here that
is relievingly to me as a mathematician settling around about a third. Now I wouldn't expect it
to be really tight on a third until I've done thousands; and it actually always surprises me
when I do models how long it takes to converge on what theories- even if I believe the theory
it takes a long time for randomness to to back it up. But it's not bad, I mean this feels like I have
rigged it, but it's settling about 0.34 - but this is backing us up which is a relief. So the two ways of
solving it don't feel like they've had any crisis of conscience and the experiment, however computer-y
remodeled it is, is is backing us up. Slightly above a third at the moment. The trouble is,
when we upgraded this problem from sticks to cake, it feels very natural for me to then in my head
upgrade from linear cake to a proper cake. So I started asking the question, what happens if you do
this on a square cake or a round cake which feels much more traditional. And I scared myself slightly.
If you'd like to see ben scaring himself with a 'proper cake' check out the links to part two of
this video. They'll be on the screen and in the description. Or if you'd like to just see more
Numberphile videos about stuff getting cut, well let's just say that's a topic we've covered before.
Cutting a round cake on scientific principles... - I mean there's more to a cake than just size.
Some people like the sort of crusty bit around the outside.. - Again, links in all the usual places.