So, you know, this is not an unrealistic situation in principal.

When false results get proved.

When false results get proved, when false results

get published, that is, because they've never been proved.

The false results get published when, what's going on then is that the referee

has gone through it and and think, and thought, everything's okay.

Okay, here it's dramatic because the answer's absurd.

And these peyton, they force.

But this is actually a not unrealistic scenario.

And then so, giving a high grade is, isn't, you know, it's not about.

I mean it's, it's, it's, I'm doing this to emphasize

the distinction between the various things we're looking at here.

And this one I think makes it clear.

That there are other issues involved other than, other than logical correctness.

But as I said, if this was a regular mathematics

course, I wouldn't be giving 16 out of out of 24.

In fact, I'm not even sure I'd give any

marks for this, if a student came up for this.

Because it really this is really a big mistake.

Okay?

But within this context, within the context of this course this

is the kind of thing we're looking for in

looking at mathematical thinking and mathematical proofs and communication.

Okay.

There we go!

Well that's the end of the, the problem set questions.

But let me leave you with one more tantalizing little puzzle.

We are probably familiar with the story of Archimedes, lived in Greece about 250 B.C

[UNKNOWN]

who was asked by the king to determine whether a crown

had been given was actually made of pure gold or not and

that involved calculating the density and he knew how to calculate to

find out the, the, the, the massively, the weights of the crown.

But the question is how do you calculate its volume?

now, now Archimedes knew lots of mathematics for calculus

in volumes, indeed he'd invented a lot of that mathematics.

He was able to calculate areas of circle

and volumes of spheres and various other shapes like

boxes and rectangles and pyramids and so forth.

He knew all of that stuff.

So he had a lot of mathematics at his disposal that he could have applied.

But it didn't seem to work for something

irregular, like a crown, or at least not easily.

But then one day when he's taking a bath, this is sort of

the story goes, when he is taking a bath, he has this amazing insight.

He says to himself, if I immerse the crown in water, it will displace some water.

In fact, the amount of water it will displace

is exactly equal to the volume of the crown.

So if I collect the water when it spills out of the bath,

and if I, when I put a crown inside it, into, into the

water, then I'll be able to just measure the volume of the water

in, in a standard way and I'll know the volume of the crown.

And as that story goes, he was so

impressed and tickled about his solution that he

jumped out of the bath and ran stark

naked through the streets of Athens crying out eureka,

eureka! Which is Greek for I found it!

I found it! now, I've known lots of mathematicians.

I certainly haven't known Archimedes, but I doubt if even a mathematician deep

in the throes of solving a problem would run naked through the streets.

However, I can imagine him being extremely pleased with himself

and having an adrenaline rush when he had that insight.

Because that's a great example of thinking outside the box.

He knew lots of techniques for calculating

volumes, he invented many of them, but on this occasion he thought

outside the box and found a really elegant solution that was different.

And the puzzle I'm going to give you is very much along those lines and it

actually is about taking a bath. And here it is if

it takes half an hour for the cold water faucet to fill your bathtub and an hour

for the hot water faucet to fill it, how long will

it take to fill the tub if you run both faucets together?

Now this looks like one of those, those frustrating little

word problems you get in, in, in, in high school.

Okay, where you, you end up, you sort of say that

the, that the rate of flow of the cold water be f.

And, and then the t.

And you, you, you write down some equations

and you, you you figure something out, okay.

I'm sure you know

how to do that.

You can apply standard technique for doing this kind of

thing involving rates of change, and you'll get the answer.

But you don't need to know any of that, you don't need

to do any calculations at all in order to solve this one.

If you think outside the box you can answer it without doing any

of those calculations based on the rates of flow or anything like that.

You simply have to think of the problem a different way.