Name: Mondrian Puzzle - Numberphile
Uploaded: 2020-03-27T18:01:22.000Z
Duration: 3 min 56 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...

The superlative

- You are Mondrian's worst art critic 100 years ago.

Modal verbs - A2

You tell him he can no longer create art where two rectangles have the same dimension.

You're going to say, okay, I am going to score each of your pieces of art.

We're going to insist that all of the rectangles have got different dimensions.

your largest area of rectangle minus your smallest area of rectangle.

Your smallest score possible is what you want to go for.

The other condition is you have to tile the whole canvas.

Brady, how big do you want this first rectangle to be?

- [Brady] Three by two.  - [Gordon] Three by two, okay.

So we can no longer make a two by three or three by two anywhere else.

- [Brady] Should we make that little square, two by two?  - [Gordon] Two by two? Yeah.

If you were to stop right now, this is area 20.

So if you were to stop right now, your score would be 20.

That's your biggest minus your smallest, four.

Your score would be 16 and that is not very good.

We could split this into an 8 here and a 12.

We could do that. And so now our score is 12 minus 4 is equal to 8.

You can tell Mondrian you are scoring this an eight.

Your largest rectangle is eight, minus your smallest, it is three.

and that is the best possible score that you can get for a six by six Mondrian art puzzle.

Of course, you have to ask yourself, what about a seven by seven?

He's proven that we have these optimal solutions, beautiful results

and great for algebra class as well  to come up with general solutions.

So for example, for an odd square, let's say that you are dealing with a square 101 by 101.

Well, at least you know for sure that you can split this nearly in two.

And you can see right away that your score for any odd square,

You can see that it's just the edge length.

So you can see can guaranteed that you can at least do 101.

You can actually do a lot better than 101.

An unsolved problem here is when this number ever goes to 0?

You wpi;d think, well, this has got to increase.

For example, 17 by 17, the score for that is only 8.

And the score for 16 by 16, 15 by 15, I think they're larger.

And it is not clear that this, for example, doesn't go to zero.

- [Brady] Presumably called the Holy Grail

is some kind of generalization for an n by n square.

The solution will always be n cubed minus four plus four.

- [Gordon] Oh, that would be a joy again!

- [Brady] You can follow it all the way through.