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

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

  • That's just wrong.

  • 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.

  • And your score is going to be equal to

  • 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.

  • So let's start out.

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

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

  • There's a three by two.

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

  • But I can make a six by one?

  • You can make a six by one.

  • In fact, why don't we do that?

  • There we go. There is a six by one.

  • What else would you like in here?

  • - [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.

  • You want low and 16 is not low enough.

  • So what we could we do?

  • 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.

  • You can score five.

  • So let me show you this one.

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

  • So that score is five,

  • 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?

  • What about an eight by eight?

  • What about an n by n?

  • What is the optimal?

  • Ed Pegg has happened to prove solutions.

  • 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.

  • So this is a rectangle 50 by 101

  • And this rectangle is 51 times 101.

  • 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.

  • But you can at least do 101.

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

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

  • But it doesn't always 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.

  • So this does not continue to increase.

  • It's a very complicated function.

  • 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!

  • But good luck.

  • (laughs) Good luck. Dream on.

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

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

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A2 初級

蒙德里安拼圖 - 數字愛好者 (Mondrian Puzzle - Numberphile)

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    林宜悉 發佈於 2021 年 01 月 14 日
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