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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.

• 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.

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

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