第一次聚餐：切蛋糕

constant · 发表于 2005-12-8 21:40:56

难度：+++

脑坛第一次聚餐时有一个正方形蛋糕。（主菜单由莲花MM负责，现在保密。）这个蛋糕要切成N个面积相同的长方形。但每个人都要求自己的长方形和别人的不相似。N等于几时才可能？

sean9991 · 发表于 2005-12-8 23:19:52

4k+1, k =0,1,2,...

constant · 发表于 2005-12-9 21:10:32

How do you do, say, 5?

sean9991 · 发表于 2005-12-13 23:46:40

Sorry for late response. Just too busy lately...

It seems 5 doesn't work, as it turns out 4 of them are actually similar. But 7 works. The idea is to try to form a spiral of rectangles from corner to center.

Let's start with a sqrt(7) square. The first rectangle starts from any corner. It has sides a=(8sqrt(7)-sqrt(133))/21 = 0.458736 and b=21/(8sqrt(7)-sqrt(133))=2.1799. The next rectangle sits at the adjacent corner. It has sides c=sqrt(7)-b and d=1/c. We can continue in this fashion.

The first rectangle could have been a=8sqrt(7)+sqrt(133))/21 = 1.55707 and b=21/(8sqrt(7)+sqrt(133))=0.64223. The spiral can be formed similarly.

constant · 发表于 2005-12-14 05:06:34

7 or more is correct

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第一次聚餐：切蛋糕

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How do you do, say, 5?

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