矩形整数边问题 from WXC

fzy · 发表于 2005-3-5 01:39:17

一个矩形被划分为若干个小矩形，每个小矩形的边平行于大矩形的边，并且每个小矩形都至少有一边长度为整数。证明大矩形至少有一边长度为整数。

Also there is a general version of the problem that is not yet solved at WXC:

Let G be a subgroup of the real numbers, such as integers, rational numbers, algebraic numbers. If a rectangle is filled with small rectangles, each has at least one side in G, then the large rectangle has at least one side in G.

乱弹 · 发表于 2005-3-5 03:01:20

Similar methods should work, right?

fzy · 发表于 2005-3-5 20:02:34

Some proofs work. The one posted at WXC does not.

乱弹 · 发表于 2005-3-6 02:45:01

I am not in the pure math field and not familiar with the Journal you mentioned in WXC (since it is so old, it should be quite hard to find). Thanks a lot.

fzy · 发表于 2005-3-7 18:44:08

This (American Mathematical Monthly ) is probably the most famous Elementery Math magazine in the world. You should be able to find it in the central library of any large city. Or if you are near a university, visit its Math Library (usually in or near the math department.) I did not find this article online, and do not know how many proofs you can find online.

sean9991 · 发表于 2005-3-10 00:41:46

by induction. we can shrink the rectangle in the following way. consider the top left corner small rectangle. if it's vertical side is integer length, we cut the large rectangle along the bottom of the small rectangle. therefore, we reduce the vertical side of the large rectangle by an integer length. if not, then the horizontal side is integer length, we cut the large rectangle along the right side of the small rectangle. we reduce the horizontal side of the large rectangle by an integer length. Note the condition still holds with the shrinked rectangle and the smaller rectangles (at least one of their sides are integer lengths). Eventually, the larger rectangle will have either vertical / horizontal side equals 1.

fzy · 发表于 2005-3-10 18:11:07

The problem with your approach is that when your cut off the top strip, there might be small squares at top whose integer side is along the top edge, and whose noninteger side is above your cut line. So the whole rectangle is cut off, and the second rectangle from top then is cut off a noninteger portion.

Hint: Convert the problem to a 一笔画问题.

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矩形整数边问题 from WXC

Similar methods should work, right?

Some do and some don't

Are there any web resources for these proofs?

回复：Are there any web resources for these proofs?

回复：矩形整数边问题 from WXC

Problem and hint