Proving the Integer Rectangle Property: A Geometric Puzzle

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Homework Help Overview

The discussion revolves around a geometric puzzle involving the properties of rectangles in a plane, specifically focusing on a large rectangle partitioned into smaller rectangles with integer dimensions. The goal is to prove that the larger rectangle also possesses integer dimensions based on the properties of the smaller rectangles.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts various methods, including summing the diagonals of smaller rectangles and working from the larger rectangle to the smaller ones and vice versa. They express difficulty in articulating their intuitive understanding. Other participants introduce the idea of integrating a complex function over the rectangle and discuss the implications of integer dimensions on the function's behavior.

Discussion Status

Participants are exploring different mathematical approaches, including integration of complex functions, to understand the properties of the rectangles. There is a recognition of an interesting connection between the integer dimensions of the sides and the behavior of the integrals, though no consensus or definitive proof has been reached yet.

Contextual Notes

Some participants mention challenges with integrating complex numbers and express a need for further reading to grasp the concepts being discussed. The original poster indicates that this is not a homework problem but rather a puzzle, suggesting a more exploratory nature to the discussion.

intrepid_nerd
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Not homework but this is probably the best suited place for a puzzle:

A large rectangle in the plane is partitioned into smaller rectangles, each of which has either integer height or integer width (or both). Prove that the large rectangle also has this property.

I've given this several attempts, starting with adding up the diagonals of each smaller rectangle, I didn't think this was good enough for a definitive proof. I've worked from the larger in towards the smaller ones and the smaller ones out to the larger ones, every time it seems so easily intuitive but the I fail to connect words to it, any help is appreciated!
 
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The easy way is to integrate e^(i*pi*(x+y))dxdy over the rectangle. Do you see what happens if a side is an integer?
 
Dick said:
The easy way is to integrate e^(i*pi*(x+y))dxdy over the rectangle. Do you see what happens if a side is an integer?

I've never tried integrating imaginary numbers so i can't really follow that function but I'm guessing that the function plots as a non-smooth curve. I've got some reading to do to try and figure this out. Thanks for the new perspective though, I'm intrigued.
 
Last edited:
Actually I flubbed the integrand. Make that exp(2*pi*i*(x+y)). It's not a hard complex integral. Integrate it just like you would integrate exp(a*x). The basic point is that exp(2*i*pi*n)=1 where n is any integer. So exp(2*i*pi*x)-exp(2*i*pi*(x+n))=0.
 
that worked well when set to zero; then only when there was a side of integer length would the equation be satisfied. What beauty!
 
Yeah it is a nice trick. Good to hear you finally got it.
 

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