Proving the Integer Rectangle Property: A Geometric Puzzle

[intrepid_nerd]

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!

 

[Dick]

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?

 

[intrepid_nerd]

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.

 

[Dick]

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.

 

[intrepid_nerd]

that worked well when set to zero; then only when there was a side of integer length would the equation be satisfied. What beauty!

 

[Dick]

Yeah it is a nice trick. Good to hear you finally got it.