Have I explained that a tiled rectangle has at least one integer side properly?

Question

I am trying to explain why if a rectangle can be tiled by smaller rectangles each of which has at least one integer side, then the tiled rectangle has at least one integer side.

My attempt:

For the integer case, when a tile is placed the tile covers a region whose squares have a summed value of $0$, whereas for the continuous case the integral of a line of integer length from $(x_1, y_1)$ to $(x_1, y_1 + \Delta y)$ or from $(x_1, y_1)$ to $(x_1 + \Delta x, y_1)$ is $0$. This is because when we integrate a line of an integer length which is parallel to either the $x$ (tile width) or $y$ (tile height) axis, it is the same thing as when we place a $1 \times h$ tile anywhere on a region of integer dimensions, recalling that the sum of the values of the squares covered by the tile is $0$. If a tiling is possible, this means the integral of the individual tile heights or widths are $0$. Then it follows that the integral of the region is also $0$ since the sum of the tile areas makes up the region, therefore the sum of the integrals of the tiles make up the integral of the region.

$$\int_{a}^{b} \int_{c}^{d} f \,dy\,dx = \int_{a}^{b} e^{2\pi ix} \,dx \int_{c}^{d} e^{2\pi iy} \,dy = \bigg(\frac{e^{2\pi ib} - e^{2\pi ia}}{2\pi i}\bigg)\bigg(\frac{e^{2\pi id} - e^{2\pi ic}}{2\pi i}\bigg)$$

I feel like it is not correctly explained and not completely explained.

Can anyone help me?

Answer 1

As you doubted, your explanation isn't quite clear and complete yet. You should start by defining the function $f$ appearing in the integral.

Also, the part:

the tile covers a region whose squares have a summed value of $0$

What are the "squares of the region"?

Maybe you want something like: The integral over the whole rectangle can be calculated by summing the integrals over the tiles, or on the other hand as a single double integral over the whole rectangle.

I guess the key words are "the integral of $f$ over a rectangle is $0$ if and only if that rectangle has an integer side". And you have already shown this by calculating the integral (one of the factors must be $0$ and this means the exponentials must be equal which in turn leads to the corresponding difference of numbers e.g. $b-a$ being an integer).