Cut Table

Impose coordinates parallel to the sides of the rectangle (units centimeters). Label the subrectangles $R_1,\ldots,R_n$ , and the original/large table $R$ .

Given a rectangle $[a,b]\times[c,d]$ , the integral

$\displaystyle\int_a^b\int_c^d e^{2\pi i(x+y)}dydx$

$\displaystyle=-\frac{1}{4\pi^2}(e^{2\pi b}-e^{2\pi i a})(e^{2\pi i d}-e^{2\pi i c})$

is zero if and only if at least one of $d-c,b-a$ is an integer - that is, if the rectangle has at least one integral side. In particular, note

$\displaystyle\int\int_{R_j}e^{2\pi i(x+y)}dA=0$ for each $j$ .

Therefore,

$\displaystyle\int\int_R e^{2\pi i(x+y)}dA=\sum_{j=1}^n\int\int_{R_j}e^{2\pi i(x+y)}dA$

$\displaystyle=\sum_{j_1}^n0=0,$ so the original rectangle (table) must have at least one integral side.

No matter how you divide up the table into subrectangles, you can always extend the sides of one or two rectangles to form 4 large subrectangles. This simplifies the problem quite a bit.

In the example image, I've extended one side of a rectangle to form subrectangles A,B,C,D. If subrectangle AC is integer vertically, than the table must be integer vertically. If subrectangle AC is non-integer vertically, then it must be integer horizontally, forcing subrectangle BD to be integer horizontally. Since an integer plus an integer is an integer, that would force the table to be integer horizontally. Therefore, the table must be integer on at least one side.

simple logic. there is one parallel vertical line in the image which is not vecrtical side of original rectangle. Constider two sub-rectangles consisting of those if vecrtical line is not integer both horizontal lines would be integer summing up to horizontal side an integer