3. A $4 \mathrm{~cm} \times 4 \mathrm{~cm}$ square is split into four rectangular regions using two line segments parallel to the sides.
How many ways are there to do this so that each region has an area equal to an integer number of square centimetres?
Solution
Let the lengths of the sides of the regions be $k \mathrm{~cm}, \ell \mathrm{cm}, a \mathrm{~cm}$ and $b \mathrm{~cm}$, as shown. We know that each of these is less than $4 \mathrm{~cm}$.
Then the regions have areas $k a \mathrm{~cm}^{2}, k b \mathrm{~cm}^{2}, \ell a \mathrm{~cm}^{2}$ and $\ell b \mathrm{~cm}^{2}$ respectively, and thus each of $k a, k b, \ell a$ and $\ell b$ is an integer.
Hence $k a+k b=k(a+b)=4 k$ is an integer, so that $k$ is an integer multiple of $\frac{1}{4}$. Similarly, each of $\ell, a$ and $b$ is an integer multiple of $\frac{1}{4}$.
However, none of $k, \ell, a$ and $b$ is an integer multiple of 4 , because each is between 0 and 4 . Hence each of them has to be an integer multiple of $\frac{1}{2}$ in order that each of $k a, k b, \ell a$ and $\ell b$ is an integer.
If both $a$ and $k$ are integers
then clearly all four rectangles have integer area. In this case we have three choices for $a(1$, 2 and 3 ) and three choices for $k$ (also 1,2 and 3 ), giving $3 \times 3=9$ choices.
If one of $a$ or $k$ is not an integer (that is, an odd multiple of $\frac{1}{2}$ )
then the other is 2 , otherwise $k a$ is not an integer. When $a=2$ it is possible for $k$ to take four such values $\left(\frac{1}{2}, \frac{3}{2}, \frac{5}{2}\right.$ and $\left.\frac{7}{2}\right)$ and similarly when $k=2, a$ could take the same four values.
Note that once $a$ and $k$ have been chosen, then these determine $b$ and $\ell$ uniquely.
Thus altogether there are $9+4+4=17$ ways.
NoтE
If the grid can be rotated or reflected, so that, for example, $a=1$ is counted as being the same as $k=1$ or $a=3$, then there are 5 ways, given by the following pairs of values of $(a, k)$ or $(k, a)$ :
$$
(1,1),(1,2),(2,2),\left(\frac{1}{2}, 2\right),\left(\frac{3}{2}, 2\right) \text {, }
$$
as shown below.