A rectangular plot is almost square: its width and length are integers in meters that differ by exactly 1 meter. The area of the plot, in square meters, is a four-digit number, with the thousands and hundreds digits being the same, and the same is true for the tens and units digits. What are the possible dimensions of the plot?
The area is a number of the form $a a b b$, where $a$ and $b$ represent digits; now recall that

$$
a a b b=1100 a+11 b=11(100 a+b)
$$

Let $x$ be the width of the plot, then

$$
x(x+1)=11(100 a+b) \quad(\mathrm{I})
$$

and we deduce that $x$ or $x+1$ is a multiple of 11. Searching for multiples of 11 that satisfy condition (I) is quite laborious, so to simplify, let's establish which values $x$ can have. We will look for the minimum and maximum values for $x$:

- Minimum: the smallest possible area is $1111$, so $x(x+1)=1111 \Rightarrow x>32$ (II).
- Maximum: the largest possible area is 9999, so $x(x+1)=9999 \Rightarrow x<100$ (III).

Now we look for $x$ and $x+1$ satisfying (I), (II), and (III).

$$
\begin{aligned}
& 33 \times 34=1122 ; 43 \times 44=1892 ; 44 \times 45=1980 ; 54 \times 55=2970 ; 55 \times 56=2970 \\
& 65 \times 66=4290 ; 66 \times 67=4422 ; 76 \times 77=5852 ; 77 \times 78=6006 \\
& 87 \times 88=7656 ; 88 \times 89=7832 ; 99 \times 100=9900
\end{aligned}
$$

We find 3 possibilities for $x: 33, 66$, and 99.