Example 1. In a mathematics competition: three math teachers made the following guesses about the results of the final for the participating students:
A: Xiao Wang first place, Xiao Liu second place;
B: Xiao Wang second place, Xiao Luo third place;
C: Xiao Li second place, Xiao Luo fourth place.

As a result, each of the three teachers' guesses was exactly half correct. What are the rankings of each person? (No duplicate rankings)
Consider a $4 \times 4$ grid array (Figure 1). For convenience, let's denote Xiao Wang, Xiao Liu, Xiao Luo, and Xiao Li as $A_{1}, A_{2}, A_{3}, A_{4}$ respectively.
If a teacher estimates that $A_{i}$ is in the $j$th place, a circle is drawn in the cell $a_{1 j}$. Thus, circles should be drawn in the cells $a_{11}, a_{22}, a_{12}, a_{33}, a_{42}, a_{s}$. Then, connect the results (circles) that belong to the same person with lines. Finally, the correct estimates are marked as solid dots. Clearly, each row and each column of the grid array should have exactly one solid dot (each person gets and only gets one rank), and each line should have exactly one solid dot (each estimate has one half correct).

First, assume that $a_{11}$ is a circle (not a solid dot), then $a_{22}$, which is on the same line as $a_{11}$, must be a solid dot. $a_{12}$, which is in the same column as $a_{22}$, must be a circle, and $a_{32}$, which is on the same line as $a_{12}$, must be a solid dot. $a_{34}$, which is in the same row as $a_{33}$, must be a circle, and $a_{42}$, which is on the same line as $a_{34}$, must be a solid dot (Figure 1). This way, the second column has two solid dots $a_{22}$ and $a_{42}$, which contradicts the reality ($A_{2}$ and $A_{4}$ cannot both be second). Therefore, $a_{11}$ must be a solid dot. Thus, $a_{12}$, which is in the same row as $a_{11}$, must be a circle, and $a_{33}$, which is on the same line as $a_{12}$, must be a solid dot. $a_{34}$, which is in the same row as $a_{33}$, must be a circle, and $a_{42}$, which is on the same line as $a_{34}$, must be a solid dot. As shown in Figure 2, the result is: Xiao Wang is first, Xiao Li is second, Xiao Luo is third, and Xiao Liu is fourth.