1. The students $A, B, C, D$ competed in a race. Each of them predicted the order in which the race would finish. Student $A$: $A B D C$, student $B$: BACD, student $C$: $C B D A$, student $D$: $D C B A$. It turned out that no one correctly predicted the order of all the racers, but only one of them guessed the position of just one racer. What was the order of the finishers in the race?
Solution. Let's write down the given predictions in the form of a square table:

$$
\begin{array}{ll}
A: & A B D C \\
B: & B A C D \\
C: & C B D A \\
D: & D C B A
\end{array}
$$

All four students predicted a different winner of the race, which means that one of them must have guessed who came first correctly. Now, from the condition of the problem, it follows that no one guessed the order of the other competitors correctly. From the table, we see that no one predicted that $D$ would be second, nor that $A$ would be third, nor that $B$ would be fourth. According to this, the first place was taken by $C$, and he alone correctly predicted his own victory. Finally, the order was: $C D A B$.