Seven consecutive integers stood in a row, arranged from the smallest to the largest. After a while, the numbers got bored, so the first one swapped places with the last one, then the middle one moved to the very beginning of the row, and finally, the largest number stepped to the middle. To its great delight, it found itself next to a number with the same absolute value. Which seven numbers could have been in the row?

(S. Bednářová)

Hint. Determine the difference between the two numbers with the same absolute value.
We will denote the numbers in ascending order as 1. to 7. Their arrangement changed as follows:

$$
\begin{array}{lllllll}
1 . & 2 . & 3 . & 4 . & 5 . & 6 . & 7 . \\
7 . & 2 . & 3 . & 4 . & 5 . & 6 . & 1 . \\
4 . & 7 . & 2 . & 3 . & 5 . & 6 . & 1 . \\
4 . & 2 . & 3 . & 7 . & 5 . & 6 . & 1 .
\end{array}
$$

Two numbers with the same absolute value can be either 3. and 7., or 7. and 5. Since these are two different numbers, one must be positive and the other negative. The positive one is the larger of the two, i.e., the 7. number.

First, let's consider that the same absolute value is shared by the 3. and 7. numbers. Their difference is 4, so the larger of the two is $4: 2=2$. The smallest number in the sequence is 6 less than the 7. number, so it is $2-6=-4$.

Now, let's consider that the same absolute value is shared by the 7. and 5. numbers. Their difference is 2, so the larger of the two is $2: 2=1$. The smallest number in the sequence is then $1-6=-5$.

In the sequence, the integers could range from -4 to 2 or from -5 to 1.

$\mathrm{V}$