To find the number of possible arrangements where the sum of the numbers at each end of the line is 4, we consider the possible pairs for the ends of the line: (1, 3), (3, 1), and (2, 2).

For each of these pairs, the remaining 5 dice can show any of the 6 faces. Therefore, there are:

\(6^5\) ways to arrange the middle 5 dice.

Since there are 3 possible pairs for the ends, the total number of arrangements is:

\(6^5 \times 3 = 7776 \times 3 = 23328\)