A column of soldiers is 25 miles long and they march 25 miles a day. One morning a messenger started at the rear of the column with a message for the guy up front. The messenger began to march and gave the message to the guy up front and then returned to his position by the end of the day. Assume that the messenger marched at the same rate of speed the whole time. How many miles did the messenger march?

By the problem, the speed of the column and thus the speed of the lead officer is 25 miles per day.

Let the messenger meet the lead officer in m days (say). Then it
follows that the speed of the messenger is (25 + 25/m) miles per day.

Since the messenger mat his lead officer and returned to his original position, it follows that he covered a total of (25+ 50m) miles in a single day.

Acoordingly, we must have:
25 + 50*m = 25 + 25/m, giving:
m = 1/v(2)

Thus, the messenger traveled a total distance of 25[1+v(2)]
= 60.36 miles