Answer
It would have to analyze all possible permutations of the 25 cities. That
would mean $25 !$ possible paths. This is a total of 1.551121$\cdot 10^{25}$ paths. By
analyzing $10,000,000$ paths per second our computer would need 1.551121.
$10^{18}$ seconds, which is around $49,185,724,100$ years.
This is a famous problem in computer science called the Travelling Salesperson.
Scientists are still trying to come up with more and more
efficient algorithms.
Work Step by Step
By using the algorithm designed by the salesperson, which is to list all
possible ways to visit the 25 cities it would have to analyze all possible
permutations of the 25 cities. That would mean $25 !$ possible paths. This is
a total of 1.551121$\cdot 10^{25}$ path. By analyzing $10,000,000$ paths per second our
computer would need 1.551121$\cdot 10^{18}$ seconds, which is around $49,185,724,100$
years.
This is obviously not a very efficient algorithm for our task, given the number
of cities that we have to analyze.
A possible way to go about it would be to start with the two closest cities.
