This is a beautiful geometry puzzle that my friend, Alok Mittal, gave to children in his Mathematical Circles class yesterday.
There are four towns at the corners of a perfect square with a side of 1 km. Your task is to connect these towns through a railroad such that all the four towns are connected such that the length of the railroad is the smallest. As an example, if you connect all the four sides of the square, then the length of the railroad is 4 km. If you connect any three sides, all the four towns are still connected, and the length is 3 km. Can you do better?
As always, please send your answers directly to me at alokgoyal_2001@yahoo.com. If you like the puzzle, please share it with others. If you have interesting puzzles to share, please send them to me at my e-mail given above.
Happy railroading!
Alok Goyal's Puzzles Page 
Two diagonals.. And answer is 2* root (2)
Good try Tarun, but there is a way to do better than this as well.
How about this:
Let the square be ABCD with centre O.
in triangle ABO, let q1 be the centroid. similarly, q2 be the centroid of CDO
Join ABO trough q1 and CDO through q2. I believe this will be the shortest and further iterations in this fashion won’t converge(not sure, have to check). So, the path will look something like >-<
ah, never mind, the solution was already out. Sorry, didn’t see it before.