A metal wire of resistance R is cut into three equal pieces that are then connected side by side to form a new wire , the length of which is equal to one third of the orignal length . The resistance of this new wire is ---------------------

Your question seems incorrect. There is a contradiction in your question. If three pieces of wire are connected side by side it means they are connected in series combination and if the length of new wire formed is one third of the original wire then three pieces must be connected in parallel combination to satisfy this condition.

I am providing you answer of both the cases.

The metal wire is cut into three equal pieces.

Let the original length of the wire be l

=> length of each of the three wires, l' = l/3

resistance of the original wire, R = ρl/A

Resistance of each of the wire, R' = R/3

Now as the three wires are connected side by side, that is, in series then

The total resistance R''= R' + R' + R'

 = 3R'

 = R

But if the wires are connected in parallel( the new wire is one third of the original length), then 

the length of each wire, l' = l/3

resistance of each wire, R'= R/3

total resistance of the arrangement, R'' = R'/3 =R/9