The sum of the lengths of the edges of a cube is equal to half the perimeter of a square. If the numerical value of the volume of the cube is equal to one-sixth of the numerical value of the area of the square, then the length of one side of the square is:

Cube has 12 sides of equal length. 
Let the length of each side be c units.
Sum of lengths of all sides of the cube = 12c units.
Let the side of the square be s units.
Perimeter of square = 4s units.
Given, $$12c = \dfrac{1}{2} \times 4s => c = \dfrac{s}{6} => s = 6c$$
Volume of cube = $$c^3$$
Area of square = $$s^2$$
Given, $$c^3 = \dfrac{1}{6} \times s^2$$
Substituting s = 6c in above equation.
$$c^3 = \dfrac{1}{6} \times 6c \times 6c => c = 6 units$$
Then, s = 6c = 6*6 = 36 units.
Therefore, side of square = 36 units.