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This problem trains for: SAT-I, GMAT, AMC-10.

The intersection of the surface of a cube of side 1 with the surface of a sphere of radius 1 that have the same center is formed of:

The sphere of radius 1 unit has a diameter of 2 units.

The cube of side 1 unit has a diagonal of √3 units.

and

Therefore, if the two solids are concentric (have the same center) the cube is entirely contained within the sphere. The intersection of the two surfaces is empty.