We first imagine the cross with the blue cubes only:
We now glue the yellow faces to the up and bottom faces of the blue cross. We require 5 yellow cubes for wrapping each of the two blue cubes. (We used transparent yellow cubes)
Now we need one yellow cube for each of the four corners:
Finally, we have 4 faces to cover, so we need 4 more squares.
Therefore, we used $5+5+4+4 = 18$ yellow cubes.
This problem is taken from the UKMT Mathematical Challenges.