Make it Pink

The same surface is painted, except for faces of the small cubes that were originally "inside" the big cube. There are 6 of these faces, each with area $1\times 1 = 1$ square inch:

so the answer is $6$ square inches.

The Surface area of a cube can be represented as 6s^2 where s is the side length. So the surface area of the blue cube is 6 x 1^2 which is 6 inches squared. For the pink cube, there are 8 cubes with half the side length, so we can represent this as 8(6((s/2)^2). Solving, half of the side length is 0.5. 0.5^2=0.25. 0.25 x 6 is 1.5, and 1.5 x 8 is 12 inches squared in total. So the blue cube surface area is 6, while the pink cube surface area is 12, leaving a difference of 6 square inches.

Looking at the smaller cubes, we can see that for each cube, 3 of the faces are pointing outwards and 3 are pointing inwards. Since paint only the outward faces must take the same amount of paint as to paint all of the larger cube, and there number of inward faces is equal to the number of outward faces, it must be that it takes twice as much paint to paint the smaller cubes as compared to the large cube. All that remains is finding the surface area of the large cube, which is trivial.

This the 1"x1x1" cube forced me to think conceptually rather than mathematically. Please bear with me as I'm answering on my phone.

Essentially, the original cube has a surface area of 6 sq in. 6 sides @ 1"X1" per side.

By cutting the cube into 4 equal pieces we produce 3 more faces on 4 cubes of half the size, thus doubling the surface area. (3 original sides plus 3 new interior sides).

This means that we doubled our original surface area. That is the total surface area is 12 sq in. However, the question asks for the difference between pink and blue paint, so the difference is 6.

surface area of cube = 6 × side²

surface area of big blue cube = 6 × 1² = 6 in²

surface area of 8 pink cube = 8 × 6 × (1/2)² = 12 in²

difference = 12 – 6 = 6 in²