The 27 small cubes that make up the larger $3 \times 3 \times 3$ can be broken into 4 categories: 1 small cube in the very centre of the larger cube (not seen in the diagram), 8 small cubes at the vertices of larger cube (an example is marked with $V$), 12 small cubes on the edges not at vertices (an example is marked with $E$), and 6 small cubes at the centre of each face (an example is marked with $F$). The centre cube contributes 0 to the surface area of the cube. Each of the 8 vertex cubes contributes 3 to the surface area of the larger cube, as 3 of the 6 faces of each such cube are on the exterior of the larger cube. Each of the 12 edge cubes contributes 2 to the surface area of the larger cube. Each of the 6 face cubes contributes 1 to the surface area of the larger cube. There are 10 small red cubes that need to be placed as part of the larger cube. To minimize the surface area that is red, we place the red cubes in positions where they will contribute the least to the overall surface area. To do this, we place 1 red cube at the centre (contributing 0 to the surface area), 6 red cubes at the centres of the faces (each contributing 1 to the surface area), and the remaining 3 red cubes on the edges (each contributing 2 to the surface area). In total, the surface area that is red is $1 \times 0+6 \times 1+3 \times 2=12$.