Unit Cubes 4 – Faces

For the number of blocks with $1$ side painted, the formula is ${ 6(n-2) }^{ 2 }$ where $n$ is the number of cubes on one row or one column of the cube. Thus, we get the answer $54$ .

The number of small cubes which have paint on only one side are those in the middle portion of each face of the large cube, not the ones on the edges or corners. So considering each face of the n x n cube, we need to leave out the 1 cube on each outside edge of the face, so, leaving out all 4 edges, we are left with each face containing (n-2) x (n-2) cubes.

E.g. for a 5x5 cube, leaving out all 4 edges of each face, we have 3x3 cubes in the internal part of each face. For 6 faces of the cube, the total is 6 x (n-2) x (n-2) or for a 5x5 cube, 6x3x3 =54.

Each side has nine cubes with one side painted . There are six sides . So the number of cubes with one side painted are :

9 * 6 = 54

The inner 9 blocks on each face , i.e 9x6 = 54

Professor's Cube Questions... 9*6=54

9 cubes from one surface of 5x5x5 will be painted exactly on on one side ,so total 9*6 =54 cubes of 1x1x1 will be painted on one side.