MarkAF wrote:
I may be over thinking this but, going with the "that" modification ...
Big cube. 6^3 = 216, hidden/interior cube 4^3, minimizing color exposed interior cube gets 64 colored leaving 96 on the outside.
The exterior surface is 6*36 = 216 but only 152 blocks ... Interesting.
There are 8 corners each accounting for 3 of the 216, use 8 non-colored for the corners.
There are 12 edges, each 4 long, each covering 2 of the 216, use the remaining non-colored for these positions.
Since everything else is colored, the needed info is available.
There are 8*3 + 12*4*2 = 24+96 = 120 non-colored of 216, factoring 24 out leaves 5/9 which is the complement of the desired percentage. 44.44% is correct.
Yes Mark, your logic is sound. You are taking the long way but focus on the surface of the big cube is warranted (in this question, it doesn't matter).
The exterior of the big cube has 216 small cube surfaces - 36 surfaces on each face. On each face, there are 4*4 = 16 cubes which have only one face exposed. So of all the cubes that make the exterior, there are exactly 16*6 = 96 cubes which have only a single surface exposed. We should use our leftover coloured cubes for these 96 cubes to minimise colour exposure.
Hence, the percentage of coloured surface = 96/216 = 44.44%
This would come in very handy if the question were a little different:
A big cube is formed by rearranging the 180 coloured and 36 non-coloured similar cubes in such a way that the exposure of the coloured cubes to the outside is minimum. The percentage of exposed area
THAT is coloured is:
Big cube. 6^3 = 216, hidden/interior cube 4^3, minimizing color exposed interior cube gets 64 colored leaving 116 on the outside. Use 96 such that only one surface is exposed. You have 20 leftover coloured cubes.
How many cubes have 2 surfaces exposed? The edges but not the corners. 4 cubes on each edge have only 2 surfaces exposed. So total 12*4 = 48 cubes have 2 surfaces exposed. Use 20 leftover cubes here so another 40 surfaces are coloured.
Total 96 + 40 of the 216 outside surfaces are coloured.
So the answer here will not be 116/216 but 136/216.
Alternatively, place the non coloured cubes on 8 vertices and leftover 36 - 8 = 28 on edges. So non coloured cubes make 8*3 + 28*2 = 24 + 56 = 80 surfaces. So coloured cubes will make 216 - 80 = 136 surfaces.