+1791 The six faces of a cube are painted black. The cube is then cut into $3^3$ smaller cubes, all the same size.
(a) How many of the smaller cubes have exactly one black face?
(b) How many of the smaller cubes do not have any black faces?
(c) One of the small cubes is chosen at random, and rolled. What is the probability that when it lands, the face on the top is black?
+174 Analyzing the Cube Cuts
Imagine the large cube is 3 units on each side. When we cut it into smaller cubes, each side of the larger cube will be made of 3 smaller cubes.
(a) Cubes with One Black Face
A cube will have exactly one black face only if it's on the edge of the larger cube but not on a corner.
Edges: There are 12 edges on a cube.
Corner exceptions: On each edge, there are 2 smaller cubes that touch a corner.
Since corner cubes will have 3 black faces, we subtract these exceptions from the total edge cubes. There are 8 corners, so there are 8×2=16 corner exception cubes.
Therefore, the number of cubes with one black face is the total number of edge cubes minus the corner exceptions: 12 edges - 16 corner exceptions = -4 cubes
This seems like a negative number of cubes, which doesn't make sense. The mistake lies in assuming all the edge cubes have one black face.
Here's the correction: We only counted the edges once, but each edge actually has 2 cubes that qualify (one on each side). So, we need to multiply the number of edges by 2:
Total one-black-face cubes = (2 cubes/edge) x (12 edges) - 16 corner exceptions = 24 - 16 = 8 cubes
(b) Cubes with No Black Faces
These cubes must be completely inside the larger cube, not touching any of the faces.
Inner core: Since each side of the larger cube is made of 3 smaller cubes, the inner core will be a cube with sides of length 1 unit less (3 - 2 = 1 unit). The volume of this inner core is therefore 1 x 1 x 1 = 1 cube.
Inner cubes: This inner core cube is itself made of smaller cubes. Each side has 1 cube, so there are a total of 1 x 1 x 1 = 1 smaller cube inside.
Therefore, there is only 1 cube with no black faces.
(c) Probability of Top Face Black
When a small cube is rolled, there are 6 possible faces that could land on top.
Out of these 6 faces, only 1 face is painted black (since we're considering the cubes with at least one black face).
Therefore, the probability of the top face being black is the number of black faces divided by the total number of faces: Probability = 1 black face / 6 total faces = 1/6.
