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The six faces of a cube are painted black. The cube is then cut into 6^3 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?

 

thanks!

 Apr 17, 2024
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Here is part a:

 

There are two ways to approach this problem:

 

Method 1: Analyzing the Cube's Layers

 

Outermost Layer:

 

Each face of the large cube contributes a single layer of smaller cubes to the final set.

 

Since each face is painted black, the outermost layer of smaller cubes will all have one black face.

 

There are 6 faces, and each face contributes a layer of 3 * 3 = 9 cubes.

 

So, the outermost layer contributes a total of 6 faces * 9 cubes/face = 54 cubes with one black face.

 

Inner Cubes:

 

The inner cubes are completely enclosed by other cubes and won't have any black faces.

 

Method 2: Identifying Specific Locations

 

Cubes on Edges:

 

Cubes on the edges of the larger cube will have exactly two black faces (one from each adjacent larger face).

 

There are 12 edges (4 on each length), and each edge contributes 2 cubes.

 

So, the edges contribute a total of 12 edges * 2 cubes/edge = 24 cubes with two black faces.

 

Cubes on Corners:

 

Cubes on the corners of the larger cube will have exactly three black faces (one from each adjacent larger face).

 

There are 8 corners (one at each vertex), and each corner contributes 1 cube.

 

So, the corners contribute a total of 8 corners * 1 cube/corner = 8 cubes with three black faces.

 

Center Cube:

 

The cube in the exact center of the larger cube won't have any black faces.

 

Total with One Black Face:

 

Since none of the inner cubes and only the outermost layer has cubes with one black face, we can subtract the cubes with multiple black faces from the total number of cubes on the faces (54) to find the ones with exactly one black face.

 

Total cubes on faces = 6 faces * 3 cubes/face * 3 cubes/face = 54 cubes Total cubes with multiple black faces = 24 cubes (edges) + 8 cubes (corners) = 32 cubes

 

Therefore, there are 54 cubes (total on faces) - 32 cubes (multiple black faces) = 22 cubes with exactly one black face.

 

Answer:

 

Both methods lead to the same answer: there are 22 cubes with exactly one black face.

 Apr 17, 2024

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