Help me I'm stuck


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?

 Oct 28, 2023

Part a :The cubes displaying a solitary black face are the ones situated within the cube's interior rather than along its edges. Removing the outermost cubes on each face reveals a \(3 \) by \(3 \) array of cubes, each featuring paint on just one side. Consequently, there are 9 such cubes per face, each exclusively black on one side, resulting in a grand total of \(9\times 6=\boxed{54}\) of the smaller cubes exhibiting a single black face.

Part b: The "outer shell" of your cube has been painted, but if you take it away, it reveals a \(3 \) by \(3 \) cube inside, then you can multiply to get 27

 Oct 28, 2023
edited by xxyoohoo  Oct 28, 2023

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