Question 27

A cube has edge length $$x$$, an integer. Three faces meeting at a corner are painted blue. The cube is then cut into smaller cubes of unit length. If exactly 343 of these cubes have no faces painted blue, then the value of $$x$$ is

Correct Answer: 8

Solution

The small cubes with no painted face form an inner cube whose side length is $$x-1$$ because the three painted faces remove one layer from each corresponding direction. Thus $$\left(x-1\right)^3=343=7^3$$. Hence $$x-1=7$$ and $$x=8$$.

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A cube has edge length $$x$$, an integer. Three faces meeting at a corner are painted blue. The cube is then cut into smaller cubes of unit length. If exactly 343 of these cubes have no faces painted blue, then the value of $$x$$ is

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