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jmc

geometry senior

Problem

A solid cube is composed of unit cubes. Each face of the large, solid cube is partially painted with gray paint, as shown.
problem
What fraction of the entire solid cube's unit cubes have no paint on them? Express your answer as a common fraction.
Solution
We know that each of the unit cubes in the cube in the center of the cube has no paint on it. On the surface of the cube, three of the unit cubes on each edge of the big cube have no paint on them, and the center unit cube of each face of the big cube has no paint on it. Since a cube has edges and faces, this makes a total of unit cubes with no paint on them. There are unit cubes altogether. The fraction with no paint is
Final answer
\frac{69}{125}