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2005 AMC 12A Problem 10

Problem 10 of 25EasierAlgebraGeometry

A wooden cube nn units on a side is painted red on all six faces and then cut into n3n^3 unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is n?n?

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Solution

The n3n^3 unit cubes have 6n36n^3 faces total. The red faces are exactly the surface of the original cube, 6n26n^2 of them. Setting the red fraction to one-fourth, 6n26n3=1n=14, \dfrac{6n^2}{6n^3} = \dfrac{1}{n} = \dfrac{1}{4}, so n=4.n = 4. Thus, the correct answer is B.

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Concepts: cube geometry · surface area · ratio and proportion

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.