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Difficulty: MediumCubes and Dice

A large solid cube of side length 6 cm6 \text{ cm} has its opposite faces painted with the same color. The colors used are red, blue, and yellow. After the paint dries, the large cube is completely cut into smaller cubes of side length 1 cm1 \text{ cm} each. How many of these smaller cubes will have exactly two faces painted, with one face being red and the other face being yellow?

Answer: 16 cubes

Answer

16 cubes
There are exactly 4 edges where a red face and a yellow face meet. Each of these edges contains exactly 4 middle cubes (excluding the corner cubes) that have exactly two faces painted. Therefore, multiplying 4 edges×4 cubes4 \text{ edges} \times 4 \text{ cubes} gives 16.

Step-by-Step Solution

1
Calculate the number of smaller cubes along one edge of the large cube.
n=6/1=6n = 6 / 1 = 6 cubes per edge.
The value nn is required to determine how many cubes lie along any given dimension.
2
Determine the number of edges where a red face and a yellow face intersect.
There are exactly 4 intersecting edges.
The cube has 2 red faces (top/bottom) and 2 yellow faces (left/right). Each of the 2 red faces meets both of the 2 yellow faces, resulting in 2×2=42 \times 2 = 4 edges.
3
Calculate how many cubes on a single edge have exactly two faces painted.
n−2=6−2=4n - 2 = 6 - 2 = 4 cubes per edge.
The cubes at the two extreme ends of any edge are corner cubes with three painted faces, so they must be subtracted from the total edge length.
4
Multiply the number of relevant edges by the number of two-face painted cubes per edge.
4 edges×4 cubes=164 \text{ edges} \times 4 \text{ cubes} = 16 cubes.
This yields the total count of smaller cubes possessing exactly one red and one yellow painted face.

Key Concept

Calculation of painted smaller cubes resulting from cutting a colored larger cube based on intersecting faces.
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Topics

Cubes and Dice

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