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Difficulty: MediumCubes and Dice

A solid wooden cube is painted Black on two adjacent faces, while the remaining four faces are painted White. The large cube is then cut into 6464 identical smaller cubes. What is the total number of smaller cubes that have at least one Black face?

Answer: 28 cubes

Answer

28
The large cube is divided into 6464 smaller cubes, which means it forms a 4×4×44 \times 4 \times 4 grid. A single face of this large cube contains 4×4=164 \times 4 = 16 smaller cubes. Since two adjacent faces are painted Black, we count the cubes on both faces. The first face has 1616 Black cubes. The second face also has 1616 Black cubes, but it shares one edge with the first face. This shared edge consists of 44 smaller cubes. If we simply add 16+16=3216 + 16 = 32, we double-count the 44 cubes on the shared edge. Subtracting the overlap gives 32−4=2832 - 4 = 28 unique smaller cubes that have at least one Black face.

Step-by-Step Solution

1
Find the edge length of the large cube in terms of smaller cubes.
n=4n = 4
The total number of smaller cubes is 6464, and the total volume of a cube is n3n^3. Therefore, n=643=4n = \sqrt[3]{64} = 4.
2
Count the smaller cubes on the first Black face.
1616 cubes
Each face of an n×n×nn \times n \times n cube contains n2n^2 smaller cubes. Here, 42=164^2 = 16 cubes.
3
Count the smaller cubes on the second Black face and identify the overlap.
The second face adds 1616 cubes, but the two faces share an edge of 44 cubes.
Since the two Black faces are adjacent, they intersect along one edge. The cubes on this edge are counted as part of both faces.
4
Calculate the total unique cubes with at least one Black face.
2828 cubes
Using the inclusion-exclusion principle: 1616 (first face) + 1616 (second face) - 44 (shared edge) = 2828.

Key Concept

Calculation of painted smaller cubes from a larger cut cube, specifically handling adjacent face intersections.
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Topics

Cubes and Dice

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