A cube with each corner labeled A, B, C, D, E, F, G and H is divided into 27 equal smaller cubes. Before dividing the cube, each of its faces has been painted with different colors... A cube with each corner labeled A, B, C, D, E, F, G and H is divided into 27 equal smaller cubes. Before dividing the cube, each of its faces has been painted with different colors. How many such small cubes will there be which are colored with more than one color?

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Understand the Problem

The question describes a cube that is divided into 27 smaller cubes. Each face of the original cube is painted a different color. The question asks how many of the smaller cubes will have more than one color on them. To solve this problem, you need to visualize the cube and determine which smaller cubes would have multiple painted faces.

Answer

20
Answer for screen readers

(B) 20

Steps to Solve

  1. Visualize the Cube: Imagine a $3 \times 3 \times 3$ cube made up of 27 smaller cubes. Each face of the larger cube is painted a different color.

  2. Corner Cubes: The 8 corner cubes will have three faces painted, each with a different color.

  3. Edge Cubes: The cubes along the edges (but not at the corners) will have two faces painted. There are 12 edges on a cube, and each edge has one such cube in this $3 \times 3 \times 3$ configuration (since the corner cubes are excluded). So there are 12 edge cubes with two painted faces.

  4. Face Cubes: The cubes in the center of each face will have only one face painted.

  5. Inner Cube: The one cube in the very center of the $3 \times 3 \times 3$ cube will have no painted faces.

  6. Cubes with more than one color: These are the corner cubes and the edge cubes. So those are the cubes with two or three colours, therefore the cubes with more than one colour.

  7. Calculate the total: Add the number of corner cubes and edge cubes: $8 + 12 = 20$.

(B) 20

More Information

A standard cube has 6 sides, 8 corners, and 12 edges.

Tips

A common mistake is to miscount the number of edge or corner cubes. Also, some might include the center cubes of each face, but those only have one color.

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