A Cube is Painted Red on All Faces. It is Then Cut into 27 Equal Smaller GMAT Problem Solving

Question: A cube is painted red on all faces. It is then cut into 27 equal smaller cubes. How many cubes are painted on only 2 faces?

1. 12
2. 8
3. 6
4. 10
5. 16

Correct Answer: A
Solution and Explanation:
Approach Solution 1:

The problem statement states that:

Given:

• A cube is painted red on all faces.
• The cube is then cut into 27 equal smaller cubes.

Find out:

• The number of cubes painted on only 2 faces.

It is given that the cube is painted on all sides and it is known that a cube has 6 sides. One face of the cube is divided into 9 parts, this gives us 9 small cubes. In total, the cube is then cut into 27 pieces.

Therefore, we can say it is cut in the sequence of 3 x 3 x 3. This implies that there is a total number of 27 cubes.

There is a cube in the middle which is not painted on any side.Therefore, out of 27 cubes, only 26 cubes are left because one cube is not painted on any side.

The corners of the cube, i.e, 8 cubes are painted on all three sides. Therefore, we need to subtract 8 small cubes from the remaining 26 small cubes. We are left with 18 small cubes.

Now, there are 6 small cubes on the surface of the larger cube which is painted on one side.Therefore, we need to subtract the 6 small cubes from the leftover 18 cubes.

As per the solution, (27 - 1 - 8 - 6) = 12 cubes will be left whose 2 sides will be painted red.

The number of cubes painted on only 2 faces = 12 cubes.

Approach Solution 2:

The problem statement informs that:

Given:

• A cube is painted red on all faces.
• The cube is then cut into 27 equal smaller cubes.

Find out:

• The number of cubes painted on only 2 faces.

We know that in total a cube has 6 sides.
Now, it is said that the cube with 6 sides will be further cut into 27 pieces, which means 9 parts on each side of the cube.

Let us take the side of the cube as 'n' and the total number of cubes is n x n x n. This shows 3 x 3 x 3 = 27 so in total there are 27 parts or small cubes.

Therefore, n= 3

As per the formula, the number of cubes painted on only 2 faces = 4n = 4 x 3 = 12

Approach Solution 3:

The problem statement informs that:

Given:

• A cube is painted red on all faces.
• The cube is then cut into 27 equal smaller cubes.

Find out:

• The number of cubes painted on only 2 faces.

Since the cube is painted red on all six faces and divided into 27 small cubes of equal parts, then it has been cut into a 3×3×3 arrangement.

Therefore, there exists 1 cube in every centre (middle of 3 in each axis direction i.e., longitudinal lateral and vertical) that holds no colour.
On each of the 6 sides of the cube, there is a central smaller cube that is painted on one face.
There are 4 cubes (at the middle of the edges) on each of the 6 sides of the cube.
These are shared with one other side of the cube (or we could just count the 12 edge lines).
Therefore, the number of cubes painted on only 2 faces = 6 * 4 / 2 = 12.

“A cube is painted red on all faces. It is then cut into 27 equal smaller”- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. This topic has been taken from the book “GMAT Official Guide 2020”. GMAT Problem Solving questions allow the candidates to assess facts given in the question to solve numerical problems. GMAT Quant practice papers assist the candidates to get familiar with different types of questions that will improve their mathematical skills.

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