The Entire Exterior of a Large Wooden Cube is Painted Red GMAT Problem Solving

Question: The entire exterior of a large wooden cube is painted red, and then the cube is sliced into \(n^3\) smaller cubes (where n > 2). Each of the smaller cubes is identical. In terms of n, how many of these smaller cubes have been painted red on atleast one of their faces?

1. \(6n^2\)
2. \(6n^2-12n+8\)
3. \(6n^2-12n+24\)
4. \(4n^2\)
5. 24n - 24

Answer:
Approach Solution (1):

Say n = 3

So, we would have that the large cube is cut into \(3^3 = 27\)smaller cubes:

Out of them only the central little cube won’t be painted red at all and the remaining 26 will have atleast one red face. Now, plug n = 3 and see which one of the options will yield 26. Only B works: \(6n^2-12n+8\) = 54 – 36 + 8 = 26

Correct Option: B

Approach Solution (2):

Number of cubes inside = \((n-2)^3\)

\((n-2)^3\) cubes have no colored faces.

Remaining cubes will have atleast one face colored red
Remaining cubes = \(n^3-(n-2)^3\)

\(n^3-(n^3-8-3*n*2(n-2))\)

\(n^3-(n^3-8-6n^2+12n)\)

\(6n^2-12n+8\)

Correct Option: B

“The entire exterior of a large wooden cube is painted red, and then the cube is sliced into \(n^3\) smaller cubes (where n > 2). Each of the smaller cubes is identical. In terms of n, how many of these smaller cubes have been painted red on atleast one of their faces?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

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