All the Numbers 2, 3, 4, 5, 6, 7 are Assigned to the Six Faces of a Cube, One Number to Each Face GMAT Problem Solving

Question: All the numbers 2, 3, 4, 5, 6, 7 are assigned to the six faces of a cube, one number to each face. For each of the eight vertices of the cube, a product of three numbers is computed, where the three numbers are the numbers assigned to the three faces that include that vertex. What is the greatest possible value of the sum of these eight products?

1. 312
2. 343
3. 625
4. 729
5. 1680

‘All the numbers 2, 3, 4, 5, 6, 7 are assigned to the six faces of a cube, one number to each face’ - is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide 2021”. 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.

Solution and Explanation:

Approach Solution 1:

This question can be solved by only one approach.
Given in the question that all the numbers 2, 3, 4, 5, 6, 7 are assigned to the six faces of a cube, one number to each face. For each of the eight vertices of the cube, a product of three numbers is computed. The three numbers are the numbers assigned to the three faces that include that vertex. It is asked to find out the greatest possible value of the sum of these eight products.
Let us assume the six faces of the cube as a,b,c,d,e, and f such that a is opposite to d, c is opposite to e, and b is opposite to f.
Now for the eight vertices the products could be
abc, abe, bcd, bde, acf, cdf, cef, and def
Now the sum of these products will be
abc + abe + bcd + bde + acf + cdf + cef + def
= b(ac + ae + cd + de) + f(ac + ae + cd + de)
= (b+f)(ac + ae + cd + de)
= (b+f)(a (c + e) + d (c + e))
= (b+f)(a+d)(c+e)
This is the required product.
It should be noted that the product is the sum of the opposite faces
There is a property that the product is maximum when the numbers are equal.
Therefore the product will be maximum when the individual sums will be maximum.
The answer will be (7 + 2)(6 + 3)(5 + 4)
= 9 * 9 * 9
= 729
The correct answer will be option D.

Correct Answer: D

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