GRE Weekly Challenge #4

Edit: This challenge is now closed

Solution

First, let’s look at the common information. If |a| > |d|, then a is further from 0 than d is, but we don’t know their signs.
Since |a|b3c2|d|e5f6g and we know that |a|, c2, |d|, and f6 must be positive (any nonzero number becomes positive when put in an absolute value sign or given an even exponent), then whatever is “left over” must be causing the entire expression to be negative.
Therefore, b3e5g < 0.
Since an odd exponent merely preserves the sign of the original number (that is, a negative number to a power of 1, 3, 5, etc. will still be negative, and a positive number to a power of 1, 3, 5, etc. will still be positive), we can ignore the exponents for the purposes of evaluating the signs of our variables:
beg < 0
For three numbers multiplied together to yield a negative, there are two possibilities:
neg x neg x neg = neg
pos x pos x neg = neg
Of course, we have no way to know which of b, e, and g are negative, but we know that either all of them are negative, or exactly one of them is negative.
Now that we have fully processed the common information, we are ready to proceed to Columns A and B. In any problem of this type, it is important to draw logical conclusions from the common information before approaching the Columns.
First, note that order doesn’t matter when multiplying, so the parentheses are meaningless, and the Columns can be re-ordered to look more similar to one another:

Column A	Column B
|a| beg	|d| beg

Although b, e, and g appear on both sides, we cannot simply ignore them—importantly, the term beg is negative. We cannot “divide out” a term from both sides unless it is known to be positive.
Let’s rephrase what we now know about the Columns:

Column A	Column B
(big absolute value)(negative)	(small absolute value)(negative)

Multiplying the negative term beg by a big absolute value will make it bigger on the negative side. Multiplying this same negative term beg by a smaller absolute value will result in a negative, but a negative closer to zero than Column A's value. Still confused? Let's illustrate with real numbers.
For instance, imagine that beg is equal to –2. The terms a and d could be something like 5 and 3, or –7 and 4, or –100 and –50, etc. (as long as the absolute value of a is bigger than the absolute value of d). So, if a and d were originally –6 and 3, for instance, then their absolute values would be 6 and 3, and when we multiplied them by –2, we’d get that Column A = –12 and Column B = –6. Since –6 is closer to zero, it is larger and the answer is B. This will work for any numbers you select that follow the rules of the problem (that is, beg is negative and |a| > |d|).
The answer is B.

The winner of this Week's Challenge is... (drum roll).... Ardarandir. Congratulations! Please send me a pm with your shipping address, and choice of Manhattan GRE Guide