2016 AMC 10B Problem 17

Pair opposite faces as $(a_1,a_2)$, $(b_1,b_2)$, and $(c_1,c_2)$. Each vertex product uses one number from each pair, so the sum of all eight vertex products is $$(a_1+a_2)(b_1+b_2)(c_1+c_2).$$ The six face labels sum to $2+3+4+5+6+7=27$, so the three opposite-pair sums have total $27$. Their product is maximized when the sums are as equal as possible, namely $9,9,9$, giving at most $9^3=729$. This maximum is attainable by pairing $2$ with $7$, $3$ with $6$, and $4$ with $5$. Hence the greatest possible sum is $729$. Thus, the correct answer is D.