2012 AMC 10A Problem 25

Real numbers x, y, and z are chosen independently and at random from the interval [0,n] for some positive integer n. The probability that no two of x, y, and z are within 1 unit of each other is greater than \frac {1}{2}. What is the smallest possible value of n?

\textbf{(A)}\ 7\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 11

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.