1993 AHSME Problem 29

Which of the following could NOT be the lengths of the external diagonals of a right regular prism [a "box"]? (An \textit{external diagonal} is a diagonal of one of the rectangular faces of the box.)

\text{(A) }\{4,5,6\} \quad \text{(B) } \{4,5,7\} \quad \text{(C) } \{4,6,7\} \quad \text{(D) } \{5,6,7\} \quad \text{(E) } \{5,7,8\}

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