18. Among all tetrahedra with edge lengths 2, 3, 3, 4, 5, 5, what is the maximum volume? Prove your conclusion.
(1983 National Competition Problem)
18. According to the principle that the difference between any two sides of a triangle is less than the third side, the other two sides of a triangle with one side of length 2 can only have the following four scenarios: (1) 3,3; (2) 5,5; (3) 4,5; (4) 3,4. For the tetrahedron in the problem, the two triangular faces sharing the edge of length 2 can have three possible scenarios: (1)(1) and (2), (2)(1) and (3), (3)(2) and (4). In (1), let \( AC = BC = 3 \) and \( AD = BD = 5 \). There is only one such tetrahedron, \( V_{1} = \frac{1}{3} CD \cdot S_{\triangle ABC} = \frac{8 \sqrt{2}}{3} \); (2) There are two such tetrahedra, \( V_{2} = \frac{1}{3} h_{2} \cdot S_{\triangle ABC} < \frac{1}{3} DB \cdot S_{\triangle ABC} = V_{1} \); (3) There are also two such tetrahedra, \( V_{3} = \frac{1}{3} h_{3} \cdot S_{\triangle ACD} < \frac{1}{3} AB \cdot S_{\triangle ACD} = \frac{5}{6} \sqrt{11} \). Comparing \( V_{1}, V_{2}, V_{3} \), the maximum is \( V_{1} = \frac{8 \sqrt{2}}{3} \).