6. If the six edges of a tetrahedron are $2,3,4,5$, 6,7, then there are $\qquad$ different shapes (if two tetrahedra can be made to coincide by appropriate placement, they are considered the same shape).
6.10.

Let the edge of length $k$ be denoted as $l_{k}(k \in\{2,3,4,5,6,7\})$. Consider $l_{2}$ and $l_{3}$.
(1) If $l_{2}$ and $l_{3}$ are coplanar, then the other side of the plane must be $l_{4}$.
(i) If $l_{2}$, $l_{3}$, and $l_{4}$ form a triangle in a clockwise direction (all referring to the direction of the three sides when viewed from inside the shape, the same below), as shown in Figure 6, then side $D A$ cannot be $l_{6}$ (otherwise, it would make the three sides of $\triangle B C D$ be $2,5,7$, which is a contradiction).
If $D A=l_{5},\{D B, D C\}=\left\{l_{6}, l_{7}\right\}$, there are 2 cases;
If $D A=l_{7},\{D B, D C\}=\left\{l_{5}, l_{6}\right\}$, there are also 2 cases.
A total of 4 cases are obtained.
(ii) If $l_{2}$, $l_{3}$, and $l_{4}$ form a triangle in a counterclockwise direction, similarly, 4 cases are obtained.
(2) If $l_{2}$ and $l_{3}$ are skew, let $A B=l_{2}, C D=l_{3}$. Then the remaining four edges, each one must be adjacent to $l_{2}$ and $l_{3}$. Therefore, the other side of the plane containing $l_{2}$ and $l_{7}$ must be $l_{6}$.
(i) If $l_{2}$, $l_{6}$, and $l_{7}$ form a triangle in a clockwise direction, without loss of generality, let $A C=l_{6}, B C=l_{7}$, as shown in Figure 7. The remaining two edges, $B D$ cannot be $l_{4}$, so only $B D=l_{5}, A D=l_{4}$, yielding 1 case.
(ii) If $l_{2}$, $l_{6}$, and $l_{7}$ form a triangle in a counterclockwise direction, similarly, 1 case is obtained.
Therefore, in this problem, there are 10 different shapes.