6. Let $P_{1}(1,2,3), P_{2}(2,4,1), P_{3}(1, k, 5)$, $P_{4}(4, k+1,3)$ be the four vertices of a tetrahedron in space with a volume of 1. Then $k=$ $\qquad$
6. -2 or 1 .

The volume of the tetrahedron is
$$
\begin{array}{l}
V=\left|\frac{1}{6}\right| \begin{array}{cccc}
1 & 2 & 3 & 1 \\
2 & 4 & 1 & 1 \\
1 & k & 5 & 1 \\
4 & k+1 & 3 & 1
\end{array}||=1 \\
\Rightarrow|6(k-2)-2(k-7)|=6 \\
\Rightarrow k=1 \text { or }-2 .
\end{array}
$$