2007 AMC 12A Problem 20

Slicing removes two equal segments of length $x$ from each edge. Each octagon then has side length $x\sqrt2,$ and the edge satisfies $1=2x+x\sqrt2,$ so $$x=\frac{1}{2+\sqrt2}=\frac{2-\sqrt2}{2}.$$ Each removed corner is a tetrahedron with three mutually perpendicular legs of length $x,$ so its volume is $\tfrac16 x^3.$ There are $8$ corners, giving total volume $$\begin{aligned} &8\cdot\tfrac16 x^3 \\ &=\tfrac43\left(\tfrac{2-\sqrt2}{2}\right)^3 \\ &=\frac{10-7\sqrt2}{3}. \end{aligned}$$ Thus, the correct answer is B.