smc

Observe the diagram above. Each dot represents one of the six vertices of the regular octahedron. Three dots have been placed exactly x units from the $(0,0,0)$ corner of the unit cube. The other three dots have been placed exactly x units from the $(1,1,1)$ corner of the unit cube. A red square has been drawn connecting four of the dots to provide perspective regarding the shape of the octahedron. Observe that the three dots that are near $(0,0,0)$ are each $(x)(\sqrt{2}$) from each other. The same is true for the three dots that are near $(1,1,1).$ There is a unique $x$ for which the rectangle drawn in red becomes a square. This will occur when the distance from $(x,0,0)$ to $(1,1-x,1)$ is $(x)(\sqrt{2}$). Using the distance formula we find the distance between the two points to be: $\sqrt{(1-x{)}^2+(1-x{)}^2+1}$ = $\sqrt{2x^2-4x+3}$. Equating this to $(x)(\sqrt{2}$) and squaring both sides, we have the equation: $2x^2-4x+3$ = $2x^2$ $-4x+3=0$ $x$ = $\frac{3}{4}$. Since the length of each side is $(x)(\sqrt{2}$), we have a final result of $\frac{3\sqrt{2}}{4}$. Thus, Answer choice $\boxed{\frac{3\sqrt{2}}{4}}$ is correct. (If someone can draw a better diagram with the points labeled P1,P2, etc., I would appreciate it).