We are given a cube of side length 2. We then slice a pyramid off each corner, as shown, so that every side length of the remaining polyhedron has the same length. Let $A$, $P$, $Q$, and $R$ be the vertices shown.
Let $x = AP$, $y = AQ$, and $z = AR$. Convince yourself that $x=y=z.$ Don't forget here that all we know is that the edges of all of the octagons are the same length. Find $x$.
