Four cubes with edge lengths $1$, $2$, $3$, and $4$ are stacked as shown. What is the length of the portion of $\overline{XY}$ contained in the cube with edge length $3$? $\textbf{(A)}\ \dfrac{3\sqrt{33}}5\qquad$ $\textbf{(B)}\ 2\sqrt3\qquad$ $\textbf{(C)}\ \dfrac{2\sqrt{33}}3\qquad$ $\textbf{(D)}\ 4\qquad$ $\textbf{(E)}\ 3\sqrt2$
Solution
With edge lengths $1,2,3,4$, set $X=(0,0,10)$ (top of the whole stack) and $Y=(4,4,0)$ (bottom-front-right of the largest cube), where the vertical extents of the cubes (from top to bottom) are $[9,10],[7,9],[4,7],[0,4]$ for edges $1,2,3,4$.
Along $\overline{XY}$, at height $z$ the horizontal position is $x=y=\frac{2(10-z)}5$; this stays within each cube's footprint throughout.
The portion inside the edge-$3$ cube corresponds to $z\in[4,7]$; its length is $3\cdot\sqrt{1+\left(\frac25\right)^2+\left(\frac25\right)^2}=3\cdot\frac{\sqrt{33}}5=\frac{3\sqrt{33}}5$. The answer is (\textbf{A}).
Four cubes with edge lengths $2,3,5,10$ are stacked in increasing order from top to bottom, with their back-left vertical edges aligned. Point $X$ is the top-back-left corner of the smallest cube, and point $Y$ is the bottom-front-right corner of the largest cube. What is the length of the portion of $\overline{XY}$ inside the cube with edge length $5$?
$\textbf{(A)}\ \dfrac{3\sqrt{33}}5\qquad$ $\textbf{(B)}\ 2\sqrt3\qquad$ $\textbf{(C)}\ \dfrac{2\sqrt{33}}3\qquad$ $\textbf{(D)}\ 4\qquad$ $\textbf{(E)}\ 3\sqrt2$ 
