Problem 291

To compare the lengths of these Paths, we begin by removing identical portions. In particular, we remove the horizontal segment of length 2, a vertical segment of length 1 from the left, and a vertical segment of length 4 from the right to obtain the following images:

[[IMAGE0]]

By removing the same lengths, we do not change the relative lengths of the Paths.

Each of the Paths still has a vertical segment of length 1, so we remove each of these segments, again maintaining the relative lengths of the Paths.

[[IMAGE1]]

Each of Path 1 and Path 3 now consists of the diagonals of two of the grid squares. Thus, their original lengths were equal and so $x=z$. This means that the final answer must equal (C) or (E), depending on whether $x=z$ is less than $y$ or greater than $y$. To answer this question, we re-draw the remaining segments of Path 1 under Path 2: [[IMAGE2]] Since a straight line path between two points is shorter than any other path between these two points, the length of the semi-circle is longer than the total length of the two straight line segments. This means that $x=z$ and $z<y$.

(As an alternate approach, can you determine the length of each of the original Paths and compare these numerically?)