Suppose you are given an $nn$ grid, with $n2,$ and you want to find

a path between horizontally and vertically adjacent vertices that starts in

the upper left corner, visits every other grid cell exactly once, and ends in

the cell immediately to the right of where you started $($in the upper left

corner$).$ The diagram below shows a solution to this problem for the case

$n=4$:

For every $n,$ either give an algorithm for finding this path, or explain why

none exists.