$($a$)$ Show that, for even positive integers n$,$ there is no Gray code which starts at

$0000$ and ends at $1111.$

$($b$)$ Find a Gray code for n $=3$ which starts at $000$ and ends at $111.$

$($c$)$ Discuss whether your construction in $($b$)$ can generalize to any odd n $>=3.$

$2.$ Describe, with proof, how to find the successor of x $=$ x$1$x$2...$ xk in the lexicographic

list of all k$-$term weakly increasing sequences with symbols in $[$n$].$