In this problem we consider the movement of the knight in a game of chess. If you are

unfamiliar with this, please see the reference material for this problem, Figure $10\mathrm{.}6\mathrm{.}5.$

The knight we are considering is on the standard $88$ chessboard. We use the standard

notation to references the spaces on this board.

For example, a knight located at $d4$ can move to any of the spaces, $c2,e2,f3,$ b$3,$ b$5,$

f$5,$ c$6,$ e$6.$

We may make a graph as follows. The vertices of this graph correspond to the $64$

spaces on the chessboard. Two vertices are connected when ever a knight on one space

may move to the other space. For example we see that there is an edge between $d4$

and $c2.$ Use this graph to answer the following questions.

$($a$)$ Can a knight that begins on b$1,$ after a series of moves, end on c$1?$ If so$,$ what is

the minimum number of moves that this would take? Explain.

$($b$)$ For a knight that begins on b$1$ is there any space on the board that it cannot

occupy after any series of moves?

More generally, for any starting position is there any position to which a knight

cannot move to from there?

$($c$)$ Let the distance between two spaces $x,y$ on the board $($relative to a knight's

travel$)$ be the minimum number of moves for a knight to move from $x$ to $y.$

Starting at b$1,$ which space$($s$)$ is$/$are the "farthest away" from here?

More generally, what is the largest distance between any two spaces on the chess$-$

board for a knight?