A chess knight, on one turn, moves either two squares vertically and one horizontally, or two horizontally and one vertically. If we consider a knight starting at the point (2,y) in ZxZ, it has eight possible moves, to (x + 1, y + 2), (x + 1, y - 2), (2-1, y + 2), (2-1, y - 2), (+ + 2, Y +1), (*+2, y - 1), (2-2, y +1), or (x - 2, y - 1). Required:
a. Prove that given any two points (2,y) and (z',y') in Zx Z, there is a sequence of knight moves from the first point to the second.
b. Let a and b be different positive naturals. An (a, b)-knight also has eight possible moves, from (x, y) to (x ± y ± b) or (x± b, y ±a). What conditions on a and b allow the (a, b)-knight to go from any point in Zx Z to any other?
c. If a and b do not meet the conditions of part (b), exactly which points can the (a, b)- knight reach from (x, y)?