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 ⟨x, y⟩ in Z × Z, it has eight possible moves, to ⟨x+1,y+2⟩,⟨x+1,y−2⟩,⟨x−1,y+2⟩,⟨x−1,y−2⟩,⟨x+ 2, y + 1⟩, ⟨x + 2, y − 1⟩, ⟨x − 2, y + 1⟩, or⟨x − 2, y − 1⟩. (a) Prove that given any two points ⟨x, y⟩ and ⟨x′, y′⟩ in Z × 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±a,y±b⟩ or ⟨x±b,y±a⟩. What conditions on a and b allow the (a, b)-knight to go from any point in Z × Z to any other? Prove your answer. (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⟩
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 ⟨x, y⟩ in Z × Z, it has eight possible moves, to ⟨x+1,y+2⟩,⟨x+1,y−2⟩,⟨x−1,y+2⟩,⟨x−1,y−2⟩,⟨x+ 2, y + 1⟩, ⟨x + 2, y − 1⟩, ⟨x − 2, y + 1⟩, or⟨x − 2, y − 1⟩.
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(a) Prove that given any two points ⟨x, y⟩ and ⟨x′, y′⟩ in Z × Z, there is a sequence of knight moves from the first point to the second.
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(b) Let a and b be different positive naturals. An (a, b)-knight also has eight possible moves, from ⟨x,y⟩ to ⟨x±a,y±b⟩ or ⟨x±b,y±a⟩. What conditions on a and b allow the (a, b)-knight to go from any point in Z × Z to any other? Prove your answer.
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(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⟩
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