Math Problem of the Fortnight: A knight on a chessboard
A knight is placed on one square of an infinite chessboard. The squares of the chessboard are labeled with ordered pairs of integers so that the knight is at the origin (0, 0), as shown below.
We define the distance between two squares (x1, y1) and (x2, y2) to be |x1 − x2| + |y1 − y2|; in other words, the distance between two squares is the number of horizontal or vertical steps required to travel from one square to the other. In particular, the distance from a square (x, y) to the origin is |x| + |y|.
As in standard chess, on each move the knight travels two squares horizontally and one square vertically, or two squares vertically and one square horizontally. Hence, one complete move of the knight looks like the letter L. However, for this problem the knight is only allowed to make moves that increase his distance from the origin.
A square X of the chessboard is called reachable if there is some sequence of these distance-increasing moves that take the knight from the origin to X; otherwise X is unreachable.
Is the set of all unreachable squares finite or infinite?
Winner
The winning solution to this problem was by Rob Brase.
