An "edge-rational" nightrider fairy chess piece can repeatedly jump at any rational slope subject to the constraint that among the squares on its path extended must be one of the squares on the edge of the board. Equivalently, it can jump to any edge square, but also stopping at squares on the way to the edge which are at the same slope. It can capture or be blocked by pieces on those intervening squares.
The edge-rational knight is like the nightrider but cannot repeatedly jump; it stops at the first square of a permitted slope.
On average, how many squares can an edge-rational nightrider reach in one move on an empty board? Knight? What is the asymptotic behavior as the board size increases?
Where can it go in two moves?
Its behavior is number-theoretically interesting, especially if it is a prime number of squares away from an edge. It can always move like a queen because 1 divides everything. Perhaps disallow those moves.
Could allow or disallow moving backwards along a path to the edge square.
Rather than defining the edge squares the region its path must "hit", we could instead define the target as some other subset of squares, e.g., just the 4 corner squares.
Could make the target the squares one square beyond the edge. This eliminates always being able to jump to any square on the edge.
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