The Number of Squares Reachable in k Moves with (1, b)-Knight’s Move where b ∈ {3, 5, 7}

Sirirat Singhun, Ratree Theprod, Ratinan Boonklurb

Abstract

The m × n chessboard is an array with squares arranged in m rows and n columns. If
m, n → ∞, then it is called an infinite chessboard. An (a, b)-knight move is the move with a squares
vertically or a squares horizontally and then b squares move at 90 degrees angle. If a = 1 and b = 2, then
it is a legal knight’s move. There are many way to discuss on the chessboard. Such as finding the way
to land on each square exactly once and then return to the starting square (shown by many researchers)
or counting the number of squares that the knight can move to in k moves. In this paper, we focus on
the latter. The formula of number of squares reachable by a knight with (1, b)-knight’s move in k moves
where b ∈ {3, 5, 7} are obtained. Moreover, the cumulative number of squares are also obtained.