Leprechauns On The Chessboard

We introduce in this paper leprechauns, fairy chess pieces that can move either like the standard queen, or to any tile within k king moves. We then study the problem of placing n leprechauns on an n x n chessboard. When k = 1, this is equivalent to the standard n-Queens Problem. We solve the problem for k = 2, as well as for k > 2 and n <= (k + 1)(2), giving in the process an upper bound on the lowest non-trivial value of n such that the (k, n)-Leprechauns Problem is satisfiable. Solving this problem for all k would be equivalent to solving the diverse n-Queens Problem, the variant of the n-Queens Problem where the distance between the two closest queens is maximized. While diversity has been a popular topic in other constraint problems, this is not the case for the n-Queens Problem, making our results the first major ones in the domain. (C) 2021 The Author(s). Published by Elsevier B.V.