Game of Moving (k,m)-Pieces where (k,m)∈{(1,2n+1),(2,2n+8),(3,4n+5),(3,4n+7)}

Abstract

Let m and k be positive integers such that m≥k. The game of moving (k,m)-pieces is a one-player game on a 1×m board involving two colors of pieces, with a total of m-k pieces. Each color has an equal number of pieces. At the beginning, we place all first color pieces from the rightmost to the leftmost followed by all second color pieces. The goal of this game is to move k consecutive pieces to the empty spaces on the board until the pieces are arranged in alternative colors placed from the leftmost position onward. In this study, for any positive integer n≥1, we obtain  1) when (k,m)=(1,2n+1), the minimum number of moving is 2n-2[n/2]  2) when (k,m)=(2,2n+8), the minimum number of moving is 2n+3  3) when (k,m)=(3,4n+5), the minimum number of moving is 4n-1 and  4) when (k,m)=(3,4n+7), the minimum number of moving is 4n+2.