%0 Journal Article
%T Counting the Number of Squares Reachable in k Knight¡¯s Moves
%A Amanda M. Miller
%A David L. Farnsworth
%J Open Journal of Discrete Mathematics
%P 151-154
%@ 2161-7643
%D 2013
%I Scientific Research Publishing
%R 10.4236/ojdm.2013.33027
%X <p>
	<span><span style=\"font-family:Verdana;\">Using
geometric techniques, formulas for the number of squares that require</span><i><span style=\"font-family:Verdana;\"> k</span></i><span style=\"font-family:Verdana;\"> moves in order to be reached by a
sole knight</span></span><span><span style=\"font-family:Verdana;\"> from its initial position on an infinite
chessboard are derived. The number of squares reachable in exactly</span><i><span style=\"font-family:Verdana;\"> k</span></i><span style=\"font-family:Verdana;\"> moves are 1, 8, 32, 68, and 96 for </span><i><span style=\"font-family:Verdana;\">k</span></i><span style=\"font-family:Verdana;\"> = 0, 1, 2, 3, and 4, respectively, and
28</span><i><span style=\"font-family:Verdana;\">k</span></i><span style=\"font-family:Verdana;\"> ¨C 20 for </span><i><span style=\"font-family:Verdana;\">k </span></i><span style=\"font-family:Verdana;\">¡İ 5. The cumulative number of squares r</span><span style=\"font-family:Verdana;\">each</span></span><span><span style=\"font-family:Verdana;\">able in </span><i><span style=\"font-family:Verdana;\">k</span></i> </span><span style=\"font-family:Verdana;\">or fever </span><span><span style=\"font-family:Verdana;\">moves are 1, 9, 41, and 109 for </span><i><span style=\"font-family:Verdana;\">k</span></i><span style=\"font-family:Verdana;\"> = 0, 1, 2, and 3, respectively, and 14</span><i><span style=\"font-family:Verdana;\">k</span></i><sup><span style=\"font-family:Verdana;\">2</span></sup><span style=\"font-family:Verdana;\"> ¨C</span></span><span> </span><span><span style=\"font-family:Verdana;\">6</span><i><span style=\"font-family:Verdana;\">k</span></i><span style=\"font-family:Verdana;\"> + 5 for </span><i><span style=\"font-family:Verdana;\">k</span></i><span style=\"font-family:Verdana;\"> ¡İ 4.</span></span><span style=\"font-family:Verdana;\"> Although</span><span> <span style=\"font-family:Verdana;\">these formulas are known, the proofs that are presented are new and more
mathematically accessible then preceding proo</span><span style=\"font-family:Verdana;\">fs.</span></span> 
</p>
%K Counting
%K Knight¡¯s Moves
%K Infinite Chessboard
%K Geometric Argument
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=34513