(*^ ::[ Information = "This is a Mathematica Notebook file. It contains ASCII text, and can be transferred by email, ftp, or other text-file transfer utility. It should be read or edited using a copy of Mathematica or MathReader. If you received this as email, use your mail application or copy/paste to save everything from the line containing (*^ down to the line containing ^*) into a plain text file. On some systems you may have to give the file a name ending with ".ma" to allow Mathematica to recognize it as a Notebook. 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This has been a question of interest for a long time [2]. It is easy to think of a search procedure that must find a path if there is one (and it turns out that for n >= 5, there are many): backtracking. The knight moves along as far as possible, and if it comes into a blind alley, it takes back some of its steps in order to follow another direction. The problem is that for big chessboards, backtracking is very slow. So Warnsdorff [3] proposed an algorithm that should find a path without taking back any step. At each position of the knight, a rating of its successors is computed. Successors of a position are those squares that have not been touched yet and can be reached by one knight's jump from this position. The rating is highest for the successor whose number of successors is least. That way, squares tending to be isolated are visited first and therefore prevented from being isolated. The time needed for this algorithm grows about linearly with the number of squares of the chessboard, but unfortunately the computer experiment (which was not available to Warnsdorff) shows that at least for chessboards bigger than 76 times 76 squares, it also runs into blind alleys. For smaller boards, it solves the problem quite nicely. Recently, another linear time algorithm has been discovered and proved to solve the problem for all n >= 5 [1]. It works by decomposition of the chessboard into smaller chessboards (not necessarily square ones) for which explicit solutions are known. But this algorithm is quite complicated because it has to deal with many special cases. So the question remains: is there an algorithm, comparable to Warnsdorff's in simplicity, that definitely solves the problem? So far, nobody knows. One might try to improve Warnsdorff's rating scheme. It has the unpleasant property that it introduces some arbitrariness if there is more than one successor with least number of successors. To amend this, I add the rule that in this case, the successor with largest euclidean distance to the center of the chessboard is chosen. The introduction of this global criterion substantially reduces the number of blind alleys: now it first occurs on a 428 times 428 chessboard, and the "probability" to run into one is less than one precent for all chessboards smaller than 2000 times 2000 squares. (All this assumes that the starting point is in a corner of the board.) Mathematica was chosen for the implementation of the improved version of Warnsdorff's algorithm because of its list processing capabilities. The implementation isn't optimized for speed but for brevity and readability. Chessboards up to 100 times 100 can be calculated in reasonable time on most workstations. References: [1] Conrad, A., Hindrichs, T., Morsy, H. & Wegener, I., Solution of the Knight's Hamiltonian Path Problem on Chessboards, Discr. Appl. Math., to appear (1992) [2] Kraitchik, M., Mathematical Recreations, Chapter 11, The Problem of the Knight (Allen & Unwin, London, 1966) [3] Warnsdorff, H. C., Des Roesselsprungs einfachste und allgemeinste Loesung (Schmalkalden, 1823) ;[s] 3:0,0;2723,1;2734,0;3450,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,9,Times,2,12,0,0,0; :[font = section; inactive; Cclosed; preserveAspect; startGroup] Implementation :[font = input; preserveAspect; endGroup] (* Improved Warnsdorff's Algorithm for the Problem of *) (* the Knight *) (* Mathematica 2.2 Version Copyright 1992-93 Arnd Roth *) knightJumps = {{2, -1}, {1, -2}, {-1, -2}, {-2, -1}, {-2, 1}, {-1, 2}, {1, 2 }, {2, 1 }}; successors[s_, n_, position_] := Block[{moves, onBoard}, moves = Map[position + # &, knightJumps]; (* moves over the edge of the chessboard are *) (* invalid *) onBoard = Select[moves, And @@ Thread[ {0, 0} < # <= {n, n}] &]; Select[onBoard, s[[Sequence @@ #]] == 0 &] ] numberOfSuccessors[s_, n_, position_] := Length[successors[s, n, position]] mina[s_, n_, position_] := (* successor(s) with least number of successors *) Block[{localSuccessors, localNumbersOfSuccessors, minimum}, localSuccessors = successors[s, n, position]; localNumbersOfSuccessors = Map[numberOfSuccessors[s, n, #] &, localSuccessors]; minimum = Min[localNumbersOfSuccessors]; localSuccessors[[Flatten[Position[ localNumbersOfSuccessors, minimum]]]] ] mabst[destination_, successorlist_] := (* successor(s) with greatest distance from *) (* destination *) Block[{vectors, distances, maximum}, vectors = Map[destination - # &, successorlist]; distances = Map[# . # &, vectors]; maximum = Max[distances]; successorlist[[Flatten[Position[ distances, maximum]]]] ] improvedWarnsdorff[n_] := (* main program *) Block[{s, destination, position, localMina}, (* initialize chessboard s *) (* s[[x, y]] == 0 ==> square (x, y) is *) (* untouched. *) (* in the end, s contains the step numbers *) (* at which every square was visited *) s = Table[0, {n}, {n}]; (* path should end near the following *) destination = {Round[n / 2] - 1 / 2, Round[n / 2] - 1 / 2}; (* starting point in the corner of the board *) position = {1, 1}; s[[1, 1]] = 1; Do[ localMina = mina[s, n, position]; If[Length[localMina] == 0, Print["Blind alley"]; (* algorithm failed *) Break[] ]; position = If[Length[localMina] == 1, First[localMina], First[mabst[destination, localMina]] ]; s[[First[position], Last[position]]] = stepNumber, {stepNumber, 2, n n}]; s ] ;[s] 3:0,0;119,1;130,0;2514,-1; 2:2,7,10,Courier,1,12,0,0,0;1,7,10,Courier,3,12,0,0,0; :[font = section; inactive; Cclosed; preserveAspect; startGroup] Examples :[font = input; Cclosed; preserveAspect; startGroup] s = improvedWarnsdorff[8]; MatrixForm[s] ListDensityPlot[s] :[font = output; output; inactive; preserveAspect] MatrixForm[{{1, 16, 39, 22, 3, 18, 49, 56}, {38, 23, 2, 17, 60, 55, 4, 19}, {15, 40, 63, 54, 21, 48, 57, 50}, {24, 37, 42, 59, 64, 61, 20, 5}, {41, 14, 53, 62, 47, 58, 51, 30}, {36, 25, 46, 43, 52, 31, 6, 9}, {13, 44, 27, 34, 11, 8, 29, 32}, {26, 35, 12, 45, 28, 33, 10, 7}}] ;[o] 1 16 39 22 3 18 49 56 38 23 2 17 60 55 4 19 15 40 63 54 21 48 57 50 24 37 42 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