Studies in Tours of Knight on Rectangular Boards
Total Page：16
File Type：pdf, Size：1020Kb
Studies in Tours of Knight on Rectangular Boards Awani Kumar (Lucknow, India) E-mail: [email protected] Abstract The author has constructed and enumerated tours of knight having various magic properties on 4 x n and 6 x n boards. 16 magic tours of knight have been discovered on 4 x 18 board, 88 on 4 x 20 board, 464 on 4 x 22 board, 2076 on 4 x 24 board, 9904 on 4 x 26 board and 47456 on 4 x 28 board. Magic tours exist on all boards of size 4 x 2k for k > 8. Quasi-magic tour exists on 6 x 11 board. 8 magic tours of knight have been discovered on 6 x 12 board and magic tours exist on all boards of size 6 x 4k for k > 2. Introduction: British puzzlist H. E. Dudeney [1] said, "The knight is the irresponsible low comedian of the chessboard." Knight is a unique piece and its crooked move has enchanted and fascinated not only chess players but also attracted attention of mathematics enthusiasts. It has been the basis for so many unusual and amusing combinatorial problems. “The oldest of knight puzzles”, asserts mathematics popularizer M. Gardner [2] “is the knight's tour”. Books, namely, by Rouse Ball [3], Kraitchik [4], Pickover [5], Petkovic [6], Wells [7] and by many others cover tour of knight problem. Some great mathematical minds, such as, Raimond de Montmort, Abraham de Moivre, and Leonhard Euler have delved into the knight’s tour problem, but mostly on 8 x 8 board. According to Jelliss [8], “oldest knight tour on 8x8 board is by Ali C. Mani, an otherwise unknown chess player, and al-Adli ar-Rumi who flourished around 840 AD and is known to have written a book on Shatranj (the form of chess then popular)”. Warnsdorf [9] proposed an interesting method – “Play the knight to a square where it commands the fewest cells not yet used” and Stonebridge [10] improved it further. Jelliss [11] has also described methods given by Roget and Collini but all these methods can’t be used to enumerate or estimate the number of tours. Recently Parberry [12] has described an algorithm for constructing closed knight’s tour on square boards. Kyek et al [13] have described an algorithm to estimate the bounds of the number of closed tours on a square board. Ismail et al [14] have given a new algorithm for constructing knight’s tour on 8 x 8 board. Although Cull [15] and McGown [16] have given proofs and examples of knight’s tours on rectangular boards and Baskoro [17] has also given a method for constructing closed knight’s tour on rectangular boards but most of the available literature on knight’s tour is related to square boards. Rectangular boards have got scanty attention. Construction of simple knight’s tours on 4 by n (n > 4) boards is relatively easy and the number of knight’s tours on smaller boards (up to 4 x 10) has been counted. But, what about knight tours on larger rectangular boards? Can we have magic knight’s tours on 4 by n boards? If yes, what is the smallest board size and how many magic tours are there? It is the tours having magic properties and the enumeration of tours, which have attracted, enchanted and fascinated innumerable people and the author plans to look into them. 1 A. Tours of knight on 4 by n boards Description: Like its conventional meaning, knight tour on a 4 x n board is consecutive knight moves where knight visits all the 4n cells without visiting any cell twice. If the initial cell of the tour and its last cell are connected by knight’s move, it is called re-entrant or closed knight tour else it is open knight tour. Generally, knight tours on larger board have fewer restrictions but, surprisingly, knight tours on 4 x n board have more restrictions than the 3 x n board. Sainte- Maire [18] proved that re-entrant knight tours are impossible on 4 x n board. However, closed knight tours are possible on 3 x n board if n is even and n >= 10. Similarly, there are 4 distinct knight tours (3 in geometrical form) on 3 x 4 board but no knight tour is possible on 4 x 4 board. Before proceeding further, it is better to explain few terms associated with knight tours. A ‘magic knight tour’ on a rectangular board 4 x n (n even) has all the 4 long rows adding to magic constant n(4n+1)/2 and the n short rows adding to magic constant 2(4n+1). There can’t be magic knight tour on a rectangular board 4 x n if n is odd because the adjacent long rows will contain an odd and an even number of odd-numbered cells. So the sum of the rows will be alternately odd and even. A ‘semi-magic knight tour’ has either all the long rows adding up to a constant sum, called magic constant (MC) or all the short rows adding up to a different magic constant. Jelliss [19] has defined two special types of semi-magic tours, namely, ‘quasi-magic’ tour in which the non-magic lines add to only two different values and ‘near-magic’ tour in which the non-magic lines add to the magic constant and two other values. Such types of semi- magic tours are of interest here because magic knight tours are not there on smaller boards. Sum of short and long rows have been shown in the figures and magic constants have been highlighted in bold font. 4 x 9 board: Boards up to 4 x 8 size have already been studied in sufficient details. Jelliss [20] mentions, “There are 112 symmetric open tours”. Figure 1 shows a semi-magic tour with short rows adding to magic constant 74. 1 18 35 20 74 1 4 19 36 60 1 18 35 20 74 34 21 16 3 74 32 35 2 5 74 34 21 2 17 74 17 2 19 36 74 3 18 33 20 74 9 16 19 36 80 22 33 4 15 74 34 31 6 17 88 22 33 10 3 68 9 14 23 28 74 7 16 21 30 74 15 8 23 28 74 32 27 10 5 74 22 29 8 15 74 32 27 4 11 74 13 8 29 24 74 11 14 23 26 74 7 14 29 24 74 26 31 6 11 74 28 25 12 9 74 26 31 12 5 74 7 12 25 30 74 13 10 27 24 74 13 6 25 30 74 161 166 167 172 151 182 151 182 159 174 159 174 Fig.1. Semi-magic tours Fig.2. Tours with long row s adding to alternate values. Discerning readers must have seen that long rows in Figure 1 are adding to four different values. This is common in tours on 4 x n rectangular boards with odd side. Figure 2 shows two tours with long (9-cell) rows adding up to only two different values. There are 1682 such tours. 2 4 x 10 board: Regarding semi-magic tours, Jelliss [19] asserts that “This case has not been looked at so far.” The author has enumerated 3102 semi-magic tours, 30 quasi-magic tours and 22 near-magic tours with long (10-cell) rows magic. They have been shown in Figure 3. 7 10 23 30 70 5 8 25 28 66 3 18 37 24 82 22 29 8 11 70 26 29 4 7 66 38 23 2 19 82 9 6 31 24 70 9 6 27 24 66 17 4 25 36 82 28 21 12 19 80 30 23 10 3 66 22 39 20 1 82 5 18 25 32 80 11 2 21 32 66 5 16 35 26 82 40 27 20 13 100 22 31 12 1 66 40 21 6 15 82 17 4 33 26 80 13 20 33 40 106 7 14 27 34 82 36 39 14 1 90 34 37 16 19 106 28 31 8 11 78 3 16 37 34 90 17 14 39 36 106 13 10 33 30 86 38 35 2 15 90 38 35 18 15 106 32 29 12 9 82 205 205 205 205 205 205 205 205 205 205 205 205 Fig.3. Semi-magic, quasi- magic and near-magic tours with long rows magic. There are 104 semi-magic tours, 12 quasi-magic tours and 16 near-magic tours with short (4- cell) rows magic. They have been shown in Figure 4. 7 10 31 34 82 3 18 39 22 82 1 20 39 22 82 30 33 8 11 82 40 21 2 19 82 38 23 18 3 82 9 6 35 32 82 17 4 23 38 82 19 2 21 40 82 26 29 12 15 82 24 37 20 1 82 24 37 4 17 82 5 14 27 36 82 5 16 35 26 82 5 16 35 26 82 28 25 16 13 82 36 25 6 15 82 36 25 6 15 82 17 4 37 24 82 7 14 27 34 82 7 14 27 34 82 22 39 20 1 82 28 31 10 13 82 28 31 10 13 82 3 18 23 38 82 11 8 33 30 82 11 8 33 30 82 40 21 2 19 82 32 29 12 9 82 32 29 12 9 82 187 199 211 223 203 203 207 207 201 205 205 209 Fig.4.