2017
INTERNATIONAL SCIENTIFIC CONFERENCE
17-18 November 2017, GABROVO
ENUMERATION OF SOME CLOSED KNIGHT PATHS
Stoyan Kapralov
Technical University of Gabrovo
Valentin Bakoev
Kaloyan Kapralov
V. Turnovo University
Skyscanner Bulgaria
Abstract
The aim of the paper is to enumerate all closed knight paths of length n over a square board of size n+1. The closed
knight paths of length 4, 6 and 8 are classified up to equivalence. We determine that there are exactly 3 equivalence
classes of closed knight paths of length 4, exactly 25 equivalence classes of closed knight paths of length 6 and exactly
478 equivalence classes of closed knight paths of length 8.
Keywords: closed knight path, enumeration, equivalence
INTRODUCTION
The inspiration for this paper comes from
the oldest lotto game in Bulgaria: “Toto 2” – 6
out of 49. The “Toto 2” game slip is a 7 x 7
grid pre-filled with the numbers from 1 to 49
starting with 1, 2, 3, … in the upper left and
ending with … 47, 48, 49 in the lower right
corner. Participants hoping to win the jackpot
try to guess which 6 of these numbers will
come up in a particular drawing.
In this paper, we consider those
combinations which can be traversed with a
closed path of the chess knight and we
enumerate them up to equivalence.
Two combinations are equivalent if one can
be obtained from the other by a sequence of
one or more of the following transformations:
• translation - slide a combination over
the board in one of the four directions
(up, down, left, right);
• 90° rotation of the board;
• reflection of the board.
In fact the last two transformations are
generators of a group of transformations which
is isomorphic to D8 - the dihedral group of
order 8. The latter group is defined as a group
of all symmetries of the square.
It is worth noting that the classic chess
knight problem is the construction and
classification of a knight’s tour: a sequence of
moves by a chess knight, which visits every
square of the board exactly once. If from the
II-416
last square visited, the knight can return in one
knight's move to the starting square, the tour is
a closed tour, otherwise it is an open tour. The
original knight's tour problem is to find a
closed tour on the classic 8 x 8 chessboard.
The first mathematical paper analyzing
knight's tours was presented by the most
productive mathematician of the eighteenth
century, Leonhard Euler (1707–1783), to the
Academy of Sciences at Berlin in 1759 (but
not printed until 1766) [2].
Plenty of information about knight's tours is
available on the internet, for example, see [3].
From a graph-theory point of view, finding
a closed knight’s tour is equivalent to finding a
Hamiltonian cycle in the graph, corresponding
to the chessboard. While it is well known that
''Hamiltonian cycle'' is an NP-complete
problem, the special properties of the
chessboard graph allow for the construction of
a knight’s tour in polynomial time [1, 6].
Although a single knight’s tour can be
constructed in polynomial time, finding the
number of all knight's tours is a difficult
problem, even in the computer age [7].
In 1996 two researchers write an algorithm
to determine the total number of Knight’s
tours [4], yet their implementation turns out to
be flawed within a year [5], when the total
number of undirected tours is pegged at
13,267,364,410,532 with 1,658,420,855,433
equivalence classes under rotation and
reflection of the board.
International Scientific Conference “UNITECH 2017” – Gabrovo
A knight’s tour visits all squares of the
board, while a knight’s path may visit a subset
of them, allowing for translation to also be
considered as a transformation which produces
equivalent paths.
In this paper we enumerate up to
equivalence all closed paths of length n = 4, 6
and 8 on a square chessboard of size n+1.
. . . o .
o . . . .
. . o . .
3: 2 11
. o . .
. . . .
o . o .
. . . .
. o . .
13 22
.
.
.
.
.
APPENDIX B
NEW RESULTS
We determine that there are exactly 3
nonequivalent solutions for closed paths of
length 4, exactly 25 nonequivalent solutions
for closed paths of length 6, and exactly 478
nonequivalent solutions for closed paths of
length 8 with a computer program whose
algorithm is outlined below.
All nonequivalent solutions of length 4 are
presented in Appendix A, and those of length
6 – in Appendix B.
The brief outline of our algorithm is as
follows.
For each pair of cells we construct all
connecting paths of length k = n / 2.
From every pair of paths we can construct a
cycle of length n, if and only if, the paths do
not intersect, that is, they have no common
point except for the two endpoints.
It is sufficient to consider only cycles,
which cannot be translated up or left, meaning
that they have at least one point in the top row
of the board and at least one point in the leftmost column.
Each of these cycles can be framed in a
rectangle. If the framing rectangles of two
cycles are not congruent, then the cycles are
not equivalent. Therefore, we only need to
examine for equivalence between paths
contained in each non-congruent type of
rectangular frame.
1:
2:
3:
4:
5:
6:
7:
8:
9:
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25:
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
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5
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10
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11
15
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5
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8
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30
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25
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26
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32
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23
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31
25
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24
25
34
17
24
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25
32
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30
39
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37
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45
ACKNOWLEDGEMENTS
The work of the first author was supported
in part by Grant 1704C/2017 of the Technical
University of Gabrovo, Bulgaria.
The second author is grateful for the partial
support by the Research Fund of the
University of Veliko Turnovo, Bulgaria, under
Contract FSD-31-653-07/19.06.2017.
REFERENCES
APPENDIX A
1: 1 8 12 19
o . . . .
. . o . .
. o . . .
. . . o .
2: 2 9 11 18
. o . . .
[1] A. Conrad, T. Hindrichs, H. Morsy and I.
Wegener, Solution of the knight's Hamiltonian
path problem on chessboards, Discrete Applied
Mathematics, 50, 1994, pp. 125-134.
[2] L. Euler, Solution d'une question curieuse qui
ne paroit soumise à aucune analyse” (Solution of a
curious question which does not seem to have been
International Scientific Conference “UNITECH 2017” – Gabrovo
II-417
subject to any analysis), Mémoires de l'Academie
Royale des Sciences et Belles Lettres, Année 1759,
vol.15, pp.310–337, Berlin 1766.
[3] G. Jelliss, Knight's Tours Notes, Available at:
http://www.mayhematics.com/t/t.htm
Accessed 10/23/2017.
[4] M. Lobbing and I. Wegener, The Number of
Knight's Tours Equals 33,439,123,484,294 Counting with Binary Decision Diagrams, The
Electronic Journal of Combinatorics, Vol. 1, Issue
1, 1996, pp. 1-14.
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[5] Br. McKay, Knight's tours of an 8 x 8
chessboard, Technical Report TR-CS-97-03,
Department of Computer Science, Australian
National University, Australia, Feb. 1997.
[6] I. Parberry, An efficient algorithm for the
Knight's tour problem, Discrete Applied
Mathematics, 73, 1997, pp. 251-260.
[7] N. J. A. Sloane, The On-line Encyclopedia of
Integer Sequences, Available at: https://oeis.org/,
Accessed 10/23/2017.
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