Next Article in Journal
An Explicit Hybrid Method for the Nonlocal Allen–Cahn Equation
Previous Article in Journal
Correction: Bianca, B.L.; Gheorghe, P.S. Unsupervised Clustering for Hyperspectral Images. Symmetry 2020, 12, 277
Previous Article in Special Issue
Fractional Riccati Equation and Its Applications to Rough Heston Model Using Numerical Methods

Closed Knight’s Tours on (m,n,r)-Ringboards

Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand
Department of Mathematics, Ramkhamhaeng University, Bangkok 10240, Thailand
Author to whom correspondence should be addressed.
Symmetry 2020, 12(8), 1217;
Received: 2 July 2020 / Revised: 21 July 2020 / Accepted: 23 July 2020 / Published: 25 July 2020
(This article belongs to the Special Issue Recent Advances in Discrete and Fractional Mathematics)
A (legal) knight’s move is the result of moving the knight two squares horizontally or vertically on the board and then turning and moving one square in the perpendicular direction. A closed knight’s tour is a knight’s move that visits every square on a given chessboard exactly once and returns to its start square. A closed knight’s tour and its variations are studied widely over the rectangular chessboard or a three-dimensional rectangular box. For m,n>2r, an (m,n,r)-ringboard or (m,n,r)-annulus-board is defined to be an m×n chessboard with the middle part missing and the rim contains r rows and r columns. In this paper, we obtain that a (m,n,r)-ringboard with m,n3 and m,n>2r has a closed knight’s tour if and only if (a) m=n=3 and r=1 or (b) m,n7 and r3. If a closed knight’s tour on an (m,n,r)-ringboard exists, then it has symmetries along two diagonals. View Full-Text
Keywords: legal knight’s move; closed knight’s tour; open knight’s tour; Hamiltonian cycle; ringboard; annulus-board legal knight’s move; closed knight’s tour; open knight’s tour; Hamiltonian cycle; ringboard; annulus-board
Show Figures

Figure 1

MDPI and ACS Style

Srichote, W.; Boonklurb, R.; Singhun, S. Closed Knight’s Tours on (m,n,r)-Ringboards. Symmetry 2020, 12, 1217.

AMA Style

Srichote W, Boonklurb R, Singhun S. Closed Knight’s Tours on (m,n,r)-Ringboards. Symmetry. 2020; 12(8):1217.

Chicago/Turabian Style

Srichote, Wasupol, Ratinan Boonklurb, and Sirirat Singhun. 2020. "Closed Knight’s Tours on (m,n,r)-Ringboards" Symmetry 12, no. 8: 1217.

Find Other Styles
Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

Article Access Map by Country/Region

Back to TopTop