Chronology of Knight's Tours

PART 4. 2000 onwards

© by George Jelliss, updated 1 September 2014.

Many thanks to Awani Kumar for bringing me up to date on current work! Work on all the pages is a continuing process. Let me know of missing references that you think should be included.
The list of Links to other websites was moved to a separate page, December 2012.


Back to: Part 3: 1900s
On this page 200020012002200320042005200620072008200920102011201220132014

« 2000

M. Loebbing and I. Wegener (2000): Branching Programs and Binary Decision Diagrams. Theory and Applications. SIAM Monographs on Discrete Mathematics and Applications Philadelphia, PA, 2000. [cited in Gordon and Slocum 2004]

M. Lee (2000): Finding Solutions to the Knight's Tour Problem Using Genetic Algorithms, Genetic Algorithms and Genetic Programming at Stanford 2000, published by Koza, J. at Stanford University, 2000. [cited in Gordon and Slocum 2004]

J. J. Watkins (2000): Knight's Tours on Cylinder and Other Surfaces. Congressus Numerantium 143, pp.117-127, 2000. [Google and PDF from A. Kumar] Besides cylinder boards this also considers existence of knight's tours on Moebius boards and Klein bottles.

« 2001

E. Bergholt (2001); The Games and Puzzles Journal, vol.2 #18 (March 2001 pp.321, 327-341) Reproduces the Memoirs 7, 8, 9 on knight's tours written in 1918, on "mixed quaternary symmetry".

G. P. Jelliss (2001); The Games and Puzzles Journal, vol.2 #18 p.345 Knightly quadrangles. More on geometry of knight's moves. p.347 Figured tour showing "intersquare numbers".

G. P. Jelliss (2001); The Games and Puzzles Journal vol.3 (online) #19 January 2001: http://www.mayhematics.com/j/gpj19.htm. 1. Magic squares using alternating moves (J. Wallis Strand Magazine 1908) and magic tours derived therefrom. 2. History of alternating tours (Suli c.900, van Dalfsen 1953). 3. Doubled alternating tours (knight + fers, alfil, or commuter). 4. Tripled alternating tours (D.E.Knuth, original). 5. Figured tours, prime numbers guarded by queen (Mike Keith 1998).

G. P. Jelliss (2001); The Games and Puzzles Journal vol.3 (online) #20 February 2001: http://www.mayhematics.com/j/gpj20.htm. 6. Longest shortest leaper paths (V. Kotesovec). 7. Nightrider tours without knight moves (J. Saukkola). 9. Trapezoidal tours (along edges of a triangulate grid).

G. P. Jelliss (2001); The Games and Puzzles Journal vol.3 (online) #21 March 2001: http://www.mayhematics.com/j/gpj21.htm. 11. Knight's tour font (D. E. Knuth). 12. Celtic tours, on square boards (with no size 1 triangles). 13. 3×n Celtic tours. 14. Fivefold intersected moves in celtic tours. 15. Knight's tour mosaics.

E. Mordecki (2001). On the Number of Knight's tours. Pre-publicaciones de Matematica de la Universidad de la Republica, Uruguay 2001/57 [cited in Gordon and Slocum 2004]. "An upper bound of the number of open tours was found to be approximately 1.305×10^35."

« 2002

G. P. Jelliss (2002); The Games and Puzzles Journal vol.3 (online) #22 May 2002: http://www.mayhematics.com/j/gpj22.htm. 20. Magic and semimagic knight's tours. (T.W.Marlow, on torus and cylinder; A. Kumar on 6×6.)

G. P. Jelliss (2002); The Games and Puzzles Journal vol.3 (online) #23 June 2002: http://www.mayhematics.com/j/gpj23.htm. 21. Equidirectional knight's tour.

A. Kumar (2002); The Games and Puzzles Journal vol.3 (online) #24 August 2002: http://www.mayhematics.com/j/gpj24.htm. 26-30. Magic knight's tours 12x12, including 13 with one diagonal magic.

C. A. Pickover (2002); The Zen of Magic Squares, Circles and Stars (Princeton University Press). An interesting mish-mash, but full of errors if the knight's tours sections are anything to go by. Has three sections containing knight's tours: (a) pp.89-94. Magic tours 00a, 27a, 00m. In the text he attributes the Beverley tour 27a to Euler (will this error ever be eliminated!) yet in the notes at the end of the book he acknowledges that W. S. Andrews (1917) says this square was made by Mr Beverley and published in Philosophical Magazine in 1848, which is correct. So why didn't he properly check his sources? Also gives a two-knight magic tour by Feisthamel (with 32-33 a 3-step rook move). He says Ronald R. Brown composes music and produces abstract art using the knight's tour. (b) pp.210-220. States wrongly that the de Moivre tour is the earliest recorded (citing Stewart 1992). Says Legendre improved on this with a closed tour (which he did, but 100 years later and long after Euler had done the same). Says “not to be oudone” Euler found a closed symmetric tour (which he did, but about the time Legendre was age six). Discusses tour on small boards. Gives Dudeney's tour of the faces of a cube (hinting that he got the idea from Vandermonde - who in fact did a 4×4×4 tour - of the cells not the faces). Then a 25×25 corner to centre tour attributed to Kraitchik (but in fact due to E. Lucas). Gives a tour on the surface of a 2×2×2 cube, with suitable reinterpretation of the knight move. Finally considers knight's tours on a cylinder, Moebius band and Klein bottle. As a post-script to this section, p.220, he gives a tour of squares and diamonds type and relates it to a magic square. (c) pp.232-238. Shows the 16×16 diagonal magic knight tour by H. E. de Vasa (attributed to Madachy 1979). Then goes on to magic king tour. Then non-magic tours by other pieces: Zebra tour 10×10 by J. Scholes, Giraffe tour 9×9, 63-cell Giraffe path on 8×8, Pythagoron (i.e. fiveleaper) tour on 8×8, all attributed to J. Saukkola. Then goes to magic Rook tour by S. Rabinowitz (1985).

G. Cairns (2002); Pillow Chess. Mathematics Magazine (vol.70, no.3, June 2002). About a type of spherical chess, but bibliography has 81 references including some on knight's tours. [google]

A. Grigis (2002); L'indice d'un tour de cavalier Comptes Rendus Académie des Sciences Paris Ser. I 335 (12) 989-992. [cited by Dehornoy 2003]

« 2003

T. W. Marlow (2003); The Games and Puzzles Journal vol.3 (online) #25 January-March 2003: http://www.mayhematics.com/j/gpj25.htm. 31. A semimagic 10×10 knight's tour with quaternary symmetry (magic in the ranks only). 34. Methods of computer search for magic tours.

T. S. Roberts (2003); The Games and Puzzles Journal vol.3 (online) #25 January-March 2003: http://www.mayhematics.com/j/gpj25.htm. 32-33. Two new 8×8 magic knight's tours, 01i and 14e. 34. Methods of computer search for magic tours.

G. P. Jelliss (2003); The Games and Puzzles Journal vol.3 (online) #25 January-March 2003: http://www.mayhematics.com/j/gpj25.htm. 35. Existence theorems on magic tours (proof of impossiblility on a board with singly even sides).

A. Kumar (2003); The Games and Puzzles Journal vol.3 (online) #26 April-June 2003: http://www.mayhematics.com/j/gpj26.htm. 36. In search of perfect magic tours of knight on 12×12 board (21 tours). 37. Four perfect magic tours 12×12. 38. A new type of magic tour (12×12 magic tours with both diagonals having the same sum, but not the magic constant).

G. P. Jelliss (2003); The Games and Puzzles Journal vol.3 (online) #26 April-June 2003: http://www.mayhematics.com/j/gpj26.htm. 39. Emperor magic tours (4×6, 8×6, 8×8, 8×10), 40. Quasimagic tours, ranks magic, columns have two values (4×6, 8×10, 12×6).

G. P. Jelliss (2003); The Games and Puzzles Journal vol.3 (online) #27 May 2003: http://www.mayhematics.com/j/gpj27.htm. 41. In the footsteps of Eratosthenes: a step is from p×q to p×(q+1) or to p×(q–1) or to (p+1)×q or to (p–1)×q, where the unchanged factor must be a prime number. Find a path of Eratosthenes from 5^3 to 7^3.

S. Korteling (2003); The Games and Puzzles Journal vol.3 (online) #30 December 2003: http://www.mayhematics.com/j/gpj30.htm. 57. Bishop shortest paths. [Solutions in vol.3 (online) #31 January-February 2004: http://www.mayhematics.com/j/gpj31.htm.]

J. Saukkola (2003); The Games and Puzzles Journal vol.3 (online) #30 December 2003: http://www.mayhematics.com/j/gpj30.htm. 58 Bishop maximummer paths. 59. Mao and Moa longest paths. [Solutions in vol.3 (online) #31 January-February 2004: http://www.mayhematics.com/j/gpj31.htm.]

J-C. Meyrignac and G. P. Jelliss (2003); The Games and Puzzles Journal vol.3 (online) #30 December 2003: http://www.mayhematics.com/j/gpj30.htm. 64. Knight's paths on shaped boards. [Solutions in vol.3 (online) #31 January-February 2004: http://www.mayhematics.com/j/gpj31.htm.]

P. Dehornoy (2003); Counting moves in knight's tours (Décompte des mouvements dans les tours de cavalier) Comptes Rendus Académie des Sciences Paris Ser. I 336 (2003) 543-548. Abstract: A knight's tour contains eight types of elementary moves. We prove that the only asymptotic constraints on the numbers of moves of each type are the trivial ones: for all proportions compatible with these constraints, there exists a sequence of tours asymptotically achieving these proportions. We deduce a positive answer to the question asked by A. Grigis (2002) about the existence of tours with an arbitrarily large index. [offprint from author]

G. Stertenbrink, J-C. Meyrignac and H. Mackay (2003): Computing Magic Knight Tours. http://magictour.free.fr/mt16. Results of distributed programming search for all magic knight's tours. Five new cases.

G. P. Jelliss (2003); Recent Advances in magic Knight's Tours. Variant Chess 6(43) November 2003 pp.40-41.

« 2004

G. P. Jelliss (2004): Introducing Knight’s Tours. Knight's Tour Notes http://www.mayhematics.com/t/ktn.htm [originally at: http://homepages.stayfree.co.uk/gpj/ktn.htm which is now closed down]. Topics covered: 1. earliest knight's tours, 2. Knight's tour as conjuring trick, 3. Squares and diamonds method, 4. Symmetry and shape in tours, 5. Cryptotours, 6. Enumeration of knight's tours, 7. Magic knight's tours, 8. Figured tours, 9. Other piece tours.

A. Kumar (2004); Studies in Magic Tour of Knight on 12×12 Board. The Games and Puzzles Journal vol.3 (online) #32 March-April 2004: http://www.mayhematics.com/j/gpj32.htm. Deals with reentrant tours that are four-fold cyclic and have one diagonal magic. Also some open tours.

Ben Hill and Kevin Tostado (2004), Knight's Tours December 18, 2004, 11pp [PDF, available from http://faculty.olin.edu/~sadams/DM/ktpaper.pdf (Sarah Adams, Discrete Mathematics, Franklin W. Olin College of Engineering, Needham, Massachussetts, USA)]

P. Hingston and G. Kendall (2004). Ant Colonies Discover Knight's Tours. AI 2004: Advances in Artificial Intelligence: The 18th Australian Joint Conference on Artificial Intelligence, Cairns, Springer. [cited by Hingston and Kendall 2005]

V. Scott Gordon and Terrill J. Slocum (2004): The Knight's Tour - Evolutionary vs. Depth-First Search. IEEE Congress on Evolutionary Computation, (CEC'04) p.1435-1440, Portland, Oregon. [PDF from A. Kumar] Uses knight's tours as a way of comparing genetic algorithms with other methods.

« 2005

J.J.Watkins (2005): Across the Board: The Mathematics of Chessboard Problems, Princeton University Press, 2005. [cited by A. Kumar 2005, 2009]

Awani Kumar (2005): Further Studies in Magic Tours of Knight on 12×12 Board. The Games and Puzzles Journal vol.3 (online) #42 November-December 2005: http://www.mayhematics.com/j/gpj42.htm.

G. P. Jelliss (2005): On the Relation of Magic Squares to Latin Squares. The Games and Puzzles Journal vol.3 (online) #42 November-December 2005: http://www.mayhematics.com/j/gpj42.htm.

Awani Kumar (2005): Studies in Magic Tour of Knight on 16×16 Board. Journal of Recreational Mathematics 34(4) pp.275-285, 2005-2006.

Philip Hingston and Graham Kendall (2005?); Enumerating Knight's Tours using an Ant Colony Algorithm. (Journal?) Total tours found on 5×5 board = 1728, on 6×6 board = 6,637,920 of which 19,724 are closed tours. [PDF from A. Kumar]

Tim S. Roberts The Discovery of Two New Magic Knight's Tours, The Mathematical Gazette, 89 (1), 22-27, ISSN 0025-5572.

« 2006

Awani Kumar (2006): Studies in Tours of the Knight in Three Dimensions. The Games and Puzzles Journal (online) #43, January-April 2006. http://www.mayhematics.com/j/gpj43.htm. Covers cuboids, 4×4×4, 6×6×6, 8×8×8,

Alexander Fischer (2006): New Records in Non-Intersecting Knight Paths. Games and Puzzles Journal (online) vol.3 #45 September-December 2006 [but publication was delayed to April 2010]. http://www.mayhematics.com/j/gpj45.htm. 183 steps on 16×16 board, 135 steps on 14×14 board.

S. K. Singh (2006): A 16×16 Diagonal Magic Knight's Tour. Games and Puzzles Journal (online) vol.3 #45 September-December 2006 [but publication was delayed to April 2010]. http://www.mayhematics.com/j/gpj45.htm.

« 2007

John Beasley (2007): A Remarkable Magic Tour in Three Dimensions. Variant Chess 8(55), September 2007, p.8. Analyses a 4×4×4 tour by Guenter Stertenbrink (2003).

Joe DeMaio (2007, may 9): Which Chessboards have a Closed Knight's Tour within the Cube. 9-page PDF. The Electronic Journal of Combinatorics 14 (2007) #R32.

« 2008

John Beasley (2008): Latin Cube Magic Tours. Variant Chess 8(56) February 2008 p.32. More on the work by G. Stertenbrink and A. Kumar.

John Beasley (2008): Another Look at 8×8 Magic Knight's Tours. Variant Chess 8(57) May 2008 pp.41, 50-53. Includes complete catalogue.

John Beasley (2008): Torus Tours. Variant Chess 8(58) October 2008 p.60.

Awani Kumar (2008): Magic knight's tours for chess in three dimensions. The Mathematical Gazette 92 (523) pp.111-115, March 2008.

Awani Kumar (2008, March 29): Non-crossing Knight's Tour in 3-Dimension. 6-page PDF.
From: arXiv:0803.4259

Hector Cancela and Ernesto Mordecki (2008, February 2): Counting Knight's Tours through the Randomized Warnsdorff (sic) Rule. 8-page PDF.
This appears to be a later version of: arXiv:math/0609009v1 [math.PR] 31 Aug 2006.

Samuel L. Marateck (2008, 30 March) How good is the Warnsdorff's (sic) knight's tour heuristic? 3-page PDF.
From: arXiv:0803.4321v1 [cs.DM] 30 Mar 2008

« 2009

John Beasley (2009): Grasshopper Tours. Variant Chess 8(61) July 2009 p.115. Reviews work by V. Kotesovec.

John Beasley (2009): Complementary Five-Leaper Tours. Variant Chess 8(62) p.131.

Awani Kumar (2009): A Study of Knight's Tours on the Surface of a Cube. Crux Mathematicorum and Mathematical Mayhem 35(5), pp.313-319, 2009.

Awani Kumar (2009): Construction of Magic Knight's Towers. Mathematical Spectrum 42(1) pp.20-25, 2009/10.

Vaclav Kotesovec (2009): Application of Graph Theory in Chess Problems (Dual-free Leaper and Hopper Tours) 76pp. [PDF from author] Text in Czech and English. [http://web.telecom.cz/vaclav.kotesovec/]

R. Borell (2009): A brute force approach to solving the knight's tour problem using proglog [PDF]

« 2010

G. P. Jelliss (2010); On Mixed Quaternary Symmetry in Knight's Tours. Variant Chess 8(63) January 2010 pp.165-169.

John Beasley (2010): Complementary Five-Leaper (and other) Tours with Rotational Symmetry. Variant Chess 8(64) pp.232-3.

« 2011

Andrew Perkis (a) Tours de Force. Games August 2011, pp.50-51 and 64; tour puzzles using special boards and move rules. (b) Cousins of the Knight's Tour. Games August 2011, pp.65-69; reproduces items on history of knight's tours and wazir tours, together with explanations of the new puzzles by the author in the preceding article. [PDF preprint from author]

G. P. Jelliss 'A Magic Tour on a 12×14 Board'. On KTN website and Jeepyjay Diary pages. A counter example to the conjecture that magic tours are possible only on rectangles 4m×4n; this is a board 4m×(4n+2). Proving The Possibility.

Adam H. Berliner (2011, April): A Knight's Tour de Force. Account of a 3D knight's tour artwork by Loren Larsen (or Larson?).
From: Math Horizons www.maa.org/mathhorizons

Joe DeMaio and Bindia Mathew (2011, January 5): Which Chessboards have a closed Knight's Tour within the Rectangular Prism? 14-page PDF.
The Electronic Journal of Combinatorics 18 (2011) #P8.

« 2012

J. D. Beasley (2012): 'Magic Knight's Tours' The College Mathematical Journal vol.43, #1 (January 2012) p.72-75.

J. D. Beasley (2012): 'Two notes on magic knight's tours' Other Games and Puzzles, JSB website [www.jsbeasley.co.uk] March 2012. A follow-up note to the above article.

Awani Kumar (2012 January 2): Magic Knights Tours in Higher Dimensions.
From: arXiv:1201.0458

Joshua Erde (2012) Knight's Tours in Higher Dimensions. 15-page PDF. Submitted to Electronic Journal of Combinatorics 9 February 2012.
From: arXiv:1202.5548v1 [math.CO] 24 Feb 2012.

Joshua Erde, Bruno Golenia and Sylvain Golenia (2012, April): The Closed Knight Tour Problem in Higher Dimensions. 22-page PDF.
From: arXiv:1202.5291v3 [math.CO] 20 April 2012.

Peter G. Brown 'The Magic Squares of Manuel Moschopoulos' Loci (July 2012) and on MAA website: http://www.maa.org/

« 2013

Nina Kamcev (2013, August 23): Generalised Knight's Tours
From: arXiv:1311.4109v3 [math.CO] 7 Feb 2014.

Nikolai Ivanov Beluhov On Crosspatch Knight's Tours. 7-page PDF.
From: arXiv:1310.3450v1 [math.CO] 13 Oct 2013

« 2014

Manuel Aaron and Vijay D. Pandit (2014): Indian Chess History 570 AD - 2010 AD. Knight's Tour is covered on pages 207-8,
including biographical details of the Rajah of Mysore (dates given as 1794 - 1868) and Awani Kumar (b. 10 August 1962).


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