© 2003 — compiled by George Jelliss.
J. J. Watkins (2000) Knight's tours on cylinder and other surfaces. Congressus Numerantium 143 (2000) 117-127. [google]
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".
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]
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]
Links to more links!
Mario Velucchi - page of links to other knight's tour websites!