Monday, 7 December 2009

Article on Mixed Quaternary Symmetry

I've completed an article "On Mixed Quaternary Symmetry in Knight's Tours" which John Beasley has accepted for the next issue of Variant Chess. It follows up the work by Ernest Bergholt written in the form of memoranda sent to H. J. R. Murray in 1918 but not published until 2001 in The Games and Puzzles Journal #18.

It is only possible to include a summary of my results in the article, and I will be putting diagrams of all the tours onto my mayhematics website. The idea of mixed quaternary symmetry is to produce tours on the 8 by 8 and 12 by 12 boards that show a combination of direct (reflective) and oblique (rotative) quaternary symmetries, since tours fully in oblique quaternary are not possible on these boards, though they are possible on the 6 by 6 and 10 by 10 boards. Tours in direct quaternary symmetry are not possible on any boards, though pseudotours with this symmetry formed of two or more superimposed circuits are.

Where my treatment differs from that of Bergholt and Murray is in separating out the moves, such as the eight corner moves, that show octonary symmetry. Such moves can be regarded as part of either the direct or oblique sets of moves, but it is not always clear to which of these sets they should be assigned.

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