Most Asymmetric Tour?

How does one assess how asymmetric a tour is? Or indeed any object? Is there an objective measure of the degree of asymmetry?

Thinking over this question the last two days I have come up with the tour shown which I mantain is the most asymmetric tour possible. There may be better, or alternative ways of measuring asymmetry, but I have concluded that in the case of a tour it is the number of the 21 geometrically distinct moves that occur an odd number of times. In this tour the total is 18, and moreover 12 of these occur only once.

The only moves that occur an even number of times are the corner moves like a1-c2 (where there are eight, which occur in every closed tour) the edge-to edge moves of the type a2-c1 (eight again), and the pair of moves a3-c2 and h3-f2 which are of the same type. Two moves are considered to be of the same type if one can be put in the position of the other by a rotation or reflection of the board.

The 12 moves that occur only once are b1-c3, d1-c3, f1-e3, b2-c4, d2-b3, d2-f3, d3-c5, e3-d5, a4-b6, e4-g5, b5-d6, e5-f7.

Warnsdorf Squares and Diamonds Tour

The first thing I noticed on seeing the figures from the Warnsdorf 1823 book, is that the final figure #96 is a tour of squares and diamonds type! This seems to have been added as an afterthought and is preceded by a diagram in which the squares and diamonds are lettered A, B, C, D using a different type style in each quarter of the board.
This confirms Murray's statement that 'The first composer to give the figure which shows that the cells of the quadrant could be filled by four closed quartes was v. Warnsdorf (1823)'. However the diagram is not in graphic form, and strangely Murray made no mention of the accompanying tour!
Warnsdorf shows tours in numerical form. Here is the tour diagram in graphic form:

This means that Warnsdorf has priority in the showing of a tour of squares and diamonds type. The previous earliest examples I was aware of were the two given by "F.P.H" in 1825 and the near-magic tours in the study by C. R. R. von Schinnern in 1826.

Warnsdorf from the original source

Thanks to the Cleveland Public Library, Ohio USA, I have now been able to study the diagrams in the 1823 book by H. C. von Warnsdorf. The figures relating to his Rule ("Always move the knight to a cell from which it has the fewest exits") are numbers 32 to 39 in Table 5. He begins by applying the rule to the 6x6 board, with four examples, then goes on to the 6x7, 6x8, 7x8 and 8x8 boards, giving one example for each.
Warnsdorf's tours are shown with the cells numbered in the sequence visited by the knight, but here we show them in geometrical form. A black dot indicates the start cell, and a dark circle the end cell. A circle round the initial dot indicates that there are choices of first move, though this may only be because of symmetry. The light circles mark points where the rule provides a choice of two or more moves, so that alternative routes could have been taken.

In the fourth 6x6 example the move e4-c3 (marked by a cross) does not conform to the rule, which requires e4-f6 as in the third 6x6 example. Similarly in Warnsdorf's 8x8 example there is a deviation from the rule at h5. The correct move should be to f4 which has access to only two exits whereas g7 and f6 have three. However these deviations only go to show the robustness of the rule, since in each case it still goes on to complete a tour.

Google Books useful in my Knight's Tour Research

I've recently found that some important and obscure texts on knight's tours are now available in Google Books. The following are links to four of them:

Warnsdorf 1823

http://books.google.co.uk/books?id=w5FZAAAAYAAJ&printsec=frontcover&dq=h.+c.+von+warnsdorf&hl=en&sa=X&ei=2QZmU9-jJIaXO8-hgKgL&ved=0CDEQ6AEwAA#v=onepage&q=h.%20c.%20von%20warnsdorf&f=false

Kafer 1842

http://books.google.co.uk/books?id=si1RAAAAcAAJ&printsec=frontcover&dq=Kafer+1842&hl=en&sa=X&ei=3nfbU9PFO8PE7AbEooGYDA&ved=0CCEQ6AEwAA#v=onepage&q&f=false

Perenyi 1842

http://books.google.co.uk/books?id=jf0UAAAAYAAJ&printsec=frontcover&dq=perenyi+1842&hl=en&sa=X&ei=dvtlU7fFNcGJPdbZgPgK&ved=0CDEQ6AEwAA#v=onepage&q&f=false

Jaenisch's Treatise 1862

http://books.google.co.uk/books?id=UMpUAAAAYAAJ&pg=PP11&dq=Jaenisch+Traite&hl=en&sa=X&ei=lfLKU-O-O-Se7Aau6YCwDA&ved=0CDAQ6AEwAw#v=onepage&q=Jaenisch%20Traite&f=false

A problem with the Warnsdorf and Kafer books is that the Google versions do not reproduce the figures!
However I found a copy of the Kafer sheet of diagrams elsewhere on the web, and I was able to obtain a copy of the Warnsdorf tours through the kindness of Cleveland Public Library, Ohio.
More on this subsequently.

There are probably may other titles to be found by a search in Google Books.
Kafer details added in update of this page 1 Aug 2014.

Google Rules the Web

Having received instructions from Virgin that my email address on virgin.net would no longer be accepted on Google sites from 7 May I have now changed the email I use for this purpose. I'm also using the Chrome browser to access Google sites, since they object to Internet Explorer. It seems that if you don't conform to what Google dictates these days you are an outcast on the web.

I also note that my Mayhematics home page now doesn't come out properly in Chrome. The six areas around my photo are supposed to be equal-sized squares, but some of them are now stretched out to the right. I suppose I will need to modify the HTML in some way.

Work on Hastings Pier goes ahead

I took this photo from the steps up to the White Rock path near Clambers.

Work so far is only on the front apron of the pier. The far end is still a ruin.

 

 

 

 

Warnsdorf Counterexample

In W. W. Rouse Ball's Mathematical Recreations and Essays (11th edition 1939, and probably earlier editions) states (p.181): "Warnsdorff [sic] added that when, by the rule, two or more cells are open to the knight, it may be moved to either or any of them indifferently. This is not so, and with great ingenuity two or three cases of failure have been constructed, but it would require exceptionally bad luck to happen accidentally on such a route." However he gives no reference to where work showing this was done, and diagrams no example of it. The diagrams shown here are the first cases I have encountered where the rule fails.

The choices of route available from f7 onwards all lead to a dead end, shown by the darker circle, that leaves two cells, marked by crosses, unvisited.