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.
Friday, 2 May 2014
Tuesday, 29 April 2014
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.
Monday, 10 February 2014
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.
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.
Friday, 7 February 2014
Another Two Warnsdorf Closed Tours
Continuing my enumeration I have found two more closed tours that follow the Warnsdorf rule and have only seven places where the rule allows two or more choices. These points of ambiguity are shown by the white circles.
The tours start at the black dot and end at the darker outlined circle (this is not a point of ambiguity). Perhaps the tours should properly be termed "re-entrant" since the move joining the end point to the first point is extra to the construction. Any link to the black dot in the course of the construction is prohibited since it would result in a closed circuit covering only part of the board. Alternative routes at the last ambiguity lead to open tours.
The tours start at the black dot and end at the darker outlined circle (this is not a point of ambiguity). Perhaps the tours should properly be termed "re-entrant" since the move joining the end point to the first point is extra to the construction. Any link to the black dot in the course of the construction is prohibited since it would result in a closed circuit covering only part of the board. Alternative routes at the last ambiguity lead to open tours.
Monday, 3 February 2014
Two More Warnsdorf Closed Tours
Here are two more closed tours formed according to the Warnsdorf rule, with minimum of seven cells where there is an alternative route. In these tours there is no choice on the first move given the initial placement of the knight at d1.
These tours are not as symmetric as the previous example and differ from each other only in the choice of routes from f5. Other choices at e4 lead to open tours. This is part of work in progress. I've not yet enumerated all the solutions with 6 ambiguities, but have reached 540 incomplete paths with 5 ambiguous points.
Thursday, 30 January 2014
Warnsdorf Tour Symmetrised
The Warnsdorf tour I published here a couple of days ago seemed remarkably symmetric compared with others of the type. So I thought I would see what it looks like when completely symmetrised.
This is done by deleting all moves that are not part of a symmetrically arranged pair - which leaves 52 moves in this case - and joining up the loose ends to make a symmetric closed tour. This can be done in two ways as shown here.
This is done by deleting all moves that are not part of a symmetrically arranged pair - which leaves 52 moves in this case - and joining up the loose ends to make a symmetric closed tour. This can be done in two ways as shown here.
Wednesday, 29 January 2014
A Twissty Tour
I'm making considerable progress with putting together a book on History of Knight's Tours and Related Problems based on my Knight's Tour Notes web pages and other unpublished notes.
The knight's tour of a Circular Chess board shown here appears in Volume 2 of Chess by Richard Twiss published in 1789, although it is shown there by numbers entered on the board. I've had to curve some of the moves for clarity.
The details of this work were sent to me by the late Ken Whyld on 4 March 2002. Twiss reports having seen the round board in a manuscript in the Cotton collection. The text reads: "The figures on this board (in the plate) show the march of the Knight in order to cover the sixty-four squares in as many moves. This I found after four or five hours trial on a slate at different times; it probably has never been done before, and will be found much more regular than any of the like marches on the square board." The visual form is certainly much more striking than the numerical version!
The knight's tour of a Circular Chess board shown here appears in Volume 2 of Chess by Richard Twiss published in 1789, although it is shown there by numbers entered on the board. I've had to curve some of the moves for clarity.
The details of this work were sent to me by the late Ken Whyld on 4 March 2002. Twiss reports having seen the round board in a manuscript in the Cotton collection. The text reads: "The figures on this board (in the plate) show the march of the Knight in order to cover the sixty-four squares in as many moves. This I found after four or five hours trial on a slate at different times; it probably has never been done before, and will be found much more regular than any of the like marches on the square board." The visual form is certainly much more striking than the numerical version!
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