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.

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together a book on History of Knight's Tours and Related Problems based on my Knight's Tour Notes web pages and other unpublished notes.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!

Tuesday, 28 January 2014

A Warnsdorf Tour with Minimum Ambiguity

In his 1823 book H. C. von Warnsdorf published a rule for constructing a knight's tour of the chessboard. This was to place the knight on any cell and to move it always to a cell from which it has the fewest exits (to cells not yet used). If there are two or more cells with the same number of exits an ambiguity occurs and you may choose any of the moves with fewest exits.

I've not seen Warnsdorf's book, but it is cited in many later publications. There are copies listed in the catalogues of the British Library, the Cleveland Public Library (USA) and the Koninklijke Bibliotheek (Netherlands). I also found a copy for sale on ABE Books from a dealer in Italy who was asking over £500.

I have been making an enumeration of Warnsdorf tours on the 8 by 8 board by hand with diagrams drawn on a computer. I suspect someone must have done it using a computer program, but I've not seen any results published. So far I've reached the fifth ambiguity and 255 diagrams.

Taking a couple of the more promising diagrams I have followed them through to the seventh or eighth ambiguity, and the tour shown here is the first closed tour I have found. So this probably has the minimum ambiguities (shown by the white cells). I don't count the initial move d4-c2 as an ambiguity, since the choice d4-b3 would merely reflect the tour in the diagonal.

I should have mentioned that d4-e2 (and its reflection d4-b5) is an alternative first move, leading to different tours. So perhaps d4 should be counted as an ambiguity as well.