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.

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.

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.

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.

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!

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.

Hastings Christmas Chess Tournament

Here is a somewhat fuzzy photo of the venue for the Hastings Christmas chess tournament,
just before the start of play on Sunday afternoon.

I was playing in the Minor section and scored 2.5/4, winning £20 prize
(£10 for joint third place, and £10 grading prize).
Should have at least drawn the last game, but was too tired at the end.
Was too tired for the New Year Morning tournament and scored only 1.5/5.
Recovered for the Weekend tournament scoring 2.5/5,
but tired at the end again.
Next year must remember not to enter the weekday event!