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!

Sunset Skies

 

Two contrasting photos from roughly the same position on the promenade in St Leonards. The first taken on 8th December the second on 25th December this year.

 

A few weeks ago I observed a very unusual sunset sky, but did not have my camera with me at the time. The sun appeared as a red hemisphere above dark clouds. Since then I have periodically walked down to the seafront around that time in the evening for a constitutional walk, but a similar sky has not recurred. I find the sky-scapes over the sea here endlessly fascinating and beautiful.

A Remembrance Day Walk

Since it was a nice sunny day this morning, I took a walk around 10:30 and ended up at the war memorial in Alexandra Park where the remembrance ceremony was in progress. It was all very much under control of the church with the usual formulaic wording. At the end there was a rendition of the national anthem, but I was surprised that none of the congregation seemed to join in the singing. Maybe Hastings is a republican place! Shortly after there was a parade led by a group in highland dress playing pipes and drums, which went from the park towards Queens Road, but nothing tuneful such as It's a Long Way to Tipperary, which I always find the main attraction of such events.

The event at the Royal Albert Hall, televised on Sunday evening, also seemed to be more under the dead hand of the church than usual, with the parade of the bible to the drum-head taking centre stage. There is a campaign for more secular representation at these ceremonies, but I can't see the church loosening it's grip, just as it is gaining more control over education, despite society itself becoming more secular.

Bournemouth Chess

1-4 November, I travelled to Bournemouth for the Dorset Chess Congress. Scored 3/5 in the Minor section. My best play was as Black in the mornings, after a good night's sleep. My best game was against an ungraded player on the Sunday morning. She sacrificed a Bishop early on for an attack on my King position, but with dogged defence I managed to beat it off and in the last few moves her position collapsed remarkably quickly with my Queen mopping up all the loose pieces.

I bought a book from the bookstall: "Black is OK for ever" by Andras Adorjan, apparently it is the third in a series he wrote. It seems quite humorous. Unlike most chess books it shows all the diagrams from Black's point of view.

A Magic Two-Giraffe Tour!

I've returned to doing some work on tours and other mathematical recreations after a break of a few months. I revisited the proofs I published in 1976 beginning in the second issue of Chessics that closed and open Giraffe tours on the 8x8 board are impossible. This led me to find two 16-move closed tours of the A1-B1 cells and two 16-move closed tours of the A2-B2 cells. In each case one tour is symmetric and the other asymmetric. There are also 16-cell open tours possible of course. Two such tours can be joined to form a 32-cell open tour. I have combined two copies of such a tour to form a Magic Two-Giraffe Tour, where the link 32-33 is a rook move:

15 46 11 42 23 54 19 50
24 53 20 49 16 45 12 41
01 36 05 40 25 60 29 64
26 59 30 63 02 36 06 39
47 14 43 10 55 22 51 18
56 21 52 17 48 13 44 09
33 04 37 08 57 28 61 32
58 27 62 31 34 03 38 07

I composed this tour just this morning (Wednesday 9 October), using the two symmetric quarter tours. Must now check whether any others are possible.

Catching Up Again

I found the heat during the summer rather exhausting, and lack of progress on much of my work has been depressing. This has also been made worse by the plastering and painting work being done on the outside and in the hallway of the building where I live which has made it difficult to get out or to concentrate on anything. Of course the place will look much better when it's all finished. One of the improvements has been the removal of a broken down old car that has been standing in the parking space alongside the house for a long while. I have a photo of its removal that I was going to post here, but am having difficulty finding it, since the 'My Pictures' to which the blog connection gives access for some reason doesn't show the file containing the photograph. This is puzzling. I may have to move the photo and come back to post it. - I seem to have managed that, but the photo appears in the top middle instead of the top left corner as it used to, and there was no longer a choice of arrangements offered.