Symmetric Tours of Squares and Diamonds

I reported starting a count of these tours a while ago. The result found was 274 tours in all of which 82 are of the double halfboard type. The first figure may still be short, and a further check is needed.

The second figure is definitely correct, checked by an independent method. There are three types of double halfboard tour according as the separation between the two links that connect the halves is 1, 3 or 5 cells. The numbers of these types are respectively 26, 28 and 28 adding to 82.

I'm continuing to work on my book on tours. At present in is in two parts each of about 250 pages. The first part being on History and the second on Theory.
 

Batten Down the Hatches!

It was disappointing to find that my chess grade has only gone up to 90. I was hoping it might get back to 100 as I had put in considerable effort and achieved some good results. It seems it is easier to slide two yards back down the greasy pole than to climb one yard up.

This afternoon I received a copy of a paper I had requested from a journal in Canada only a few days ago. On the other hand a cheque I sent to a company in Kent two weeks ago disappeared in the post, and I had to report the details to the bank to ensure it is not passed for payment.

When I went for an evening walk along the front yesterday evening for exercise and to see the sunset, I happened to see that a shop selling carpets was open, in the parade beneath the Marina building. This reminded me that I needed a mat to go under my chair to protect the fitted carpet from wear. I bought a nice colourful rug, about 1 by 2 metres, for £25. This has brightened up the room considerably.

It was windy out today and apparently the winds are going to get stronger during my 76th birthday on Monday, so it doesn't look as though I will be going out on any trip as I had hoped. Time to batten down the hatches and try to get some work done on my books.

Hastings Chess Congress

The Hastings Chess Congress occupied most of my time from 28 December to 5 January, since I entered all four of the supporting events. Just in the lowest graded sections of course, not the Masters! Scored 3/5 in the Xmas AM event (winning £15 grading prize), but only 1/5 in the Xmas PM event. Then in the Weekend event scored 3/5 again, against stronger players (winning £25 grading prize). Only managed 1.5/4 in the New Year event. 

These results continue my improved play over the past six months, with similar 3/5 results at Thanet in August and Bournemouth at the end of October. I'm also doing well in the internal Hastings Chess Club Rush Cup event. I hope all this will restore my ECF grade to near 100 again when the new gradings are announced later this month.

Symmetric Rhombic Tours

In November I looked at the problem of enumerating the symmetric tours of squares and diamonds type.  I found there were four distinct ways (1, 2, 3, 4) of arranging the diamonds in a quarter board, and twenty ways these could be arranged together on the board, namely in the ten pairs 11, 22, 33, 44, 12, 13, 14, 23, 24, 34, each either in direct (=) or oblique (~) symmetry. This enumeration is not yet complete but I found unique solutions in the 22= and 44~ cases, both being double halfboard tours.

A Metasquare Figured Tour

T. R. Dawson composed a figured tour with what he called the 'double triangle numbers' of the form n.(n+1) namely 2, 6, 12, 20, 30, 42, 56 in a figure of eight formation that was published in the London Evening Standard in 1932 (B17 in my Figured Tours booklet). I like to call these numbers 'metasquares'.

Waiting for a council inspector to call yesterday I though to while away the time by composing a figured tour and recalled the above task. There are too many or too few metasquares to arrange in a metasquare. It occurred to me to include the number 0 = 0.1 in place of 64 which would give a set of eight metasquares, all even numbers that might be arranged on a diagonal.

This is the result, composed 25 November 2015. The numbers are in cyclic order.

Chess in Dorset

I had an enjoyable weekend at the Bournemouth Chess Congress. Scored 3 wins and 2 losses.
Should have been 4 wins but I slipped up at the end of the third game out of sheer tiredness.
Probably went for too long a walk to Shelley Park and back through Boscombe Chine. Also took a walk round the pier there, where they have a number of musical instruments of xylophone type on display for people to play. This may be an idea Hastings Pier could take up.

I also slipped up at the end of the fourth game but it was already lost, being several pawns down.
Took another long walk on the Sunday, taking in some plaice and chips at Harry Ramsden's near Bournemouth Pier. Got back just in time for the fifth game, but slipped up immediately at the start by giving away a knight, yet still somehow won.

The hotel I stayed in wasn't as quiet as usual due to a Halloween night party going on till late.

Symmetric Chessboard Knight Tours

I've just completed an extensive piece of work collecting and constructing symmetric knight's tours on the normal chessboard. There are 183 tours on the page, showing all possible central angle formations.

http://www.mayhematics.com/t/8s.htm

Here is one group of tours from the page, (hh), (hi), (hj):

The first two are my own constructions the (hj) is from Wenzelides in Schachzeitung 1849.

It was only a month or so ago that I was able to see the work by Brede 1844 that contains 24 tours, all but #14 being in 180 degree rotational symmetry. It has become available due to digitisation. This was evidently an important pioneering work that seems to have been under-appreciated. The tours are at the end of the book:

https://books.google.co.uk/books?id=qf5dAAAAcAAJ&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false

There is also a semimagic tour formed of two copies of a 4x8 semimagic tour joined together.