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.

Long Walk to the East Hill

Went for a good long walk from around 11:30 to 3 pm to get away from the noise of plastering work being done at last on the building where I live. Went into the Hastings old town, and up the steps to the West Hilll Cafe for fish and chips (though it wasn't really any better than I could cook for myself at home). From there I decided to follow the contour round with a view to ending up at East Hill. This took me eventually to steps leading up to what is shown on the map as Pinders Walk. From there to the beacon on East Hill and down all the steps and along the seafront. It was a lovely sunny day, but surprisingly few people about for most of the time.

This is the longest bit of walking I have done since  went to Ramsgate for the Thanet Chess Congress back on 21-24 August. It was sunny for most of the time there, and I walked every day the two miles to the venue and back, from the Oak Hotel by the Ramsgate Harbour. The walking seems to have cleared my mind too since I played quite well, scoring 3.5/5 in the Minor, I won the first three games, so at that stage was leading the tournament, but couldn't keep it up. I suppose I should have reported it here on my Diary at the time, but can't seem to keep it going regularly.

Interlacing Tour

Here is an image of a tour formed from four copies of a circuit that does not cross itself. One move in each circuit is deleted and the loose ends joined up to convert the pseudotour into a tour, a process I call simple-linking. This is not a new result.

However the paths have been made to interlace, passing alternately over and under each other. This is the new aspect, and accomplishing this was the difficult part of the drawing. Other colour schemes might prove more attractive, but for this first attempt I have used primary colours, with the links in grey. Making a tour of this type was suggested in part by correspondence with Harold Cataquet.

Magic Empress Tour

A new Diagonally Magic Empress (Knight + Rook) Tour

Numbered from c8:

31 08 01 52 29 48 41 50 
02 53 30 07 42 51 28 47 
09 32 03 54 27 46 49 40 
04 55 60 43 06 21 26 45 
61 10 05 22 59 44 39 20 
56 33 62 11 38 19 16 25 
63 12 35 58 23 14 37 18 
34 57 64 13 36 17 24 15 

While going through my notes on knight's tours and checking the file on magic two-knight tours (which are really empress tours since the connecting moves are rook moves) I found this diagonally magic tour which is not among the 14 with quaternary symmetry listed in the manuscript by H. J. R. Murray, detailed on this page of my website:

http://www.mayhematics.com/t/pi.htm#(5)

It is also not in the Glasgow Weekly Herald column of 1873-4 where examples of such tours were first published, unless it is in a part that I didn't get to see when I consulted it at Colindale Newspaper Library in 1996.

I can only conclude that it must have been found when I made a check of Murray's work by searching through the list of knight circuits in quaternary symmetry that was provided to me by T. W. Marlow in 1985 when he used a computer to check the 1882 listing by Paul de Hijo (alias the Abbe Phillippe Jolivald).

I have only now realised that it must be a new discovery.

Building Work in Progress?

This is the building where my flat is. Scaffolding has been up several weeks now.

The plaster has been removed on most of the walls, and a lot of it has not been cleared.
A drainpipe has been disconnected and is allowing water to run down the street.
When the work is going to be recommenced there's no telling.

Non-crossing Tours with Quaternary Symmetry

Over the past few days I have looked at non-intersecting (also known as non-crossing or self-avoiding) knight tours on smaller boards.

Beginning with this result of 4x7 = 28 moves on the 8x8 board.This is clearly the best possible since the maximum for a closed tour without the symmetry condition is known to be 32. It would surprise me if this has not been published somewhere before, but no-one has so far found an earlier reference.

Next I found a 4x13 = 52 move tour on the 10x10 board. This is also probably the maximum since the best closed tour known takes 54 moves.

Finally the best known closed tour on the 12x12 board uses 86 moves so I tried to form a quaternary tour with 4x21 = 84 moves, but could only manage three examples using 4x19 = 76 moves.

Can anyone fit in the extra two moves in each quarter, or is that impossible?

Marine Court Hastings at Sunset

A colourful and dramatic sky behind the Marine Court building this evening.

Snapped this as I was beginning a brisk stroll along the seafront to Sea Road. Further than I usually go for my constitutional so must have been feeling energetic.
Or maybe just happy to get away from the noise of work being done on the house all the morning.
The landlord is having all the plaster taken off and replaced.

Nonintersecting Knight Path 28 by 28

Another example in birotary symmetry (i.e. invariant to 90 degree rotation)

The number of moves is 4 x 145 = 580 and the board size is 28 x 28 = 784
So the proportion of board used is 580/784 = 0.739 that is about 74%.

Non-Intersecting Knight Paths

I've just updated the page of my website on this subject:

http://www.mayhematics.com/t/2n.htm

Here are two new results I've found.
Non-intersecting knight's paths of 164 moves on the 16 by 16 board
showing 180 degree and 90 degree rotational symmetry.

Symmetry in such paths has not been much studied.
Can these be improved upon? That is do the same with more knight moves?

View from my Window

The photo is a typical view from one of the windows of my flat at present.

There is scaffolding all round the building as the landlord is having the upper walls cleaned and repainted.

Sorry I've not been keeping this diary up to date.
I've been concentrating on trying to finish my book on Knight's Tours.
More on this soon.