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?