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.

Magic Rectangle 7 by 9

The first rectangle here shows a King Tour in which the ranks sum to all the successive values from 285 to 291 and the files to the successive values from 220 to 228.

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

The middle rank and file are naturally magic, consisting of pairs of complements (adding to 64) plus the middle average number 32, giving the required totals of 288 and 224.

This type of King Tour with consecutive rank and file totals seems to be possible on any odd-sided oblong where the sides have no common factor. I've not seen this result published anywhere before. The moves are completely regular, being diagonal except where they meet a board edge when the king takes a lateral step along the edge and then resumes its diagonal moves as if reflected from the edge.

I used this regular numbering of the cells to construct the following magic rectangle by a series of interchanges of pairs of entries.

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

This began with the interchange of 56 with 59 and 5 with 8 which fixed the top and bottom ranks and the second and eighth files (without altering the total of the middle file). Then the interchange of 16 with 18 and 46 with 48 fixed the second and sixth ranks and the third and seventh files(without altering the total of the middle rank). After that it became a bit more difficult to find suitable changes that did not disrupt the previous ones. The resulting tour uses ten different types of move instead of just two. Can it be done with less disruption?

Here is an earlier example I found based on a knight tour.

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

This uses 16 different types of move, so can hardly be called a "tour" at all. It is also not symmetric (or "associated" as magic square devotees term it) since the pairs of complements 55-9, 41-23, 53-11 and 13-51 lie along the second and sixth ranks instead of being diametral.