Tuesday, 7 February 2017

Knight's Tour Notes Update

I've been asked to give an update on the progress of my work on knight's tours that I have been trying to put into book form. Back in October last year I reported having the material in the form of eight monographs each of about 100 pages.  These soon combined to form four volumes, each of about 200 pages. The latest development is that these have spontaneously rearranged themselves to form three volumes each of around 260 pages.

Volume 1 covers History, consisting mainly of an update of the Chronological Bibliography that I produced in 1990, in 25-year stages, including diagrams of magic tours, separated by essays on methods of construction and ending with a catalogue of quaternary pseudotours.

Volume 2 is on Symmetry and Shape in Knight's Tours and consists of enumerations of tours on small boards, square, oblong or shaped, plus examples on larger boards. It includes a catalogue of all the tours on the 6x6 board, and an account of Mixed Quaternary Symmetry on 8x8 and 12x12 boards.

Volume 3 whose title is undecided is on Theory of Moves and of Magic Tours in general, together with catalogues of tours by Leapers from Wazir to Antelope, and Multi-Movers from King to Wizard. The later sections include Magic Squares using up to seven move types. An appendix lists all the 280 magic knight tours in arithmetical form.

This seems to have reached a stable configuration, so I am hopeful of completing it soon.

Thursday, 19 January 2017

Magic Wizard Problem Continued

Magic Wizard Tour Problem Continued

The original double latin square by E. T. Parker in the 1960 paper, differs from the version shown as the frontispiece in the Coxeter/Ball Mathematical Recreations, in being more geometrically regular:

This has only 15 'Witch' moves that pass over the centres of cells: 06-07 (N), 09-10 (R), 25-26 (N), 32-33 (N), 36-37 (B), 40-41 (N), 48-49 (C), 54-55 (B), 57-58 (C), 62-63 (Z), 69-70 (R), 86-87 (A), 89-90 (N), 96-97 (N), 98-99 (B). Where the letters indicate tthe directions of the moves: A = antelope, B = bishop, C = camel, N = knight, Z = zebra.

Working from this by permuting the ranks and files several times I have arrived at the following square:

This has only 4 Witch moves: 44-45 (N), 51-52 (Z), 94-95 (B), 99-00 (N), shown by the straight lines. One of these is the closure move 99-00. So regarded as an open tour it uses only three Witch moves. Can this be further improved?

What is in effect the same double latin square can be presented in other forms by permutation of the left or right digits (since a latin square is just a pattern independent of the actual symbols used). But whether a better permutation can be chosen to make a Wizard tour more likely is not clear to me.

It seems that every version will have one vertical and one horizontal move lke 09-10 and 69-70 in these examples, or 49-50 and 79-80 in the previous examples.

Thursday, 29 December 2016

Magic Wizard Problem

(a) 10x10 magic square formed of two orthogonal latin squares. Found by Ernest Tilden Parker 1960. (See Frontispiece of Rouse Ball's Mathematical Recreations 12th and later editions)
(b) Permuted ranks and files so diagonal is 00, 01, ..., 08, 09.


(1) Find a permute that has the minimum number of different move types
between successive cells (00-01, 01-02, ..., 98-99).

Is a Magic Wizard tour possible? (No moves crossing an intermediate cell)
The 49-50 and 79-80 moves would have to be wazir steps {0,1}. Moves like {2,2}, {2,4}, {2,6}, {3,3}, {3,6} crossing other cell centres would be avoided. Maybe keep to moves of type {1,n} and {n,n+1}? i.e. just "off" being lateral or diagonal.

In the first try below the following 13 moves fail:16-17, 21-22, 33-34, 35-36, 51-52, 53-54, 64-65, 67-68, 88-89, 91-92, 93-94, 97-98, 99-00.






Is a Magic Queen tour possible? (all moves lateral or diagonal) Probably not.
Is a Magic Witch tour possible? (all moves crossing an intermediate cell)

Saturday, 24 December 2016

Missing Magic Empress Tour

An email from Jaime Gutierrez Salazar 21 Dec 2016 enclosed the following image:


This magic Empress tour (or Magic Two-Knight Tour if you prefer) was apparently included in the collections of magic knight tours published by General Parmentier in the 1890s, but it is missing from the page of such tours on the KTN website. Whether this was in the H. J. R. Murray 1951 manuscript and I missed it out I'm not sure. There are subtle differences in the tours.

There is also another that I reported here in 2015:
http://jeepyjay.blogspot.co.uk/2015/08/magic-empress-tour.html
So I must get round to updating the page.

Note added 19 Jan 2017: I now realise that the above array from Jaime is the reverse numbering of tour O as listed on the Knight's Tour Notes web-page.

Thursday, 22 December 2016

Spectacular Trouble

One support arm of my glasses fell off this morning, so I had to take them along to the opticians (now Boots who have taken over Dolland and Aitchison). They couldn't repair it so I have had to pay £25 to leave the glasses with them. Apparently they need to find another frame of the same type.

I asked about re-using one of my old frames and was told it would cost £95 for reading glasses and £250 for varifocals, but apparently it would be cheaper just to buy a new pair! How does this make any sort of sense? It doesn't, as far as I can see, without my glasses.

This is a policy that will result in all unused frames ending up in landfill.

I'm not keen on varifocals, and always had bifocals before the present pair. However it seems they don't make bifocals any more. This doesn't seem like progress to me. Probably they sacked the expert bifocal makers and replaced them by machines.

When I had an eye test recently the optometrician said he would recommend I go in for an operation to remove a cataract in the left eye.  Apparently this means replacing the material in the lens with something else, which can be done in a way that is minimally invasive.

I'm getting grumpy at the way everything seems to be breaking down at present.

Friday, 9 December 2016

Figured Tour with Legendre Numbers

At the August Bank Holiday Rapid-Play at the Hastings Chess Club I somehow did well enough to win a £5 book voucher. I spent it on "Professor Stewart's Incredible Numbers". On page 46 there is an account of numbers that cannot be expressed as a sum of three squares. A-M. Legendre found the formula (4^k)(8n + 1) for all such numbers. Here is a figured tour that I just constructed this afternoon showing all the Legendre Numbers less than 65 on the diagonals.
They consist of the eight odd numbers that are one less than a multiple of eight (that is 8n - 1) and the two even numbers 28 and 60. They occur in pairs that differ by 32 thus allowing a symmetric arrangement in a symmetric knight tour. The reader may like to try constructing a similar tour with the odd numbers in a different sequence or the even numbers on different cells.

Sunday, 4 December 2016

Atheism is Not a Religion!

On Friday 2 December I went to London to attend two meetings held at the University College London (UCL) Institute of Archaeology, 31-34 Gordon Square, on Unbelief and Atheism.

The meetings were sponsored by the Templeton Foundation, which automatically makes me distrustful of the speakers, since its purpose is promote supposed scientific study of religion.

The first part was an "SSNB Roundtable: Who Cares about Unbelief?" SSNB stands for "Scientific Study of Nonreligious Belief". The most coherent of the speakers to my way of thinking was Dr Judith Everington from University of Warwick. In the Q&A session at the end I tried to point out that even people with Religious beliefs also have a background Nonreligious Worldview. It is the factual knowledge they have of the real world, that they rely upon to conduct their everyday life. It is what they are left with if  they lose their religion. So there is no great mystery about it.

The second part was the "NSRN Annual Lecture: Is Atheism a Religion?" A lecture, with three speakers. The third speaker, Prof Chris French from Goldsmiths College, London, gave the straight answer No! And was quite amusing, but the other two speakers I found illogical.

The first speaker Jonathan Lansman, from Queens University Belfast, thought the subject was absurd, but mainly because he could not find coherent definitions of Religion and Atheism. He maintained that they are like the concept of Weeds which has no scientific meaning in botany, despite being useful in Gardening. With this sort of vagueness I don't see how it is possible for him to think clearly about anything. He thought however that there were some "natural types" like Water that did have a definite meaning. This type of terminology made me wonder if he might be a Creationist, since they think of animal species in this way.

The second speaker Miguel Farias from Coventry University, seemed to favour the proposition, but his arguments I now only vaguely recall, and bore little relevance to my own experience.

I recognised and spoke to Norman Bacrac from Conway Hall and David Pollock from the Rationalist Association and BHA. Rupert Sheldrake was also pointed out to me in the audience but I didn't get into conversation with him.