Monday, 24 April 2017

More 6x6 Dawsonian Tours

Here is another knight tour on the 6 by 6 board with the square numbers in a closed knight circuit.

These four use the same circuit but in a different position on the board, and numbered from a different point.
They differ only in the route taken from 25 to 36.

Saturday, 15 April 2017

Dawsonian 6x6 Tour

Back in Chessics #22 (1985) I published a solution to the 1881 Carpenter problem of making a tour with the square numbers in sequence along the first rank, and found the solution was unique.

However it has only recently occurred to me to try the similar Dawson problem of a tour with the square numbers in a closed knight circuit. Here is a solution I found late last night, or early this morning, after examining nearly 70 arrangements of the numbered circuits.

It has been known for a long time (apparently in Chess Amateur though I've not located the exact reference) that there are 25 such geometrically distinct circuits. One is too large to fit on the 6-board. The others can be placed on the board in various positions (81 at my last count), and the numbers can be placed on them in up to 12 ways, though many cases are easily eliminated. For instance in the above diagram the node at f5 must be an end-point (1 or 36) since there is only one other move available there.

Addendum: Found another solution mid-day today. This one uses a symmetric circuit.

The occurrence of two-move straight lines in both solutions is surprising.

Saturday, 25 March 2017

Tom Marlow

I regret that I have just learnt that my long-term correspondent Thomas W. Marlow died in September 2011. My first contact with him was around 1980 when he sent me new results on the "Rook around the Rocks" problem that I published in the Problemist November (1979). Our combined results appeared in Chessics #12 (1981). Another of his interests was in Grid Dissection problems (polyominoes) Chessics #23 (1985) p.78-9.

In 1985 as reported in Chessics #24 p.92 he made a computer check on the de Hijo (1882) enumeration of 16-move knight paths in direct and oblique quaternary symmetry, which I recently (Sept 2016) published in diagram form:

He also did significant work on Fiveleaper tours including 52 magic tours, which have a page to themselves in the Knight's Tour Notes:

Possibly his most notable work was his enumeration of all the "regular" magic knight tours on the chessboard (that is those formed of Square, Diamond and Beverley quartes) which was reported in the Problemist January 1988 (p.379) with diagrams of five new magic tours, the first discovered since the work of H. J. R.Murray published in Fairy Chess Review in 1939.

Although we corresponded over several decades we never met in person. I had the impression that he was younger than me, mainly in view of his expertise with computers, but perhaps I was mistaken. I will update this page as further details come to light.

UPDATE 26 April 2017:

I should also have mentioned his discovery in 2003 of the unique 10 by 10 semimagic knight tour with quaternary symmetry that I published in Games and Puzzles Journal (online):

From the new information below it seems that TWM was a good deal older than I thought, and also a rather adventurous individual!

I have heard from Mrs Dorothy E. Marlow as follows:


"Tom  ... was a very quiet, reserved, private person, and would not relish any publicity, but I will provide you with a few details.

He was Thomas William Marlow, born 24 January 1927. Apart from his National Service, which he spent mainly in Burma, his working life was with the LCC/GLC, taking early retirement when the GLC was abolished. He then became an advisor for the Citizens Advice Bureau, which for more than 20 years he found stimulating and rewarding.

His personal hobbies/interests were anything and everything concerning Mathematics. I always said that he was a human computer! He also liked travelling - as independently as possible as he disliked organised groups, although sometimes they were necessary. Most of our holidays were of the walking / trekking / backpacking type in such places as the Himalayas, the Andes, New Zealand, etc. In the UK it was anywhere among the mountains - Snowdonia, the Lake District and the Munros in Scotland. As the years started to take their toll, it became long-distance footpaths & coastal paths. I have some wonderful memories.

Another hobby, which I nearly forgot to mention, was gliding. He had been a glider pilot for about 60 years and kept his own glider at the London Gliding Club at Dunstable."


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:
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.