I have been looking through the issues of Le Siecle in which H. J. R.Murray reported that A Feisthamel published the magic knight tours as they were discovered. However I find that most of the tours published there are of the two or four knight variety that I prefer to call Emperor tours, since they use rook moves to link the ends of the knight sections. The tours appear mostly in the Wednesday section of Feisthamel's puzzle column. However they are not presented as tour diagrams, instead they are all cryptotours whose solutions, besides delineating the tours, also serve as a word puzzle in themselves such as an anagram, though these puzzles all seem pretty feeble to me. The issues can be accessed via Gallica here:
http://gallica.bnf.fr/ark:/12148/cb32868136g/date
In the issue for 3 Feb 1883 Feisthamel also lists a number of other French newspapers or magazines in which toors were published. These include the titles National, Globe, Soir, Telegraphe, Gil Blas, Gaulois, Clairon, Etoile francaise, Paris Journal and two others where single examples appeared. So far I have only located one of these online:Le Gaulois, where the puzzles appear on Mondays and are in two separate series (Jeux D'Esprits and Passe-Temps Hebdomadaire) which alternate fortnightly.
http://gallica.bnf.fr/ark:/12148/cb32779904b/date.langFR
This includes King tours by R. Dubief and by a Monsieur Galtier, the latter being magic tours.
Much scope for further historical research here!
Thursday, 22 June 2017
Wednesday, 10 May 2017
Final Dawsonian 6x6 Tours
This set of four Dawsonian tours on the 6x6 board complete my search for solutions.
As with the previous batch of four they differ only in the path taken by the 25-36 linkage.
Since my search has been made "by hand" not by computer programming,
it is quite possible I may have missed one or more solutions,
I also recorded a number of "near-misses" where one of the knight moves
is replaced by some other, such as a wazir, rook or zebra move,
bit these don't seem to be of any particular interest.
As with the previous batch of four they differ only in the path taken by the 25-36 linkage.
Since my search has been made "by hand" not by computer programming,
it is quite possible I may have missed one or more solutions,
I also recorded a number of "near-misses" where one of the knight moves
is replaced by some other, such as a wazir, rook or zebra move,
bit these don't seem to be of any particular interest.
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.
These four use the same circuit but in a different position on the board, and numbered from a different point.
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.
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: http://www.mayhematics.com/t/qp.htm
He also did significant work on Fiveleaper tours including 52 magic tours, which have a page to themselves in the Knight's Tour Notes: http://www.mayhematics.com/t/pf.htm
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):
http://www.mayhematics.com/j/gpj25.htm
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."
====================
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: http://www.mayhematics.com/t/qp.htm
He also did significant work on Fiveleaper tours including 52 magic tours, which have a page to themselves in the Knight's Tour Notes: http://www.mayhematics.com/t/pf.htm
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):
http://www.mayhematics.com/j/gpj25.htm
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:
====================
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.
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.
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.
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