Magic Knight Tours 8x10 and 8x14

On my 80th Birthday, and shortly after, I was sent a series of 80-cell magic knight tours on the 8x10 rectangle by Guenter Stertenbrink and later some by Awani Kumar. 

Some of the tours in the Stertenbrink list are closed tours with a complete braid along the bottom border. This can be extended, in four different ways, to cover a further four rows forming a magic 8x14 tour. Here is one example that I sent to these correspondents on 18 February.

These tours answer the final missing cases of Magic Knight Tours on rectangular boards.

07 62 07 52 09 56 01 58

06 51 06 61 04 59 10 55

49 08 63 08 53 12 57 02

64 05 50 05 60 03 54 11

09 48 65 04 27 30 83 86

02 67 46 11 82 85 28 31

47 10 03 66 29 26 87 84

68 01 12 45 88 81 32 25

13 44 99 70 23 34 89 80

00 69 14 43 90 79 24 33

41 16 71 98 35 22 77 92

72 97 42 15 78 91 36 21

17 40 95 74 19 38 93 76

96 73 18 39 94 75 20 37

Underlined: add 100.

The formations on the fifth to eighth ranks down are familiar: a double Beverley quad, and a Snake-head formation as in the 12x12 magic tour by the Rajah of Mysore.

I am not able to do geometrical diagrams as yet on my new computer, but have made some progress with installing HTML-Kit and Cute-FTP which will enable me to update some of the pages on my website. I have already begun to update the History section, where a lot of the links no longer work, but have not yet uploaded this work to the website. .

Next Weekend 13 -15 March I am due to play in the Blackpool Chess Congress, though I've been having doubts whether to go in view of the Coronavirus outbreak.

Break Building

I spent most of the Easter and May holiday week-ends watching the snooker world championship from Sheffield, in the hope that newcomer Judd Trump might win over the old guard John Higgins, and he came quite close.

Rather than abolish the May Day holiday break, surely it is time to fix the date of Easter closer to the Spring equinox, in March, so that the two holidays don't come so close. It's getting like Christmas and the New Year.

Much of the rest of my time was spent revising and checking the section on King tours on the Knight's Tour Notes pages, particularly the enumeration of the tours on boards of two ranks. The totals are given by recursion relations and by formulae involving the Fibonacci numbers. This proved quite troublesome, but maybe that's because my brain isn't working a smoothly as it did even a few years ago.

On the 2x8 board there are 128 closed tours, which is easy to verify, but the number of open tour diagrams works out to the surprisingly large, and surprisingly round number T = 32000. I still wonder whether I've got this right, but the other figures seem to be consistent with this. On the same board there are 584 reentrant tours, and G = 8176 geometrically distinct open tours, of which S = 352 are symmetric. These figures are related by G = (T + 2S)/4.

Magic Rook Tours

The image shows three diagonally magic rook tours that I constructed on 20-21 May 1986 but then seem to have forgotten about. (It was a busy year for me!) The tours have biaxial symmetry and each quarter tour is a bisatin (i.e. uses two cells in each rank and file). If the numbering is shifted by one quarter, so 17 becomes 1, 18 becomes 2 and so on, the rank and file magic property is automatically retained. However for the diagonals to remain magic there must be two numbers less than or equal to 16 in each (i.e. the bisatin has to be diagonal as well). The work was inspired by a less structured diagonally magic rook tour by J. Brugge that appeared in the German chess problem magazine Die Schwalbe in August 1985. Anyone care to enumerate all the diagonal bisatins that could be used?

Tours and Twitters

I've just been trying to remove two blogs from the list of those that I "follow", but there doesn't seem to be any way to do it. The instructions explain how to do it, and they are supposed to lead to a page where you can click a "stop following this blog" button. But they don't!

A day or so ago I uploaded a new page on King and Queen tours to my Knight's Tour Notes pages on the Mayhematics site. This is a page that has been waiting to be uploaded to the old KTN site for several years. There's stil a lot of other material, and updates to existing material, to be added. It's a continuous process.

I've been getting more involved with Twitter. I now have about 15 followers, have sent about 30 tweets and follow about 70 people and organisations. Mostly these are to do with mathematical recreations and humanism. One message led me to this site on mathematics by John D. Cook where the old subject of the misattribution of Beverley's magic tour to Euler came up, and I've contributed to the discussion.

Honeycomb Tours

I have added a new page about tours of Honeycomb Boards to the Knight's Tour Notes pages. This collage shows a selection of results. Some date back to 1974 when I sent an example tour to W. Glinski whose Hexagonal Chess on a 91-cell board was popular. Others are new results. I've included wazir, king and knight tours that show various different types of symmetry.

A Magic Knight Rectangle

Back in 2003 I was able to prove that magic knight's tours were not possible on boards 4n+2 by 4m+2, but a proof for the 4n by 4m+2 case eluded me. I now see that that is because there is no such proof! Thanks to a suggestion by John Beasley, that since there is a simple magic knight+wazir tour on the 2x4 board, a magic knight tour should be possible on a sufficiently large 4n by 4m+2 board, I looked at the subject again and found two 12x14 examples last night, of which this is the first:

141 122 143 038 139 124 127 042 045 030 131 026 047 028
144 037 140 123 128 039 044 125 130 041 046 029 132 025
121 142 035 138 119 126 129 040 043 050 031 134 027 048
036 145 120 063 034 137 014 155 032 135 106 049 024 133
011 064 061 118 013 154 033 136 015 156 051 108 105 158
146 117 012 151 062 059 016 153 110 107 018 157 052 023
065 010 115 060 149 152 111 058 017 020 109 054 159 104
116 147 150 009 114 057 094 075 112 055 160 019 022 053
091 066 007 148 093 074 113 056 095 076 021 162 103 078
006 069 092 073 008 003 082 085 168 161 096 077 100 163
067 090 071 004 083 088 167 002 081 086 165 098 079 102
070 005 068 089 072 001 084 087 166 097 080 101 164 099

It is constructed by the "rolling pin" method that I devised for 12x12 magic tours. It's surprising I hadn't thought of trying this before. It's just a matter of widening the board. The files add to 1014 = 169x6 and the ranks add to 1183 = 169x7. Each file consists of three pairs adding to 127 and three pairs adding to 211. The ranks are made up of pairs of complements adding to 169.

Knight's Tour Notes website gone

I've just noticed that my old Knight's Tour Notes website has vanished into the ether. This is not unexpected, since it was housed on a dial-up site which I have been unable to access for several years. I will now have to relaunch it on my Mayhematics site, or start up a new URL. This may take some time. I will probably want to put it into a revised format and improve some of the diagrams.

Archiving Websites

I've received requests from the British Library to allow my Knight's Tour Notes and Mayhematics websites to be "archived". I think this is a result of requests sent to them by John Beasley with regard to the archiving of the Chess Variants material and the BCVS site.

The address of the archive is: http://www.webarchive.org.uk/ukwa/. I've received emails acknowledging receipt of the online forms that I filled in, but I expect it will be some time before the sites actually appear in the archive. I took the precaution of removing some of the illustrations from the Biographies section of the Mayhematics site, in case they do not meet the copyright requirements.

Five-Directional Knight's Tour Problem

It's over a month since I posted a message on this Diary, but since I seem to be the only person to whom it is of any interest that probably won't have been noticed.

The most interesting occurrence of the past month was that one of my correspondents. Harold Cataquet, sent me details of a new knight's tour discovery by Jonathan Welton from Crowthorne, Berks. This can be presented in the form of a puzzle:

To construct a closed knight's tour of the standard chessboard that uses moves in exactly five of the eight possible knight-move directions.

There is just one solution, apart from rotations and reflections. I was able to confirm this by trying to construct a tour. It required only one page of workings on a sheet of squared paper, and involved 21 diagrams though this is far more than is really necessary. I won't publish the answer here, since Mr Welton will probably want it to appear somewhere in print first.

However here is an open-tour solution that I found, starting and finishing on adjacent cells.

17 32 21 28 47 52 45 50
20 29 18 15 22 49 26 53
33 16 31 48 27 46 51 44
30 19 34 23 14 43 54 25
09 38 13 42 55 24 59 04
12 35 10 07 60 03 64 01
39 08 37 56 41 62 05 58
36 11 40 61 06 57 02 63

This was my first attempt at finding a solution on 25 October.

Archiving the Ether

I've been trying to sort out all the files in the "My Documents" section of the computer. These included several lists of "Old Favourites" from previous computers. Naturally a lot of these coincided with those on my current list, but a surprisingly large number of old links have just vanished into the ether. There were quite a number on the "geocities" site which it seems Yahoo took over and closed down. There are now a number of archiving sites where old web pages are kept, for instance the British Library now has a webarchive, and there is an American internet archive based in San Francisco.

This week I received another letter from Professor Donald Knuth, enclosing four chapters on knight's tours from his forthcoming "Fun and Games" book. They cover non-intersecting knight paths, celtic tours (which include no minimal triangles), tours on three-rank boards, and longer leaper tours. As might be expected his idea of "fun and games" extends to some quite complicated mathematics. I'm naturally pleased to see that quite a number of my own results are quoted.

Mathematical Art: A Chessboard Mosaic

Over the bank holiday weekend I spent some time drawing and colouring the pattern shown here. It is a "Chessboard Mosaic" of the type I described in The Games and Puzzles Journal (Vol.1 No.4, March-April 1988, p.64) but on a larger scale. Varied patterns of this type can be formed by first numbering the cells of a chessboard 1 to 64 in some fashion. This example is derived from the numbering of the first Magic Knight's Tour, discovered by William Beverley in 1848. It is a 64 by 64 matrix; when numbered 1 to 64 along the top and left edges, a mark in the square where the r row meets the s column indicates that a piece can move from cell r to cell s. In this example the dark cells indicate rook moves, the yellow cells bishop moves and the red cells knight moves. The pattern is symmetric about the principal diagonal since these moves are all reversible. The rook move pattern is symmetric about the secondary diagonal, but the knight and bishop patterns deviate slightly from this symmetry, due to the nature of the Beverley numbering. The red railway-line pattern down the main diagonal is the result of using a knight's tour numbering, since r is always connected to r+1 by a knight move. The apparent figure "8"s on the main diagonal derive from several zigzag pattern of knight moves in the tour wheren the first and third and second and fourth cells are in the same rank or file and thus connected by a rook move. I'm wondering whether I should exhibit this at the Art Forum; apparently any member can exhibit one item free; but is it really Art?

Edit: I've replaced the original image by an enhanced version, since it came out too gray. The background I used was in fact a sheet of white card.

Chess and Mathematics

Last Monday I took part in a chess match, playing on behalf of the Hastings club against a team of four from Kent. I was on the third board. The time allowance was quite generous, which suits my slow play, and I managed a draw. Most of the time I was a knight down, but with the advantage of a passed pawn, so it was a matter of trying to get the pawn promoted. There were a lot of interesting tactical situations that arose the game. The opposing team won overall by 2.5 to 1.5.

On Wednesday I received a letter from Professor Donald E. Knuth of Stanford University. We have previously corresponded on knight's tours, but I hadn't had a letter from him for several years. He is putting together a book of his Selected Papers on Fun and Games which will include several chapters on tours, among much else. I had to look up what "potrzebie" was all about. It seems it's a Polish word adopted by MAD Magazine as a running joke back in the 1960s.

The topic Prof Knuth was asking about concerned the results obtained by Robin H. Merson on non-intersecting knight's paths. As a result I have now placed PDF versions of Robin Merson's two main letters to me, dealing with open and closed paths, on the knight's tours page of my mayhematics website. They haven't scanned very clearly; for instance the background graph lines have not come out, but that's the best I can do at present.

Prof Knuth also likes to collect the middle names of everyone whose work he cites, but I was unable to locate what Robin Merson's "H" stood for. He worked for the Royal Aircraft Establishment at Farnborough on the use of satellites for mapping the Earth, among other activities.

More Random Thoughts

I spent most of Thursday journeying to and from Uckfield to attend the East Sussex SACRE meeting on behalf of Hastings Humanists. Since I have a Senior Railcard and a Buspass this was not expensive, just time-consuming. When I returned, and all next day, I had a headache. Whether this was due to bumping about on the bus, or waiting for it in the cold and wet, or some other cause I'm not sure, but at least it seems to have cleared up today. At any rate it stopped me going to the chess club on Friday evening.

While in Uckfield I chanced to go into a Health Food shop and bought a jar of Barley Cup as a possible substitute for drinkng too much Coffee. It doesn't have any distinctive taste that I can detect, just a smooth texture. I did try flavouring it with some Malt Extract, bought at the same shop, but Honey would probably be better. Since I arrived in plenty of time for the meeting I looked around to see what cafes were available and ended up in a Poppins restaurant, which provided a nice lesagna with baked potato and salad.

I'm still working on the knight's tours book. I had hoped to get it finished for my 70 th birthday, but there is still a lot to be done. At present I'm on the chapter dealing with tours on oblong boards. I completely rechecked the tours on the 3x9 board, finding 146 as reported on the KTN website back in year 2000, although there is a minor misprint there, the number of {1,1} tours, with ends a diagonal step apart, is 28 not 29. The next section to check is on the 4xn boards, where I did some work trying to generate recursion relations for the numbers of half-tours, which has never yet been reported on the KTN site.

Newton Day

I spent most of 25 December, Newton Day, appropriately doing some mathematics.

I've put together all my pages of notes on knight's tours to see if I can fit them into a reasonably sized book of around 250 pages. At present it is at around 350 pages, but further selection and editing should get it down to the right size. The pages are formatted to standard A4 size with inch margins, which allows four chessboard diagrams across the page.

The main problem I've had with the whole idea is to get the right balance between History and Theory. I now start with a chapter headed Chessics which introduces the theory, follow this with a chapter on lateral and diagonal movers, i.e. wazir and king tours, and then get to the knight, with more theory and tours by longer leapers and other pieces coming after. I'm not sure I like splitting the Theory section like this, but I do feel I'm making progress.

I had some items on knight's tours back in the first issue of Chessics published in 1976, which is now 33 years ago. I'm rather a slow worker.

Article on Mixed Quaternary Symmetry

I've completed an article "On Mixed Quaternary Symmetry in Knight's Tours" which John Beasley has accepted for the next issue of Variant Chess. It follows up the work by Ernest Bergholt written in the form of memoranda sent to H. J. R. Murray in 1918 but not published until 2001 in The Games and Puzzles Journal #18.

It is only possible to include a summary of my results in the article, and I will be putting diagrams of all the tours onto my mayhematics website. The idea of mixed quaternary symmetry is to produce tours on the 8 by 8 and 12 by 12 boards that show a combination of direct (reflective) and oblique (rotative) quaternary symmetries, since tours fully in oblique quaternary are not possible on these boards, though they are possible on the 6 by 6 and 10 by 10 boards. Tours in direct quaternary symmetry are not possible on any boards, though pseudotours with this symmetry formed of two or more superimposed circuits are.

Where my treatment differs from that of Bergholt and Murray is in separating out the moves, such as the eight corner moves, that show octonary symmetry. Such moves can be regarded as part of either the direct or oblique sets of moves, but it is not always clear to which of these sets they should be assigned.

Tidying Up

Tidying up did cheer me up a bit. A bit more progress on sorting my knight's tour material might help. The problem is that I keep changing my mind about what is the optimal method of arrangement. The way it is going seems to be to place emphasis on the different types of symmetry rather than on the board shapes. Thus instead of having a section on rectangular boards it seems better to classify rectangular tours across several sections according to their symmetry, and whether closed or open.

I tried switching on one of my convection heaters last week but it produced too much smell of burning dust. So today I took it all to pieces and cleaned it internally. I'm sure it could be made to come apart more easily. So many screws to take out, and so many pieces, like the wheels, to remove before the actual bodywork could be opened to give access to the oil-filled radiator! Fortunately it all fitted back together again and it is now working OK without the smell. It was all a bit like playing with the Meccano that I used to enjoy as a boy.

There is a central heating system and radiators, which came with the flat, but I'm not at all certain how that works. I use it to provide hot water for washing, but hesitate to use it for the radiators, since I'm not sure how much gas it is likely to use. I expect a cold spell will be needed to stimulate me to try it out.

DNA folds into a 3D Wazir Tour!

According to a recent article in Science Daily, or at least the illustration accompanying it, DNA folds into a 3D wazir tour, similar to the 2D examples shown in my drawing alongside. These fractal-type diagrams occur at the end of a section of my Knight's Tour Notes website on Wazir Tours. For those not familiar with the terminology, a "wazir" is a chess piece that moves just to the adjacent square, like a single-step rook, or as in a king's non-diagonal move. A tour of course is a journey or path that visits all the squares of the board once only, without backtracking over any squares already visited.

Knight's Tours Again

I've been making some progress in organising my files on Knight's Tours into publishable form. It is taking the form of a series of separate "Studies", each of about 60 pages, covering a particular topic. These are at present: History, Theory of moves, Leaper tours, Knight on 6x6 board, Knight on 8x8 board, Knight on other rectangular boards, Shaped and holey boards, Figured and magic tours. I should be able to make these available in PDF form, if not as a book.

We had two three or four-hour power cuts here over the past two weeks. The first one on Thursday 3rd September 5-8pm was without warning but the other on Friday 11th September 11am-3pm was notified by the electric company EDF. I took the opportunity to give my refrigerator a thorough defrosting and clean. Fortunately I had kept a supply of matches to light the gas, and have a supply of candles somewhere in case of emergency, but not needed on this occasion.

Euler and Me

As noted before I'm in process of tidying up my notes on knight's tours. One of the sections I've not previously published, as far as I recall, concerns the enumeration of all the smallest knight-tourable boards, at least up to 12 cells. The illustration shows all the centrosymmetric tours on boards of 12 cells.

The first two of these diagrams are not merely centrosymmetric, in the sense of being unchanged by a 180 degree rotation, but are also axially symmetric about the two diagonal axes. The first of these was published by the famous mathematician Leonhard Euler in what waa probably the first scientific paper on knight's tours, presented in 1759. The second is one of my own favourite discoveries, it is the 12-cell tour whose containing rectangle is the largest possible, the 6 by 6 square.

Knight's Tour Notes

I've been making progress in sorting out all my notes on knight's tours, with the idea of putting the material into book form. So far it consists of a mere 75 chapters covering 500 pages! And there are still a lot of files I've not incorporated, plus material not yet put into electronic form. The secret to making progress I've found is to stick to one simple standard layout (A4 pages with 1 inch borders and diagrams with cells 1/5 inch width.) In the past I've tried out numerous different forms, with consequent difficulty in editing them all together. Once I've put everything in it will be time to be more selective in deciding the final contents. It may of course make several books, aimed at different readerships.

On Sunday Radio 3 was devoted to a series of programmes on Haydn, and I listened to quite a lot of them, particularly the symphonies and quartets. His music went well with a warm sunny day. I think he had a sunny disposition.