It is time I gave an update on my work in putting my Knight's Tour Notes webpages into book form. This is a project I have been working at for years. The current proposal is to put all the material into a series of eight separate monographs, each of around 100 pages.
The subjects of the studies are roughly as follows. The numbering is provisional. 1. History of Tours. 2. Knight-move Geometry, 3.Small-Board Knight Tours. 4.Symmetry in Knight Tours. 5. Simple Linking of Pseudotours. 6.Shaped Board Tours. 7.Leaper Tours. 8. Magic Tours.
The first six titles are almost entirely about knight tours. The last two include tours using pieces with other moves. The magic tours end with magic squares that use up to four different types of move.
If my health holds out I aim to get this completed by the end of the year. I've given deadlines before and not met them, but this time I'm more confident.
Thursday, 13 October 2016
Monday, 13 June 2016
A magic 5 by 7 rectangle tour
This 5 by 7 symmetric magic tour using six move types was composed by me this evening (12 Jun 2016 - our Queen's official 90th birthday) and is the best so far found on this board. The magic constants are 90 and 126 (i.e. 5×18 and 7×18). It uses six different types of move (my previous best was eight). {0,1}{0,2}{1,1}{12}{1,3}{1,4}
This was derived in part from the "arithmic" king tour on this board, but also used the "method of complementary differences" by Charles Planck, as described in Andrews (Magic Squares and Cubes 1917) p.257-262. The example by Planck shown there uses ten move types.
This was derived in part from the "arithmic" king tour on this board, but also used the "method of complementary differences" by Charles Planck, as described in Andrews (Magic Squares and Cubes 1917) p.257-262. The example by Planck shown there uses ten move types.
Monday, 30 May 2016
The Tree
It's along while since I posted a photo of the oak tree opposite my front door.
Strong contrasts of black and green in the sunlight yesterday.
Strong contrasts of black and green in the sunlight yesterday.
Saturday, 5 March 2016
Maximal Bergholtian Symmetry
Bergholtian symmetry is centrosymmetry of the type that passes twice through the centre,
and is possible only on boards with one side odd and the other singly even (e.g. 6×7 or 5×10).
When numbered from the ends of the central cross it has the property that diametrally opposite numbers add to a constant sum. This is in contrast to the more common tours with Eulerian symmetry in which opposite numbers have a constant difference.
On other boards it is possible to construct tours which have partial Bergholtian symmetry, not all of the pairs adding to the constant. An example of this was sent to me by Prof D. E. Knuth, which inspired the following examples, which I think show the maximum amount of Bergholtian symmetry on the 8×8 and 10×10 boards.
The dots which, form two rhombs, mark the ends of the symmetric parts. One rhomb is connected in the opposite way to the other and this is the only asymmetry. If one of the rhombs is reversed this results in a tour with Eulerian symmetry.
The geometrical forms:
The arithmetical forms:
Diametrally opposite numbers add to 62
with the exception of the three pairs in bold.
47 52 45 26 05 24 21 28
44 03 48 53 64 27 06 23
51 46 01 04 25 22 29 20
02 43 54 49 30 63 12 07
55 50 31 62 13 08 19 60
42 33 40 37 58 61 16 11
39 56 35 32 09 14 59 18
34 41 38 57 36 17 10 15
crosslinks c6-e7-f5-d4-f3 and b3-d2-c4-e5-g6
03 14 53 16 55 34 37 40
52 17 02 13 42 39 56 35
31 04 15 54 33 36 41 38
18 51 32 01 12 43 62 57
05 30 19 50 61 64 11 44
24 21 26 29 08 47 58 63
27 06 23 20 49 60 45 10
22 25 28 07 46 09 48 59
crosslinks: d5-f4-h3-g5-e4 and d3-b4-a6-c5-e6
Connecting instead 01-48, 00-47 or 51-98, 50-97
results in a tour with Eulerian symmetry
10×10 example. This linkage will not work on the 8×8 board,
since moves through two corners are prevented.
17 22 89 26 45 38 87 34 31 36
90 25 18 01 88 27 44 37 86 33
21 16 23 46 39 48 03 32 35 30
24 91 00 19 02 43 28 57 04 85
15 20 93 40 47 56 49 84 29 06
92 69 14 99 42 51 58 05 78 83
13 94 41 70 55 96 79 50 07 74
68 63 66 95 98 59 52 75 82 77
65 12 61 54 71 10 97 80 73 08
62 67 64 11 60 53 72 09 76 81
Diametrally opposite numbers add to 98 with the exception of the three pairs in bold.
00 underlined indicates 100.
crosslinks: 01-00-99-98-97 and 47-48-49-50-51
and is possible only on boards with one side odd and the other singly even (e.g. 6×7 or 5×10).
When numbered from the ends of the central cross it has the property that diametrally opposite numbers add to a constant sum. This is in contrast to the more common tours with Eulerian symmetry in which opposite numbers have a constant difference.
On other boards it is possible to construct tours which have partial Bergholtian symmetry, not all of the pairs adding to the constant. An example of this was sent to me by Prof D. E. Knuth, which inspired the following examples, which I think show the maximum amount of Bergholtian symmetry on the 8×8 and 10×10 boards.
The dots which, form two rhombs, mark the ends of the symmetric parts. One rhomb is connected in the opposite way to the other and this is the only asymmetry. If one of the rhombs is reversed this results in a tour with Eulerian symmetry.
The geometrical forms:
The arithmetical forms:
Diametrally opposite numbers add to 62
with the exception of the three pairs in bold.
47 52 45 26 05 24 21 28
44 03 48 53 64 27 06 23
51 46 01 04 25 22 29 20
02 43 54 49 30 63 12 07
55 50 31 62 13 08 19 60
42 33 40 37 58 61 16 11
39 56 35 32 09 14 59 18
34 41 38 57 36 17 10 15
crosslinks c6-e7-f5-d4-f3 and b3-d2-c4-e5-g6
03 14 53 16 55 34 37 40
52 17 02 13 42 39 56 35
31 04 15 54 33 36 41 38
18 51 32 01 12 43 62 57
05 30 19 50 61 64 11 44
24 21 26 29 08 47 58 63
27 06 23 20 49 60 45 10
22 25 28 07 46 09 48 59
crosslinks: d5-f4-h3-g5-e4 and d3-b4-a6-c5-e6
Connecting instead 01-48, 00-47 or 51-98, 50-97
results in a tour with Eulerian symmetry
10×10 example. This linkage will not work on the 8×8 board,
since moves through two corners are prevented.
17 22 89 26 45 38 87 34 31 36
90 25 18 01 88 27 44 37 86 33
21 16 23 46 39 48 03 32 35 30
24 91 00 19 02 43 28 57 04 85
15 20 93 40 47 56 49 84 29 06
92 69 14 99 42 51 58 05 78 83
13 94 41 70 55 96 79 50 07 74
68 63 66 95 98 59 52 75 82 77
65 12 61 54 71 10 97 80 73 08
62 67 64 11 60 53 72 09 76 81
Diametrally opposite numbers add to 98 with the exception of the three pairs in bold.
00 underlined indicates 100.
crosslinks: 01-00-99-98-97 and 47-48-49-50-51
Sunday, 21 February 2016
Symmetric Tours of Squares and Diamonds
I reported starting a count of these tours a while ago. The result found was 274 tours in all of which 82 are of the double halfboard type. The first figure may still be short, and a further check is needed.
The second figure is definitely correct, checked by an independent method. There are three types of double halfboard tour according as the separation between the two links that connect the halves is 1, 3 or 5 cells. The numbers of these types are respectively 26, 28 and 28 adding to 82.
I'm continuing to work on my book on tours. At present in is in two parts each of about 250 pages. The first part being on History and the second on Theory.
The second figure is definitely correct, checked by an independent method. There are three types of double halfboard tour according as the separation between the two links that connect the halves is 1, 3 or 5 cells. The numbers of these types are respectively 26, 28 and 28 adding to 82.
I'm continuing to work on my book on tours. At present in is in two parts each of about 250 pages. The first part being on History and the second on Theory.
Saturday, 6 February 2016
Batten Down the Hatches!
It was disappointing to find that my chess grade has only gone up to 90. I was hoping it might get back to 100 as I had put in considerable effort and achieved some good results. It seems it is easier to slide two yards back down the greasy pole than to climb one yard up.
This afternoon I received a copy of a paper I had requested from a journal in Canada only a few days ago. On the other hand a cheque I sent to a company in Kent two weeks ago disappeared in the post, and I had to report the details to the bank to ensure it is not passed for payment.
When I went for an evening walk along the front yesterday evening for exercise and to see the sunset, I happened to see that a shop selling carpets was open, in the parade beneath the Marina building. This reminded me that I needed a mat to go under my chair to protect the fitted carpet from wear. I bought a nice colourful rug, about 1 by 2 metres, for £25. This has brightened up the room considerably.
It was windy out today and apparently the winds are going to get stronger during my 76th birthday on Monday, so it doesn't look as though I will be going out on any trip as I had hoped. Time to batten down the hatches and try to get some work done on my books.
This afternoon I received a copy of a paper I had requested from a journal in Canada only a few days ago. On the other hand a cheque I sent to a company in Kent two weeks ago disappeared in the post, and I had to report the details to the bank to ensure it is not passed for payment.
When I went for an evening walk along the front yesterday evening for exercise and to see the sunset, I happened to see that a shop selling carpets was open, in the parade beneath the Marina building. This reminded me that I needed a mat to go under my chair to protect the fitted carpet from wear. I bought a nice colourful rug, about 1 by 2 metres, for £25. This has brightened up the room considerably.
It was windy out today and apparently the winds are going to get stronger during my 76th birthday on Monday, so it doesn't look as though I will be going out on any trip as I had hoped. Time to batten down the hatches and try to get some work done on my books.
Wednesday, 6 January 2016
Hastings Chess Congress
The Hastings Chess Congress occupied most of my time from 28 December to 5 January, since I entered all four of the supporting events. Just in the lowest graded sections of course, not the Masters! Scored 3/5 in the Xmas AM event (winning £15 grading prize), but only 1/5 in the Xmas PM event. Then in the Weekend event scored 3/5 again, against stronger players (winning £25 grading prize). Only managed 1.5/4 in the New Year event.
These results continue my improved play over the past six months, with similar 3/5 results at Thanet in August and Bournemouth at the end of October. I'm also doing well in the internal Hastings Chess Club Rush Cup event. I hope all this will restore my ECF grade to near 100 again when the new gradings are announced later this month.
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