Strong contrasts of black and green in the sunlight yesterday.
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.
Wednesday, 2 December 2015
Symmetric Rhombic Tours
In November I looked at the problem of enumerating the symmetric tours of squares and diamonds type. I found there were four distinct ways (1, 2, 3, 4) of arranging the diamonds in a quarter board, and twenty ways these could be arranged together on the board, namely in the ten pairs 11, 22, 33, 44, 12, 13, 14, 23, 24, 34, each either in direct (=) or oblique (~) symmetry. This enumeration is not yet complete but I found unique solutions in the 22= and 44~ cases, both being double halfboard tours.
Thursday, 26 November 2015
A Metasquare Figured Tour
T. R. Dawson composed a figured tour with what he called the 'double triangle numbers' of the form n.(n+1) namely 2, 6, 12, 20, 30, 42, 56 in a figure of eight formation that was published in the London Evening Standard in 1932 (B17 in my Figured Tours booklet). I like to call these numbers 'metasquares'.
Waiting for a council inspector to call yesterday I though to while away the time by composing a figured tour and recalled the above task. There are too many or too few metasquares to arrange in a metasquare. It occurred to me to include the number 0 = 0.1 in place of 64 which would give a set of eight metasquares, all even numbers that might be arranged on a diagonal.
This is the result, composed 25 November 2015. The numbers are in cyclic order.
Waiting for a council inspector to call yesterday I though to while away the time by composing a figured tour and recalled the above task. There are too many or too few metasquares to arrange in a metasquare. It occurred to me to include the number 0 = 0.1 in place of 64 which would give a set of eight metasquares, all even numbers that might be arranged on a diagonal.
This is the result, composed 25 November 2015. The numbers are in cyclic order.
Subscribe to:
Posts (Atom)