On Wednesday 18th May I went along to the White Rock Theatre to attend the Mayor-Making ceremony (the present Mayor was re-elected) and presentation of the Freedom of Hastings to former MP Michael Foster. Having to keep on "be upstanding" every time the mayor entered or exited was a bit wearing. It was curious to note that the conservative councillors still observe the ceremonial aspects by wearing reddish-coloured robes, while the labour side were in normal dress. The "processing" in and out could well also be done away with. The best part of the show were the performances by children from local schools, though one of the songs was mildly religious. The dance by the older girls was apparently to celebrate the suffragettes.
On Saturday 21st May I accepted an invitation to attend a meeting of the Freemasons Lodge formed by Old Olavians from my former school. This was held at the Freemasons Hall in Great Queen Sreet in London. This is a very large building, in art deco style, built as a memorial after the first world war. We were admitted to the so-called "Indian Temple" which is apparently only one of the smaller temples in the building, and treated to a talk by one of the Masters. The temple was a large rectangular room with domed roof. The masons in their ceremonial aprons sat on thrones, three at each end, and one in the middle of each side, with lighted candles in front of them.
According to the talk the masons are now purely a philanthropic organisation, raising money for work done by other charities like the red cross, or to supply equipment to local hospitals, although they also provide relief for old masons fallen on hard times. However they require applicants to declare belief in a "supreme being" to become a member, and swear their oaths on a bible, which rules me out. Though doesn't this now contravene discrimination laws, if they are not an explicitly religious organisation?
I'm afraid I found their costumes and their "processing" only rather comical. They made me think of Tweedledum and Tweedledee, especially as most of the members were of portly proportions. This may be a consequence of their dinners, there was one following the meeting which declined to attend, since it cost £35, a price that represents a whole week of dinners for me.
On Monday I'm due to attend the AGM of the Rationalist Association. I wonder if they go in for robes and ceremonials? I hope not.
Sunday, 22 May 2011
Friday, 6 May 2011
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.
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.
Monday, 25 April 2011
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?
Sunday, 10 April 2011
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.
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.
Thursday, 7 April 2011
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.
Saturday, 26 March 2011
A Disorienting Day
There was a concert of light music, of which I am a fan, due to be held at St George's Church in Beckenham this evening. So I thought I would look into the possibility of travelling there by public transport. According to the National Rail Enquiries website, which I've found to be reliable previously there was a train at 4:55 that would get me to Beckenham Junction by 17:15 with one change at East Croydon.
However, when I tried to buy such a ticket at Hastings station I was told such a trip was "impossible"! Also that the National Rail Enquiries were now run by a bus company who didn't understand the railways. Apparently the link from East Croydon to Beckenham Junction is via a Tram line. This appears to be correct: it is on something called the London Tramlink, which I've never heard of before, though it has been around for ten years.
So, lacking a clear route, I aborted the trip and decided to listen to a concert on Classic FM. This was a Prokofiev concert introduced by Howard Goodall. The first item announced was his Classical Symphony, number 1. However the music played bore little resemblance to previous performances of that work I had heard. I suspect it was some other of his symphonies, in far more modern style. The second item was announced as the music from Romeo and Juliet, including the "Dance of the Knights". However, unless I fell asleep, this was not the music played.
Tomorrow apparently it's time to put the clocks back, or is it forward? I'm not sure if I'm coming or going.
However, when I tried to buy such a ticket at Hastings station I was told such a trip was "impossible"! Also that the National Rail Enquiries were now run by a bus company who didn't understand the railways. Apparently the link from East Croydon to Beckenham Junction is via a Tram line. This appears to be correct: it is on something called the London Tramlink, which I've never heard of before, though it has been around for ten years.
So, lacking a clear route, I aborted the trip and decided to listen to a concert on Classic FM. This was a Prokofiev concert introduced by Howard Goodall. The first item announced was his Classical Symphony, number 1. However the music played bore little resemblance to previous performances of that work I had heard. I suspect it was some other of his symphonies, in far more modern style. The second item was announced as the music from Romeo and Juliet, including the "Dance of the Knights". However, unless I fell asleep, this was not the music played.
Tomorrow apparently it's time to put the clocks back, or is it forward? I'm not sure if I'm coming or going.
Tuesday, 8 March 2011
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.
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.
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