tag:blogger.com,1999:blog-91044690966921922832024-03-07T01:58:18.108+00:00Jeepyjay DiaryGeorge Jellisshttp://www.blogger.com/profile/14912766967103087963noreply@blogger.comBlogger329125tag:blogger.com,1999:blog-9104469096692192283.post-20188569942829297032024-02-28T18:26:00.004+00:002024-02-28T18:26:26.120+00:008x12 Symmetric Magic Knight Tour<p> <span style="font-size: 16px;">Guenter Stertenbrink reports that after a computer search </span></p><p><span style="font-size: 16px;">there are no symmetric magic knight tours on the 8x10 rectangle.</span></p><p style="font-size: 16px; margin-bottom: 0px; margin-top: 0px;"> <a href="https://www.chessproblem.net/viewtopic.php?t=1150">8x10 magic knight tours - ChessProblem.net</a></p><p style="font-size: 16px; margin-bottom: 0px; margin-top: 0px;"><br /></p><p style="font-size: 16px; margin-bottom: 0px; margin-top: 0px;">He provides this example of a rotationally symmetric magic tour 8x12.</p><p style="font-size: 16px; margin-bottom: 0px; margin-top: 0px;"><br /></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEggJUYy0w72uJ_FDTNUjVYkCjl7i8ceZhyphenhyphenU3wAGEoBDhw_abpUEaE8WWa2OV2D-ypNmJjPAl_Cv1y8AxdT95_wQKxd8naxT5Pebq2_I84LqJ5-wJgAaLMCpjO94SPDp2Uf38MjD4WBRekrHgArdPtwLqSbpSURQYZil1_7dTLHGpB04B5n6e8oiZPiAxN4k/s416/8x12%20symm%20magic.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="144" data-original-width="416" height="111" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEggJUYy0w72uJ_FDTNUjVYkCjl7i8ceZhyphenhyphenU3wAGEoBDhw_abpUEaE8WWa2OV2D-ypNmJjPAl_Cv1y8AxdT95_wQKxd8naxT5Pebq2_I84LqJ5-wJgAaLMCpjO94SPDp2Uf38MjD4WBRekrHgArdPtwLqSbpSURQYZil1_7dTLHGpB04B5n6e8oiZPiAxN4k/s320/8x12%20symm%20magic.jpg" width="320" /></a></div><br /><p style="font-size: 16px; margin-bottom: 0px; margin-top: 0px;"><br /></p><p style="font-size: 16px; margin-bottom: 0px; margin-top: 0px;">First half numbered 1 to 48 (instead of 0 to 47).</p><p style="font-size: 16px; margin-bottom: 0px; margin-top: 0px;"><br /></p><p style="font-size: 16px; margin-bottom: 0px; margin-top: 0px;">Rank sum 582 (6x97). File sum 388 (4x97).</p><p style="font-size: 16px; margin-bottom: 0px; margin-top: 0px;"><br /></p>George Jellisshttp://www.blogger.com/profile/14912766967103087963noreply@blogger.com0tag:blogger.com,1999:blog-9104469096692192283.post-80947143513139397982024-01-02T11:25:00.000+00:002024-01-02T11:25:10.555+00:00New Years Moan<p>The Christmas to New Year period is my least favourite time of the year. </p><p>Over Christmas all the radio stations I normally listen to are taken over by Christian propaganda in the form of carols, which being a secular humanist unbeliever I find annoying. I tend to go into hibernation and put on my own CD music collection. I'm also not keen on the consumerist advertising, since I like a simple life of plain food without alcohol. </p><p>The other main annoyance is that everything closes down every few days, not excluding strikes by doctors and train drivers. This morning I was thinking of changing my method of payment of my energy supply from pay-as-you-go to direct debit, since finding a functioning Pay Point anywhere near is increasingly difficult. However I couldn't find any way of doing this online. I suppose I will have to phone tomorrow, if it is possible to get through. </p><p>I also checked my bank account online, but found I was not allowed access until they had phoned me and I had given them a four-digit code (this was after signing in with my ID, Password and four letters of my Special information). I tried opening a saver account that was on offer, but got a message that something went wrong and I should try visiting the bank! I clicked on a message that was indicated, but it resulted in signing me out. So I couldn't make the transfer of money from my current account to my ISA that I intended. So it was all a waste of time and effort. </p><p>It seems that BT is threatening to close down land lines. Since I don't have a mobile phone this would leave me incommunicado. I rely on it for communication with the medical profession. Now it seems it would also cut me off from my bank account.</p><p>Happy New Year?</p><p><br /></p>George Jellisshttp://www.blogger.com/profile/14912766967103087963noreply@blogger.com1tag:blogger.com,1999:blog-9104469096692192283.post-40111909451393866972023-07-07T16:49:00.000+01:002023-07-07T16:49:06.045+01:00Multi-storey car park<p> It seems the new building by the bus station is to be a multi-storey car park! I would have thought that another car park is the last thing Crewe needs. Or does it mean all the other car parks on the other side of Delamere street will be closed - and maybe built on or landscaped into gardens? I doubt it. </p>George Jellisshttp://www.blogger.com/profile/14912766967103087963noreply@blogger.com0tag:blogger.com,1999:blog-9104469096692192283.post-21271273524871059402023-05-19T20:11:00.000+01:002023-05-19T20:11:14.790+01:00More Road Works in Crewe<p> Here are some more recent photos of the works in Barnard Street, and at the end of Delamere Street where it meets Flag Lane and in the town centre.</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhNV6-FlUu3FqeCDJ6ZJUTSPuIOAIXAjePe3ENZoVP6pOGfqx3PB9kVk8t458-AlsNwdii_R7ePtmcBYDUTIMQhhTO2DndM2I5gs6b02bqXArwTLEilFXBfCQGNhysxEhzM2NvrgQ0z03qVLaqM0f6Td0odWJS184saaVvFGjPPJlAxhOK4XI0Y5GXl-Q/s2048/P1010681.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1536" data-original-width="2048" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhNV6-FlUu3FqeCDJ6ZJUTSPuIOAIXAjePe3ENZoVP6pOGfqx3PB9kVk8t458-AlsNwdii_R7ePtmcBYDUTIMQhhTO2DndM2I5gs6b02bqXArwTLEilFXBfCQGNhysxEhzM2NvrgQ0z03qVLaqM0f6Td0odWJS184saaVvFGjPPJlAxhOK4XI0Y5GXl-Q/s320/P1010681.JPG" width="320" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgKKJCuhJxaQ-BtIZcT1pBy_BotbzBFxgbQeJlwGY_MlxIVu1nRhFumh8GeyB9sd96n6bX9hVH0EBFvKwVHkEno4vQgNsLKjJisHKT_1Ivpm2mZHe4CTULuYYnulg1LfOGxLcX_S1ekJTg01zS84lLKxYKXbB-9LqsDH6FWuMzkhjIpuUSNA_knA9dISA/s2048/P1010684.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1536" data-original-width="2048" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgKKJCuhJxaQ-BtIZcT1pBy_BotbzBFxgbQeJlwGY_MlxIVu1nRhFumh8GeyB9sd96n6bX9hVH0EBFvKwVHkEno4vQgNsLKjJisHKT_1Ivpm2mZHe4CTULuYYnulg1LfOGxLcX_S1ekJTg01zS84lLKxYKXbB-9LqsDH6FWuMzkhjIpuUSNA_knA9dISA/s320/P1010684.JPG" width="320" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhQE5oXaEzguVzM9CEJIPi571ogDSAbPz77DxEKVfL4-MPdf65SVK0Vfo5XuFjENf2P_MtGdC_VAdhvMDedT0K8aKxsj_LZ-uaMnt5KhGLEpHDTnF9tVZ1Johgwsvgmg_Z2CaygSFw-5js1QxWDF23ziMNOu8Ewrz8w3NVJuu3eU-gshS1gdCnVLqYqUg/s2048/P1010685.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1536" data-original-width="2048" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhQE5oXaEzguVzM9CEJIPi571ogDSAbPz77DxEKVfL4-MPdf65SVK0Vfo5XuFjENf2P_MtGdC_VAdhvMDedT0K8aKxsj_LZ-uaMnt5KhGLEpHDTnF9tVZ1Johgwsvgmg_Z2CaygSFw-5js1QxWDF23ziMNOu8Ewrz8w3NVJuu3eU-gshS1gdCnVLqYqUg/s320/P1010685.JPG" width="320" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiETSGYwx9aJVS7rxaKCn8wmDj--sPP9BJchGIN3YOZA4ypPZf40QwPlD9B2AegufW0CSfxNvlbGkMKBBmu3G8_nYItxr0wgknptimivVbHfl8vPU51zVY8MkleljXJ0nHWoG6aeY9CVs61aFKdeIo6clXItr4ynKKI-j749uDvvoFVH4XcrVMI5IOdKg/s2048/P1010686.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1536" data-original-width="2048" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiETSGYwx9aJVS7rxaKCn8wmDj--sPP9BJchGIN3YOZA4ypPZf40QwPlD9B2AegufW0CSfxNvlbGkMKBBmu3G8_nYItxr0wgknptimivVbHfl8vPU51zVY8MkleljXJ0nHWoG6aeY9CVs61aFKdeIo6clXItr4ynKKI-j749uDvvoFVH4XcrVMI5IOdKg/s320/P1010686.JPG" width="320" /></a></div><br /><p>I'm not sure what the building is by the crane in the town centre but hope it is just a temporary structure and not an example of the new architecture to be built there!</p>George Jellisshttp://www.blogger.com/profile/14912766967103087963noreply@blogger.com0tag:blogger.com,1999:blog-9104469096692192283.post-45494440115587238562023-04-05T19:54:00.001+01:002023-04-05T19:54:13.980+01:00Solutions to Tour Puzzles<p>I've published a set of tour puzzles on the Knight and Leaper Tours subforum. Moves at the top of the 6x7 board (Diagonal, Lateral or Right angles) are given and the puzzle in each case is to complete a symmetric closed tour. </p><p><a href="https://www.chessproblem.net/viewforum.php?f=49">Knight and Leaper Tours - ChessProblem.net</a></p><p>To avoid having the solutions on the same page I thought it might be better to publish them here and provide a link to this page, to allow time for people to solve the puzzles.</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhSji6A4t9i4Tb--se3W5pgtpWvWpHhDShqKaDR0_gwZPyPe6V94TVNAxbBYq6oawWTR-JA-h7FhlJpFNqdrT6RyGonR1r-Z8hEyzMVrHnkS9MppmYJ132RpAx-xLuf3hpZsKubiCS9ABsGLY4zhDGcaa_Hv-wGGMFEwI5F5g3bsDj-s1Jcvf_VyOEeRw/s448/Toursolutions.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="408" data-original-width="448" height="291" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhSji6A4t9i4Tb--se3W5pgtpWvWpHhDShqKaDR0_gwZPyPe6V94TVNAxbBYq6oawWTR-JA-h7FhlJpFNqdrT6RyGonR1r-Z8hEyzMVrHnkS9MppmYJ132RpAx-xLuf3hpZsKubiCS9ABsGLY4zhDGcaa_Hv-wGGMFEwI5F5g3bsDj-s1Jcvf_VyOEeRw/s320/Toursolutions.jpg" width="320" /></a></div><div><br /></div>The solutions can all be reasoned out step by step. For instance the paths through the corner cells are fixed, and in most cases this is true of other cells. Another factor to watch for is where putting in a pair of opposite moves (to satisfy the symmetry condition) results in a short circuit (i.e. not a complete tour).<div><br /></div>George Jellisshttp://www.blogger.com/profile/14912766967103087963noreply@blogger.com0tag:blogger.com,1999:blog-9104469096692192283.post-39660836855926802452023-04-04T10:48:00.000+01:002023-04-04T10:48:07.880+01:00Chess at Great Yarmouth<p> On Friday 31 March I took to the trains to Great Yarmouth, travelling via Nottingham and Norwich, to take part in the chess congress. This was probably a bit too far to travel in one day and left me quite tired. The bed and breakfast accommodation I had booked was comfortable however.</p><p>The weather on Saturday was wet and windy and cold, and getting to the Conference Centre venue, at the race course, was a long walk. Nevertheless I began with a good win against a 1500 rated opponent. This was followed by a draw. I was unable to find a place nearby for a decent meal and had to settle for a take-away coffee and pasty. </p><p>By the time of the third round that evening I was quite tired and found I had to play a strong Junior. Unfortunately she kept coughing at every move, which I complained about to the organisers, and questioned the state of her health, but they said she only had a cold, not Covid. It was a long game and in the end I ran out of time.</p><p>On Sunday morning the weather was much better, sunny but still with a strong wind. I lost my morning game by playing all the wrong opening moves. Fortunately this gave me time to catch a No.8 bus into the town centre where I was able to get some cod and chips at the Fish O'Licious by the bus station. Surprisingly for a seaside town there didn't seem to be many fish and chip shops, but maybe it was out of season. Most places along the front were closed up.</p><p>I caught the bus back to the venue, arriving in good time, and in a good mood, to win the final game, also against a 1500 rated player, though I think he was very tired at the end. </p><p>So my score was 2.5 out of 5, which I consider my standard result. The chess results website rates my performance at the 1460 level, well above my current rating of 1311. </p><p>The journey home was uneventful but I needed a good sleep that night. I will have to look at the chess calendar to see whether there is another event I could enter before the Crewe congress at the end of June. Maybe I should try some rapid play. </p><p><br /></p>George Jellisshttp://www.blogger.com/profile/14912766967103087963noreply@blogger.com0tag:blogger.com,1999:blog-9104469096692192283.post-59540251118692121482023-03-05T21:22:00.001+00:002023-03-05T21:22:35.961+00:00Crewe Town Centre<p> Here is a photo I took of the latest works in the town centre where the bus station used to be. </p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjjNnOxT5I09uAm_sggOpwWxkGSFmsVVDZzhdjrsOoBSucYs5r7GUx9ycgt44VBgJaz8egJWFjheKvI5ysEzW8DHQu2-KCjYy2dSfghT4AXQwq6LQdivpCYrYZC541LW7samXwJ1nJLGyBSE-1zXQC6QEyPHQnLHsi1pROQUkWgqWVurkvTuY0nl2wFZQ/s2048/P1010677.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1536" data-original-width="2048" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjjNnOxT5I09uAm_sggOpwWxkGSFmsVVDZzhdjrsOoBSucYs5r7GUx9ycgt44VBgJaz8egJWFjheKvI5ysEzW8DHQu2-KCjYy2dSfghT4AXQwq6LQdivpCYrYZC541LW7samXwJ1nJLGyBSE-1zXQC6QEyPHQnLHsi1pROQUkWgqWVurkvTuY0nl2wFZQ/s320/P1010677.JPG" width="320" /></a></div><br /><p>It is beginning to be possible to see across the whole area, were it not for the piles of rubble in between.</p><p><br /></p>George Jellisshttp://www.blogger.com/profile/14912766967103087963noreply@blogger.com0tag:blogger.com,1999:blog-9104469096692192283.post-2631357334750577042023-02-24T20:04:00.000+00:002023-02-24T20:04:57.424+00:00Subforum on Tours<p> Alexander George who runs the ChessProblem.com forum has kindly allowed me to set up a subforum in the Mathematical Problems section for discussion of Knight and Leaper Tours. </p><p>Guenter Stertenbrink is also involved as a Moderator, under the user name gsgs. </p><p>Link <a href="https://www.chessproblem.net/viewforum.php?f=49">Knight and Leaper Tours - ChessProblem.net</a></p><p>There is already a column of topics, many relating to magic tours. Some of the posts from GS are rather mathematical and technical, concerning computer programming. </p><p>I hope this will continue to serve as a forum for knight and leaper tours and related subjects when I have gone. It will be in place of my current email exchanges with a wide range of correspondents.</p><p><br /></p>George Jellisshttp://www.blogger.com/profile/14912766967103087963noreply@blogger.com0tag:blogger.com,1999:blog-9104469096692192283.post-76558508773709693342023-02-14T19:39:00.000+00:002023-02-14T19:39:39.245+00:00Blackpool Chess<p>I took the train to Blackpool for the chess congress on Friday 10 February. The train I was due to catch was however cancelled due to a fatality on the line. I was directed onto the Glasgow train, which was diverted via Manchester to stop at Preston, to get a connection to Blackpool. </p><p>The Hotel where I stayed, really a B&B, was halfway between the station and the chess venue. I was the only client there, and was put in the room decorated with an Egypt theme, other rooms still being prepared for the coming season. The lack of hot water in the tap was a problem I've encountered before in hotels, but the full English breakfast was OK, and the landlady was a genial conversationalist.</p><p>The chess congress (or 'Conference' as it is officially known) was at the Imperial Hotel. Some photos I took of the playing areas have come out very blurred, so I can only show an external view.</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjdhu411Vkx5GJ75Fy6hoHn-gNMtHw0IFDa85TURWrtMw-yOp7Yx7ywQe7mPMWX_CNOMEfqdKDvpEBOqBwsgYUxxUHlkk-eR8sm_eJHH62xH-D33H2dJbnOw2fgb7Jnlay2Uf712kdPytCvymxq82ftzs5iqCXVtYAn8GgzUlamEv4uZJ7FRb5v7_IAxA/s2048/P1010654.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1536" data-original-width="2048" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjdhu411Vkx5GJ75Fy6hoHn-gNMtHw0IFDa85TURWrtMw-yOp7Yx7ywQe7mPMWX_CNOMEfqdKDvpEBOqBwsgYUxxUHlkk-eR8sm_eJHH62xH-D33H2dJbnOw2fgb7Jnlay2Uf712kdPytCvymxq82ftzs5iqCXVtYAn8GgzUlamEv4uZJ7FRb5v7_IAxA/s320/P1010654.JPG" width="320" /></a></div><br /><p>My final score was only 1.5 out of 5, consisting of a win, a draw, and three losses. Of these the losses were the longest and most interesting games. Two of the losses were against young women and went to over 70 moves with only kings and pawns on the board at the end. The win on Sunday morning was a surprisingly quick game of only 15 moves against a stronger graded man. </p><p>This is the first time I have travelled any distance for three years, and may have reinforced my confidence to travel more later this year, preferably when it is a bit warmer. Walking around Blackpool in the cold wind, when most of the attractions are closed was not particularly pleasant, but I did get some good fish and chips. </p><p><br /></p>George Jellisshttp://www.blogger.com/profile/14912766967103087963noreply@blogger.com0tag:blogger.com,1999:blog-9104469096692192283.post-15280919362999134992023-02-07T20:23:00.000+00:002023-02-07T20:23:42.775+00:00Beautiful Crewe<p> On Thursday 2 February I took a walk through Valley Park from Flag Lane to Walthall Street, and noticed that the Brook was full of discarded supermarket trolleys. I did the same walk on Sunday 5 February and they were still there. I took the following photos, and others, which I posted on Twitter under the title of #BeautifulCrewe. </p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjaUrYgzgEFSpH4QWkj_XSu41NQOJmq2KLQ0st8wkCC7d1QmfKo6lyrym-WZnINWtt3qqwDbEf5vBRBYM29ETD5Mrcshn9CcsEVRwAawLglGGuvK9_XzW4yLZEkICb9sMud3bCHBbgWrORPucWFRGCYThxzxF79s1OaN1yqg7nzwr67aocAjF5gxLCGgQ/s2048/P1010640.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1536" data-original-width="2048" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjaUrYgzgEFSpH4QWkj_XSu41NQOJmq2KLQ0st8wkCC7d1QmfKo6lyrym-WZnINWtt3qqwDbEf5vBRBYM29ETD5Mrcshn9CcsEVRwAawLglGGuvK9_XzW4yLZEkICb9sMud3bCHBbgWrORPucWFRGCYThxzxF79s1OaN1yqg7nzwr67aocAjF5gxLCGgQ/s320/P1010640.JPG" width="320" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj4INW_AomJyw48-UXXr7qstfp67vZbKj3NMtW3LcFy1NoghHYaESkOx6bnk2Nhi1kfOPzSiUqmq8HD9Zm0cNWMeMt9ORKKh9jYC3YLA7W1oJYkIeH6T5-b26Kgo9KdUJFPcoA3xP3HFT8hry-5gh5OL4KuVodYAcOZc1mro2czX8HVqCXV5g7-DInsBw/s2048/P1010642.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1536" data-original-width="2048" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj4INW_AomJyw48-UXXr7qstfp67vZbKj3NMtW3LcFy1NoghHYaESkOx6bnk2Nhi1kfOPzSiUqmq8HD9Zm0cNWMeMt9ORKKh9jYC3YLA7W1oJYkIeH6T5-b26Kgo9KdUJFPcoA3xP3HFT8hry-5gh5OL4KuVodYAcOZc1mro2czX8HVqCXV5g7-DInsBw/s320/P1010642.JPG" width="320" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjOnbvostrVso-An1wePvLxqLsojKFBaOsc1fnNrGAvEfEqCZwV4L3mz0LEvZSL0vhreMntYs6uLzbuk7SmGqOp7-zf0E5JVL57OOtDnANbwy0o69mpjQRcJ25dJFzCgcbwt_jRMHfHirKAdLM5o-Glmxe0hzPXNUEIdewMYdBUi7e1_jujxBasX_LxJg/s2048/P1010644.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1536" data-original-width="2048" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjOnbvostrVso-An1wePvLxqLsojKFBaOsc1fnNrGAvEfEqCZwV4L3mz0LEvZSL0vhreMntYs6uLzbuk7SmGqOp7-zf0E5JVL57OOtDnANbwy0o69mpjQRcJ25dJFzCgcbwt_jRMHfHirKAdLM5o-Glmxe0hzPXNUEIdewMYdBUi7e1_jujxBasX_LxJg/s320/P1010644.JPG" width="320" /></a></div><br /><p>The old ten-pin bowling hall which has been closed since I moved here in 2019 has been reopened, alongside the Odeon, as shown in this further photo.</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjxb9D_J5-d8H5QIbebcDCfIGGPPpGySYVNH7yVQzFRUdevX0he66EQ8fLWq_saqNY3QFcaAcEyqhZ2oAfMcKsQ0MAIGqHfvwCJL8NtuTqXulmvslpSzDQ6kcU0M0rLsaPd3wkXHXQeLErIKM1OPa-mlbIM0cBWNl48_THGgHxwZ79bXljXmsC_vvzXSg/s2048/P1010652.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1536" data-original-width="2048" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjxb9D_J5-d8H5QIbebcDCfIGGPPpGySYVNH7yVQzFRUdevX0he66EQ8fLWq_saqNY3QFcaAcEyqhZ2oAfMcKsQ0MAIGqHfvwCJL8NtuTqXulmvslpSzDQ6kcU0M0rLsaPd3wkXHXQeLErIKM1OPa-mlbIM0cBWNl48_THGgHxwZ79bXljXmsC_vvzXSg/s320/P1010652.JPG" width="320" /></a></div><br /><p><br /></p>George Jellisshttp://www.blogger.com/profile/14912766967103087963noreply@blogger.com0tag:blogger.com,1999:blog-9104469096692192283.post-46086959333433639102023-01-21T16:06:00.000+00:002023-01-21T16:06:10.424+00:00Road Closed<p> There was more snow overnight and in the morning on the 18-19th, but it thawed again quite quickly. </p><p>Delamere Street is closed to traffic at the end, where it meets Flag Lane, though cars were still driving through ignoring the signs last night and today. It looks as though work will be going ahead on the section of road through to Harrison Drive and Dunwoody Way, that has been planned for some months. </p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjgNJJzCIIGETQykFJNdTQoeWk0sFz41skz7ylIWaqup27RfUv2Qx8EkkBSpYPnNjGIylSfmETHRaydUM1Xlgcj64B6DJv3DSgb_erUrrHZH8IYv4nHVSnnc80Y_KlueRetp8PYCM_4Ed4fEmWZUBWt4-qoEiAks-YQbNpb1kc8XWXYKMDy995ln5ZMIw/s2048/P1010639.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1536" data-original-width="2048" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjgNJJzCIIGETQykFJNdTQoeWk0sFz41skz7ylIWaqup27RfUv2Qx8EkkBSpYPnNjGIylSfmETHRaydUM1Xlgcj64B6DJv3DSgb_erUrrHZH8IYv4nHVSnnc80Y_KlueRetp8PYCM_4Ed4fEmWZUBWt4-qoEiAks-YQbNpb1kc8XWXYKMDy995ln5ZMIw/s320/P1010639.JPG" width="320" /></a></div><br /><p><br /></p>George Jellisshttp://www.blogger.com/profile/14912766967103087963noreply@blogger.com0tag:blogger.com,1999:blog-9104469096692192283.post-23550062690808876122023-01-18T22:53:00.000+00:002023-01-18T22:53:12.958+00:00Snow, Chess and Knight Tours<p> We had our first snow of the year last night, but by this afternoon it had all thawed, before I could take a photo to prove it. So I had no trouble getting out to Crewe Chess Club this evening, to play third board in an F-team match against Alsager, which I managed to win against a young opponent. </p><p>I'm thinking of entering the Blackpool Chess event next month, but I'm not sure I fancy traipsing about the seafront in the icy wind, and the Hotel costs are getting higher, and there are still threats of rail strikes, which put me off entering the Hastings tournament earlier this month. </p><p>Professor Knuth sent me an email that included a knight's tour sent to him for his birthday on January 10. This is a 10x10 tour of the type showing a queen on cell 85 (his age) and guarding all the cells with prime numbers, and various other features. I will put it in the Figured Tours page shortly. </p><p>I've been spending an inordinate amount of time checking up on the 6x7 symmetric knight tours. There was an error in the 266 diagrams of the Sulian (axial symmetric) tours, and it took me quite a while to find the duplicate among them, since the total should be 265 according to Knuth. I did this by classifying them by geometrical properties like number of straight line moves. It turned out that tour 166 in the list was the same as 165. </p><p>Similarly when I drew out diagrams for the Eulerian tours I found 262, which was one short. I had to try check most of the enumeration again before I found the missing one. Now I have to arrange them to appear on the corrected website pages, but I'm somewhat undecided on the best arrangement method. Probably I will group together those that have the same angles in the centre cells.</p><p>Here is a puzzle to solve based on the search. Complete the tour, given the four pairs of moves in the top corner-adjacent cells. I published this also in Mathstodon but it attracted no interest there. There are twelve such choices for the near-corner angles that have a unique solution.</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhyC03rmX-QuSp7MAG-i3wi10sRqTcXD7KJvO3zSldgh0r11n8LMQcdgQ9YKpzWQ1pH2ILjQD7KoO-6a_XCimSYti2YaYWsWMv9lkUoJ7Kc0m6_fWGkYa8BDZdPOz85ie3Gz2VwwczHtNRZlp7NL3cfNmEJjEtON-Iz7bYWwSZw3JRsTDJ8ZtlB1u0dVA/s170/Puzzle%201.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="170" data-original-width="149" height="170" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhyC03rmX-QuSp7MAG-i3wi10sRqTcXD7KJvO3zSldgh0r11n8LMQcdgQ9YKpzWQ1pH2ILjQD7KoO-6a_XCimSYti2YaYWsWMv9lkUoJ7Kc0m6_fWGkYa8BDZdPOz85ie3Gz2VwwczHtNRZlp7NL3cfNmEJjEtON-Iz7bYWwSZw3JRsTDJ8ZtlB1u0dVA/s1600/Puzzle%201.jpg" width="149" /></a></div><br /><p><span>Some cases have no solution. Others have up to 30 solutions, so this method of enumerating the tours does not provide a good way of classifying them.</span><br /></p><p><br /></p><p><br /></p><p><br /></p>George Jellisshttp://www.blogger.com/profile/14912766967103087963noreply@blogger.com1tag:blogger.com,1999:blog-9104469096692192283.post-82667750316247602792023-01-05T11:32:00.000+00:002023-01-05T11:32:11.215+00:00New Bus Station Open<p>The new bus station began operation this weekend. There is obviously still a lot of work to be done on it. The old bus station has been closed off ready for demolition. The new arrangement is much more compact. There is one entry and one exit for buses. This means the buses coming from the south need to go round the houses to get to the entrance, but this may only be temporary. The buses have to reverse out of the bays which looks as if it could be awkward if there are other buses coming in from behind. Here are some photos I took. </p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhboqVSkKK94AhwgMy4PwwElIcw2GNTPIqcYPlhlLD4gWgVekXPFMs9z5z6BG01ShIZTyliIejvksyKxDlYaD7scKjuv2uzB1NtKOl7LpY2HMMvC2c5PuByiOFXEP8HiyLVu66H2nA6ndRE8zqCtePbomt6ybtyBapkW9n-Q0-L4Tkrx8Ma_xwvEffpxg/s2048/P1010632.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1536" data-original-width="2048" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhboqVSkKK94AhwgMy4PwwElIcw2GNTPIqcYPlhlLD4gWgVekXPFMs9z5z6BG01ShIZTyliIejvksyKxDlYaD7scKjuv2uzB1NtKOl7LpY2HMMvC2c5PuByiOFXEP8HiyLVu66H2nA6ndRE8zqCtePbomt6ybtyBapkW9n-Q0-L4Tkrx8Ma_xwvEffpxg/s320/P1010632.JPG" width="320" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEghM-4kY0sCUwXD1ASEnPgTXt232jIyN6DvLIol6RhFnkQyvbKSIlyUU-bJY-cWpGRGwJoC5A1-pkxngBOwYEh6uOAcIUSh14WDaqc7-TGBxyfQZGSJ9nyE-ZzBe8-FA0IJQJDQPk0tjdIDtPOHXh_SUeFNNBnINbm1ko2SybqvJTSKy4N-RA7ha1sFRA/s2048/P1010633.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1536" data-original-width="2048" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEghM-4kY0sCUwXD1ASEnPgTXt232jIyN6DvLIol6RhFnkQyvbKSIlyUU-bJY-cWpGRGwJoC5A1-pkxngBOwYEh6uOAcIUSh14WDaqc7-TGBxyfQZGSJ9nyE-ZzBe8-FA0IJQJDQPk0tjdIDtPOHXh_SUeFNNBnINbm1ko2SybqvJTSKy4N-RA7ha1sFRA/s320/P1010633.JPG" width="320" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhcBXEVZM7jbegCGhH_Ik_vSRZIWnPUa4iUBA7EeUP2ko6-Jj83-ReDFlZhfOth4t82KkaXI8Wj8KC25nFgfUTrGtahggPLSWIfbyTVgjH5sXRdUrTiWW5N6mFmtCF5eTNczu1MPb3atyMlZ9Gj3HKPKnLyJJVnmqKfp17gw2d74VOE0yhJjJh2GJEQ3g/s2048/P1010634.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1536" data-original-width="2048" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhcBXEVZM7jbegCGhH_Ik_vSRZIWnPUa4iUBA7EeUP2ko6-Jj83-ReDFlZhfOth4t82KkaXI8Wj8KC25nFgfUTrGtahggPLSWIfbyTVgjH5sXRdUrTiWW5N6mFmtCF5eTNczu1MPb3atyMlZ9Gj3HKPKnLyJJVnmqKfp17gw2d74VOE0yhJjJh2GJEQ3g/s320/P1010634.JPG" width="320" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEipuBBmDD1nJJTF8KSB7d4e_3CxqdI2NHtuL5Ekzh4ioYb0SEtqSmbPogOrseELyvZ2yMygoq0nK_4CcrmHgul__dkEO45DMmvLZw6emjfxhdiA5W5Ui6hsCZDt61PdAawF-LbnMHcfSL2zLE8ECOEaShO4xQRY0bPA8A3yHR_VcNXjpoeTjMVLxACvAA/s2048/P1010635.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1536" data-original-width="2048" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEipuBBmDD1nJJTF8KSB7d4e_3CxqdI2NHtuL5Ekzh4ioYb0SEtqSmbPogOrseELyvZ2yMygoq0nK_4CcrmHgul__dkEO45DMmvLZw6emjfxhdiA5W5Ui6hsCZDt61PdAawF-LbnMHcfSL2zLE8ECOEaShO4xQRY0bPA8A3yHR_VcNXjpoeTjMVLxACvAA/s320/P1010635.JPG" width="320" /></a></div><br /><p>The first and fourth photos are from the Delamere street exit. The second and third are from the Victoria street entrance.</p><p>I took a number 31 bus up to Leighton Hospital, to see if I could get a box of catheters to keep me supplied over the next weekend as my order had not yet arrived. This was successful. The A&E building seems to have expanded considerably, into what used to be a large car park, since I was there last. Then I took a number 12 bus back to Crewe. </p><p><br /></p>George Jellisshttp://www.blogger.com/profile/14912766967103087963noreply@blogger.com0tag:blogger.com,1999:blog-9104469096692192283.post-35242209378230941532022-08-08T07:49:00.000+01:002022-08-08T07:49:09.447+01:00Updating Website<p> New results sent to me on enumeration of the smallest board tours by Andrew Usher in the USA and of queen tours by Walter Trump in Germany have led me to upgrade the relevant Knight's Tour Notes pages on my website. The trouble is that each of these updates has led to the realisation that all the other related pages also need updating. First I did the Shaped Boards section. Then the Walker Tours section (on pieces with lateral and diagonal moves. </p><p>A result about the impossibility of magic queen tours with rotary symmetry on the 6x6 board has led me to put new results into the Magic Theory page, and that led me to reinstate the page on 4x4 Magic Squares, which had become mixed up with the page on Multimover tours. Now I am recasting the whole section on Other Pieces. A lot of the pages are out of date compared with the PDF versions I issued in 2019.</p><p>The whole of the History pages also need to be updated, due to newer information, some of which was put into the 2019 PDF versions. So it's proving a big task. After that I will have to try to put everything into book form again. Or maybe I need to start a dedicated website for Knight's Tour Notes alone. All this leaves little time for other original work.</p><p><br /></p>George Jellisshttp://www.blogger.com/profile/14912766967103087963noreply@blogger.com1tag:blogger.com,1999:blog-9104469096692192283.post-69604474011744130452022-07-14T18:53:00.003+01:002022-07-14T18:53:22.343+01:00The Onitiu Problem Published<p> I've now completed my account of the Onitiu Problem, which has featured in these diary pages over the past two years. It forms a third page in the Figured Tours section on my Knight's Tour Notes pages.</p><p><a href="https://www.mayhematics.com/t/f3_Onitiu%20Problem.htm" target="_blank">https://www.mayhematics.com/t/f3_Onitiu%20Problem.htm</a><br /></p><p>Quite a lot of work there that I'm pleased with, though still a number of gaps, and constructing solutions on larger boards is beyond the limits of my drawing program.</p>George Jellisshttp://www.blogger.com/profile/14912766967103087963noreply@blogger.com0tag:blogger.com,1999:blog-9104469096692192283.post-75343061344524740502022-06-28T08:46:00.000+01:002022-06-28T08:46:26.585+01:00Giraffe Move Pattern<p> This is a differently coloured and enlarged version of the Giraffe Move Pattern that I published here in 2021. It shows the successive cells reached from the centre cell in successive numbers of Giraffe {1,4} moves. The colouring is still based on the rainbow sequence, but achieves more contrast between areas, with the orange and yellow areas coming between the darker areas. Is it too fanciful to imagine it as an architectural plan of some ancient temple?</p><p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi_whCbrlYQ1UVysZLnxpr5szBz_YTUQI_CxC5KZHW9Lu7V98hgV8H4BzF_pQsrckijMH-51fp4cYbBLJjzKNRRCmB93AIgghXsJUE87-sVF9Ct8kVx2uPNHWTdWuXhR-ApPS6eyXtHtt6kCcPRwyTYdd9LmaVk3Tf7IRgcC8uWK-01UpK2bB5RufcyWw/s640/Giraffe%20Move%20Pattern%202.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="640" data-original-width="592" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi_whCbrlYQ1UVysZLnxpr5szBz_YTUQI_CxC5KZHW9Lu7V98hgV8H4BzF_pQsrckijMH-51fp4cYbBLJjzKNRRCmB93AIgghXsJUE87-sVF9Ct8kVx2uPNHWTdWuXhR-ApPS6eyXtHtt6kCcPRwyTYdd9LmaVk3Tf7IRgcC8uWK-01UpK2bB5RufcyWw/s320/Giraffe%20Move%20Pattern%202.jpg" width="296" /></a></div><br />It would be interesting to expand the pattern outwards further, repeating the colour sequence, so black cells wold represent 9, red cells 10 and so on, but this is close to the limits of my drawing programme.. <p></p><p><br /></p>George Jellisshttp://www.blogger.com/profile/14912766967103087963noreply@blogger.com1tag:blogger.com,1999:blog-9104469096692192283.post-30458456359771669622022-06-10T18:40:00.003+01:002022-09-05T19:53:30.778+01:00More Demolition at the Bus Station<p> I seem to have missed a month. I've been busy updating my website pages dealing with Knight's Tours on Shaped and Holey Boards. This was stimulated by new enumerations of 11-cell and 14-cell tours sent to me by Andrew Usher from the US. I've found it rather laborious, partly because I'm experiencing eye strain, and have to ration the time I spend online, to avoid getting a headache. </p><p>There have also been developments concerning magic queen tours on odd-sided boards. This is in work by Walter Trump in Germany. I suppose I will have to update the magic tour pages next.</p><p>Meanwhile in Crewe town centre work on clearing the site for the Bus Station is under way very slowly.</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEghuoZ8pon9s72xcy9uXoJ16g2RXNbhAQQQg4XeZghaImSbCMkaKU4G6PP7W2owUdpjtkz1ZqtoKLfUKNipCvLR2KnLU3gSPYLVJv1fcmP8nYMbQO-XCWU0zkGjiHfeJmAjknFOX-wPCHgf271VWO2YxwdIsrjliWbunPGzafUlq0vf4vzCsequTtW1TQ/s2048/P1010609.JPG" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1536" data-original-width="2048" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEghuoZ8pon9s72xcy9uXoJ16g2RXNbhAQQQg4XeZghaImSbCMkaKU4G6PP7W2owUdpjtkz1ZqtoKLfUKNipCvLR2KnLU3gSPYLVJv1fcmP8nYMbQO-XCWU0zkGjiHfeJmAjknFOX-wPCHgf271VWO2YxwdIsrjliWbunPGzafUlq0vf4vzCsequTtW1TQ/s320/P1010609.JPG" width="320" /></a></div><br /><p>Here is another photo</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjysGOftxILmDgYR3rWj2nhtoNXgqQVpLXEI_NGlenUlXTVbvp7ZzX9Uw-QR-oN2M3DneylZ_Ddwqoci5dz9E4jSvlcKPRLE5IoPfXl6zZ_y23MfBUHoNh5ielg9AI-BUu41stZKy-ZsQLjPnD9xPwtS7plW9gj6L_2HEElmm1rqQRoeDMJTcjL_MiRXg/s2048/P1010611.JPG" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1536" data-original-width="2048" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjysGOftxILmDgYR3rWj2nhtoNXgqQVpLXEI_NGlenUlXTVbvp7ZzX9Uw-QR-oN2M3DneylZ_Ddwqoci5dz9E4jSvlcKPRLE5IoPfXl6zZ_y23MfBUHoNh5ielg9AI-BUu41stZKy-ZsQLjPnD9xPwtS7plW9gj6L_2HEElmm1rqQRoeDMJTcjL_MiRXg/s320/P1010611.JPG" width="320" /></a></div><br /><p>Latest photo of the demolition work, taken 14 July, below: </p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjUkpZbZVl4t-lRYKn9bQSGtSKv3R4QuO29nO-cx6PeXgSHI4YiHERujGo4A9_QLAWYhNj27AhASYeFj26foViEx_Hran64nn_hubQxW2eeYhlEgB95O7EMEV_s0R7wH57u4eMuWedHV_LZ5ilkX794M6tCj4Cqrnn4uGvuKJsOeXy8E2ar678WbIkB9w/s2048/P1010616.JPG" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1536" data-original-width="2048" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjUkpZbZVl4t-lRYKn9bQSGtSKv3R4QuO29nO-cx6PeXgSHI4YiHERujGo4A9_QLAWYhNj27AhASYeFj26foViEx_Hran64nn_hubQxW2eeYhlEgB95O7EMEV_s0R7wH57u4eMuWedHV_LZ5ilkX794M6tCj4Cqrnn4uGvuKJsOeXy8E2ar678WbIkB9w/s320/P1010616.JPG" width="320" /></a></div><br /><p>Beautiful blue sky today and cool in the shade, </p><p>but I'm not looking forward to the heat wave forecast for next week.</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiQAyXkOxOh3LqU61TBKdcKUgHvi88uGiZ7saGOJ3mfCcdmn_4NpsTpp-hbf5fEh8qqevaQCkLqkDPwEkQTFZS1yP4W1rNuhpKR2IyxLGud8jzPvnNTMHWEogYs8jM2dTTsDy0fga16vgt6I3AV8lamxqmc3sHdiTZH-hkCE2_W28TBad0hUdNiiV5TUg/s2048/P1010622.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1536" data-original-width="2048" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiQAyXkOxOh3LqU61TBKdcKUgHvi88uGiZ7saGOJ3mfCcdmn_4NpsTpp-hbf5fEh8qqevaQCkLqkDPwEkQTFZS1yP4W1rNuhpKR2IyxLGud8jzPvnNTMHWEogYs8jM2dTTsDy0fga16vgt6I3AV8lamxqmc3sHdiTZH-hkCE2_W28TBad0hUdNiiV5TUg/s320/P1010622.JPG" width="320" /></a></div><br /><p>Demolition work now complete (4 September 2022). Fortunately three trees along Delamere Street have survived the devastation so far.</p><p><br /></p><p><br /></p><p><br /></p><p><br /></p>George Jellisshttp://www.blogger.com/profile/14912766967103087963noreply@blogger.com0tag:blogger.com,1999:blog-9104469096692192283.post-15328292836538978192022-04-19T08:09:00.000+01:002022-04-19T08:09:07.165+01:00Numbers and a Tour<p> I have a new publication! <i>Directory of Numbers</i>. It is a 24-page listing of the numbers up to 9999 with prime factorisations, except that four-digit numbers divisible by 2 or 5 are omitted. These are easily divided by 2, 5 or 10 and the factors of the resulting smaller number can then be looked up.</p><p>https://www.mayhematics.com/p/p.htm</p><p>I have also found a solution to the 18x18 Onitiu Problem of a knight tour with 180 degree rotational symmetry. This just fills a gap in my work, since I have not been able to find a solution to the 90 degree rotation on this board. Here is the new result:</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgVnU66FbltOty1lv0YCTJQtsxgVjH4Xj-pfdXWu4sIA1yqonWN-IXS9npsHJaLoTmsNbGiSzJb2MIu4tIbg2hUQq2bHqSR7hIMe3ZJAF4De-p3WrMqAcBcMhVYFpQ_T0e8MwS8PXrcmaPJsMB9bEAKWW9r4G4AiYH7Qp3DpB8KblMR6oVvYLOG1MiQmQ/s416/Onitiu%2018R.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="416" data-original-width="405" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgVnU66FbltOty1lv0YCTJQtsxgVjH4Xj-pfdXWu4sIA1yqonWN-IXS9npsHJaLoTmsNbGiSzJb2MIu4tIbg2hUQq2bHqSR7hIMe3ZJAF4De-p3WrMqAcBcMhVYFpQ_T0e8MwS8PXrcmaPJsMB9bEAKWW9r4G4AiYH7Qp3DpB8KblMR6oVvYLOG1MiQmQ/s320/Onitiu%2018R.jpg" width="312" /></a></div><br /><p>The square numbers 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324 are shown on the red line. The blue line is a rotation of the red. There are no intersections between the red and blue lines, but single links 63-64 and 225-226, shown as heavy black moves. </p><p><br /></p>George Jellisshttp://www.blogger.com/profile/14912766967103087963noreply@blogger.com0tag:blogger.com,1999:blog-9104469096692192283.post-601390522721615162022-03-26T16:06:00.002+00:002022-03-26T16:06:54.026+00:00Tree Cutting in Crewe<p> </p><p>There has been much cutting down of trees around the bus station in Crewe. </p><p>This is presumably to clear the way for new entrances and exits to the new bus station that is planned. </p><p>Though I have not seen any publicity of the design of the new station.</p><p>I took these three photos.</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhI1dRZR5jkclKhVbRbEbTWQbye9FuwkCc1QRLZQkGT1sQDSfZk_nTpf2YaG8b_Yu2unkRnrleHR60EjRb0kE1uyt5B-KR3nB-GgwYBotYsMvhxxp4mjK_--rRTwVI9xduhxhMi_br9HT2AksJKFupvZow2NbjX1FRHQBRJdbenTQ6BWVqMwmvD-l6P9g/s2048/P1010597.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1536" data-original-width="2048" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhI1dRZR5jkclKhVbRbEbTWQbye9FuwkCc1QRLZQkGT1sQDSfZk_nTpf2YaG8b_Yu2unkRnrleHR60EjRb0kE1uyt5B-KR3nB-GgwYBotYsMvhxxp4mjK_--rRTwVI9xduhxhMi_br9HT2AksJKFupvZow2NbjX1FRHQBRJdbenTQ6BWVqMwmvD-l6P9g/s320/P1010597.JPG" width="320" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjpwx5V1OPWLzifV4QbziTicp7HwRQOryEBLwOnJsTvb67owZjzJOR3mCrOE_PTgzRUgjiFjGClNiTcNdB-e9Z82iN6jRg0jCvJuU9UoZUP0Es46f2BEQE8fzH0jPm1LZx5x62tzHectu3beOjiZQZZCSJIv4X9AF-83QOVz6i8NDH9ZwHiMBUCR7502A/s2048/P1010598.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1536" data-original-width="2048" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjpwx5V1OPWLzifV4QbziTicp7HwRQOryEBLwOnJsTvb67owZjzJOR3mCrOE_PTgzRUgjiFjGClNiTcNdB-e9Z82iN6jRg0jCvJuU9UoZUP0Es46f2BEQE8fzH0jPm1LZx5x62tzHectu3beOjiZQZZCSJIv4X9AF-83QOVz6i8NDH9ZwHiMBUCR7502A/s320/P1010598.JPG" width="320" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjcK4J4ke1D9d-0xPfwVfPh0ty4vh4pMqfp-Q3fxY2K8zPrGw7ZM8PvoUid6AOM9daMgUECT7U9azu8b8FxB0QBv8cmCoYgSieaV35a-jhvEBm5o0AyA8Ov-zL4fxt67StEL1J8AOstuazuJezXObIYl5lWKNHLuknGiqVNs2SLEDAQhKxvXN_SJoIpZQ/s2048/P1010600.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1536" data-original-width="2048" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjcK4J4ke1D9d-0xPfwVfPh0ty4vh4pMqfp-Q3fxY2K8zPrGw7ZM8PvoUid6AOM9daMgUECT7U9azu8b8FxB0QBv8cmCoYgSieaV35a-jhvEBm5o0AyA8Ov-zL4fxt67StEL1J8AOstuazuJezXObIYl5lWKNHLuknGiqVNs2SLEDAQhKxvXN_SJoIpZQ/s320/P1010600.JPG" width="320" /></a></div><br /><p>Some of the trees cut down were substantial, so I hope others can be planted to replace them.</p><p>There was also substantial tree-felling of a row of trees on Godard Street. </p><p>This is probably to clear the way for a new housing development. </p><p><br /></p>George Jellisshttp://www.blogger.com/profile/14912766967103087963noreply@blogger.com0tag:blogger.com,1999:blog-9104469096692192283.post-74902249773562447592022-03-14T10:47:00.000+00:002022-03-14T10:47:12.571+00:00Fiery Sky<p> </p><p>A Fiery Sky at sunset yesterday evening.</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEgz7t8bO3oLaw1NztpXYULBZM5LxvIdPtno0JuHcjhKZdfT3SpcHbyYq6yinY28bJx3tRpErsORxd9LCS6HV5VxMR04WTKpwIF5ST71LUk5evFgtEiDs0aOtkT2OhulS6aAGy2jc8vZupm0n0rx3RkXsDQuzruBmxoTkwMF1ZlYDeuAgYnYdEZtGs978Q=s2048" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1536" data-original-width="2048" height="240" src="https://blogger.googleusercontent.com/img/a/AVvXsEgz7t8bO3oLaw1NztpXYULBZM5LxvIdPtno0JuHcjhKZdfT3SpcHbyYq6yinY28bJx3tRpErsORxd9LCS6HV5VxMR04WTKpwIF5ST71LUk5evFgtEiDs0aOtkT2OhulS6aAGy2jc8vZupm0n0rx3RkXsDQuzruBmxoTkwMF1ZlYDeuAgYnYdEZtGs978Q=s320" width="320" /></a></div><br /><p>view taken from my bay window.</p><p><br /></p>George Jellisshttp://www.blogger.com/profile/14912766967103087963noreply@blogger.com0tag:blogger.com,1999:blog-9104469096692192283.post-41842988712312301762022-02-26T19:26:00.002+00:002022-02-26T19:26:32.688+00:00Work in Slow Progress<p><br /></p><p>Despite not posting anything here I have been busy this month, but not making much progress. </p><p>Partly this is because I've had trouble with blisters on a leg and losing a couple of front teeth, so have been kept occupied visiting dentist and GP surgeries and hospital for checkups.</p><p>I'm still trying to complete my study of the Onitiu Problem but am stuck on the 14x14 case. </p><p>I'm also trying to edit my 12 PDFs on Knight's Tours into a shorter book. </p><p>Another project is a book on Numbers, which I may call Numerology or maybe Arithmology, with the subtitle The Wisdom and Folly of Numbers, .since it includes chapters on Arithmosophy and Numeromancy. It also includes my notes on Figurate Numbers as well as basic Arithmetic. </p><p><br /></p>George Jellisshttp://www.blogger.com/profile/14912766967103087963noreply@blogger.com0tag:blogger.com,1999:blog-9104469096692192283.post-15563731974601104582022-01-18T08:15:00.000+00:002022-01-18T08:15:17.395+00:00Red Sky<p> A weirdly mottled red sky over Crewe this morning. </p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEipynsd5sSo4gjVwlmJ6B50Yq7QnEI0mS-iq1GhcUF-HpldQjhhV8QAAHV6H8DcHXPvHuWRYfc63sd1C7LgOaUyHroWkZ5UUh_tgA2OStMunNe96kqWJJlqvVYL60U9d-GVTFKqTqbYXZiCkCBAYFEVBeSFK5hm_ZTZ4lJS7gcHPcil0sRmvyvdTOUv8w=s2048" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1536" data-original-width="2048" height="240" src="https://blogger.googleusercontent.com/img/a/AVvXsEipynsd5sSo4gjVwlmJ6B50Yq7QnEI0mS-iq1GhcUF-HpldQjhhV8QAAHV6H8DcHXPvHuWRYfc63sd1C7LgOaUyHroWkZ5UUh_tgA2OStMunNe96kqWJJlqvVYL60U9d-GVTFKqTqbYXZiCkCBAYFEVBeSFK5hm_ZTZ4lJS7gcHPcil0sRmvyvdTOUv8w=s320" width="320" /></a></div><br /><p>I went out specially to catch the image before it faded.</p><p><br /></p>George Jellisshttp://www.blogger.com/profile/14912766967103087963noreply@blogger.com0tag:blogger.com,1999:blog-9104469096692192283.post-14910204497609227122022-01-13T16:07:00.000+00:002022-01-13T16:07:20.089+00:00On Sacred Geometry<p> On Sacred Geometry</p><p><br /></p><p>I've been looking at various YouTube sites that have videos on "Sacred Geometry". Some are completely vacuous waffle to me. However a few do contain genuine arithmetic and geometric results that seem of interest from a mathematical viewpoint. In particular there is a series from the "Jain Academy" fronted by an affable Australian lecturer who calls himself "Jain108". His mathematics is mostly correct, except where he obsesses about the "true" value of pi being 3.144... The Jain Academy deals in what I can only call "New Age Eclecticism". In other words it takes bits from everywhere, Hinduism, Kabbala, Christianity, Islam, Buddhism, Astrology, and so on, regardless of dogma. </p><p><br /></p><p>The number 108 is apparently ubiquitous in Hindu mysticism. It is said in several sources to be the ratio of distance to diameter for both the Sun and the Moon. For any celestial body this ratio would be near enough the cotangent or cosecant of its apparent angular diameter. The angle whose cotangent or cosecant is 108 turns out to be 31 minutes and 50 seconds to the nearest second. In Patrick Moore's "Atlas of the Universe" (1994), which I happen to have to hand, the mean apparent diameter of the Sun is given as 32' 1" and the mean apparent diameter of the Moon as 31' 6". So the number 108 seems to be a reasonably good estimate. The closeness of the apparent diameters of Moon and Sun is of course why Solar Eclipses can be so spectacular.</p><p><br /></p><p>Another obsession of the Sacred Geometers is, as might be expected, the dimensions of the Great Pyramid of Khufu at Giza. The Jain Academy makes much of the triangle formed by the pyramid as seen at a distance from a side. Taking its base width to be 2 units it is claimed that its slope length is the golden ratio 1.618 and its height is the square root of the golden ratio. According to Wikipedia the pyramid's original height was 280 cubits and its base 440 cubits. The slope length is thus the square root of (280^2 + 220^2) = root 126800 = 356.09 cubits. The ratio of slope to half base is thus 356.09/220 = 1.61859, and the golden ratio is 1.6180339. So again a plausible approximation. The angle of slope is 51 degrees 5 minutes so the visible angle at the summit would be 77 degrees 50 minutes.</p><p><br /></p><p>When people make a model of a pyramid they nearly always take the faces to be exact equilateral triangles, but this gives a pyramid whose slope height is half of root three and whose height is half of root two. If the slope height of the pyramid is the golden ratio then this must be the altitude of the triangular face shape. The base angle then works out at 58 degrees 17 minutes and the apex angle as 63 degrees 23 minutes Differing from the equilateral by 3.6 degrees at the apex. </p><p><br /></p><p>The symbol used for the golden ratio in Sacred Geometry is phi (though mathematical texts often use tau). The claim about the new value for pi is that is should be 4/(root phi) = 3.144605...as opposed to 3.14159... Or equivalently that 4/pi = 1.2732395... should be root phi = 1.2720196... Whether this means that the Laws of Nature themselves are going to change when the New Age dawns, or only human consciousness of them is not at all clear.</p><p><br /></p>George Jellisshttp://www.blogger.com/profile/14912766967103087963noreply@blogger.com0tag:blogger.com,1999:blog-9104469096692192283.post-26525865934282528212021-12-23T14:09:00.003+00:002021-12-24T23:37:29.787+00:00Fibonacci and Combinations<p><br /></p><p> I've been looking through my books on number theory and combinatorics, but none of them seem to mention anywhere the following simple relationship of Fibonacci sequence to combinations. In the following I use the notation nCr = n!/(n-r)!r! for the number of ways of choosing r from n. </p><p>An explicit sum for F(n) can be found by summing the upward diagonals of the combination table that lists n against r and is sometimes presented in the form of Pascal's Triangle. </p><p>It can be expressed in the general form:</p><p><span style="white-space: pre;"> </span>F(n) = (n-1)C0 + (n-2)C1 + (n-3)C2 + (n-4)C3 + ... + (n-k)C(k-1) </p><p><span> </span><span> </span><span> </span><span> </span>= Summation (r=1 to k) (n-r)C(r-1).</p><p>Where k = n/2 or (n+1)/ when n is even or odd respectively. </p><p>In the symmetric form of Pascal's Triangle the upward diagonals are transformed into knight-lines.</p><p>Instead of the above elementary formula for F(n) that uses only addition, subtraction, multiplication and division of whole numbers, the mathematical texts focus on the exotic Binet formula that expresses F(n) in terms of root five, or root five and the golden ratios, which are irrational numbers that all cancel each other out.</p><p><i>A First Course of Combinatorial Mathematics</i> by Ian Anderson (Oxford University Clarendon Press (1979) page 43 derives from the Binet formula a more complicated relationship involving the summation of expressions 5^r.(n+1)Cr divided by 2^n, but does not simplify it to the above form. This is the nearest I have found in my limited sources.</p><p>Of course there is also a simple relationship of 2^n to the combinations:</p><p> <span style="white-space: pre;"> </span>2^n = nC0 + nC1 + nC2 + ... + nCn.</p><p>Does anyone have other sources they could direct me to? (see my Knight's Tour pages for my email)</p><p>Does anyone have a concrete combinatorial explanation for the formula?</p><p>CORRECTION: I have since noticed that Anderson does mention the above expression briefly, without further explanation, in Exercises 4.2 (4) on page 44.</p><p>FURTHER: I have found another reference in <i>Principles of Combinatorics</i> by Claude Berge (Academic Press 1971) page 31, though it has n+1 instead of n. It implies that (n-k)Ck counts the number of ways of choosing k items from a row of n, but with no two items being adjacent.</p><p>FURTHER: I have modified the equation, since there seems to be some divergence in the way that the Fibonacci numbers are labelled. I take F(0) = 0 and F(1) = 1, so that F(2) = 1 and F(3) = 2 and so on, whereas other sources start from F(0) = 1, F(1) = 1, F(2) = 2, etc. I hope it is now correct!</p><p><br /></p>George Jellisshttp://www.blogger.com/profile/14912766967103087963noreply@blogger.com0tag:blogger.com,1999:blog-9104469096692192283.post-92086577998678688922021-12-07T21:34:00.003+00:002021-12-07T21:35:19.789+00:00Pentagonal Numbers are Trapezoid Numbers<p><br /></p><p> I will prove here a theorem I have not seen before: </p><p>The sum of an arithmetic progression with common difference k </p><p>is the sum of k triangular numbers of two sizes. </p><p><br /></p><p>Consider the arithmetic progression h, h + k, h + 2k, h + 3k, ..., h+(n-1)k which has n terms. </p><p>The summation of this is the series: hn + k(1 + 2 + 3 + ...+ (n-1)) </p><p>= hn + kn(n-1)/2 = [kn^2 + (2h - k)n)]/2 </p><p><br /></p><p>This formula can be expressed as (k-h)n(n-1)/2 + hn(n+1)/2 </p><p>= (k-h)T(n-1) + hT(n). </p><p>That is as the sum of k triangles, h of side n and (k-h) of side (n-1). </p><p>In the case when h= 0 or h = k we get k triangles all of the same size.</p><p><br /></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEgPYHFBgjFd0W1sydg11E9N02C7d_z15fwGIlTxif2mG-wILQY3rGqOs7j9TgU_zoEBjWWeFlMo31YPiGk3m-5FGhiRcpgESZLS-6q028aFWw3ybnTzOLbXxdSW9F16jLIf45lLow5Ak2s_bp5Ksnmb88WuFbog6_iSDNzmWNL2F7K7Cl2uodKIIypulg=s432" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="352" data-original-width="432" height="261" src="https://blogger.googleusercontent.com/img/a/AVvXsEgPYHFBgjFd0W1sydg11E9N02C7d_z15fwGIlTxif2mG-wILQY3rGqOs7j9TgU_zoEBjWWeFlMo31YPiGk3m-5FGhiRcpgESZLS-6q028aFWw3ybnTzOLbXxdSW9F16jLIf45lLow5Ak2s_bp5Ksnmb88WuFbog6_iSDNzmWNL2F7K7Cl2uodKIIypulg=s320" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;">Figurate representation of the numbers</div><br /><p>When h = 1 and k = 3 we get the sequence 1, 4, 7, 10, 13, 16, 19, 22, 25, ...</p><p>its summation is the series: 1, 5, 12, 22, 35, 51, 70, 92, 117, ...</p><p>which is traditionally known as the 'pentagonal numbers'. </p><p><br /></p><p>When h = 2 and k = 3 we get the sequence 2, 5, 8, 11, 14, 17, 20, 23, 26, ... </p><p>its summation is the series: 2, 7, 15, 26, 40, 57, 77, 100, 126, ... </p><p>which are known as the 'pentagonal numbers of the second kind'.</p><p><br /></p><p>When h = 0 (or 3) we get the sequence 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, ... </p><p>its summation is the series 0, 3, 9, 18, 30, 45, 63, 84, 108, 135, ... </p><p>which are triple triangle numbers. </p><p><br /></p><p><br /></p><p><br /></p><p><br /></p>George Jellisshttp://www.blogger.com/profile/14912766967103087963noreply@blogger.com0