tag:blogger.com,1999:blog-91044690966921922832022-05-15T22:49:18.398+01:00Jeepyjay DiaryGeorge Jellisshttp://www.blogger.com/profile/14912766967103087963noreply@blogger.comBlogger312125tag: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.com0tag:blogger.com,1999:blog-9104469096692192283.post-72822995459036968642021-11-30T09:58:00.003+00:002021-12-10T11:56:29.529+00:00Negative Test for Covid<p>It was reported that a couple of members of Crewe Chess Club had tested positive for covid19, and one was in hospital. So I tested myself last night using a Lateral Flow Test that I obtained from a local pharmacy last week. Fortunately the result was negative.</p><p>I tried reporting the result to the NHS as is required but the website said it needed to know a mobile phone number, and I don't have a mobile phone. So I phoned 119 this morning. It required answering an awful lot of multiple choice questions before I could get through to the right place.</p><p>It seems both chess club members are recovering, though one is still in hospital for other health reasons. The club is closed this week, which is probably the right decision. I was due to take part in a team match. </p><p>I've also put off going to a meeting of a Writer's group at the local Library for the present. It seems best to wait until the news about the new omicron variant becomes more definite. At least I now know how to carry out the test when necessary.</p><p>UPDATE. Regrettably the club member who was in hospital, Les Hall, died on 8th December, though his cause of death may not have been covid19. He had diabetes and other conditions. Les Hall will be well known to chess players who have attended the Crewe Congress in former years. He was a larger than life personality, and one of the stalwarts behind the day-to-day running of the club. We will miss him.</p>George Jellisshttp://www.blogger.com/profile/14912766967103087963noreply@blogger.com0tag:blogger.com,1999:blog-9104469096692192283.post-71789450347026048902021-11-29T13:25:00.000+00:002021-11-29T13:25:06.568+00:00First Snow of the Season<p> Snowy scene from my back window yesterday. </p><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-vq0PPd8ioBE/YaTUaS82qQI/AAAAAAAAA5U/g8FkS_TYqq4qLrV8y9Xnf7jvcCXy2tCugCLcBGAsYHQ/s2048/P1010587.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://1.bp.blogspot.com/-vq0PPd8ioBE/YaTUaS82qQI/AAAAAAAAA5U/g8FkS_TYqq4qLrV8y9Xnf7jvcCXy2tCugCLcBGAsYHQ/s320/P1010587.JPG" width="320" /></a></div><br /><p>Put my winter boots on this morning to trek to the supermarket and make sure I have enough coffee in stock until the new year.</p><p><br /></p>George Jellisshttp://www.blogger.com/profile/14912766967103087963noreply@blogger.com0tag:blogger.com,1999:blog-9104469096692192283.post-74390001556158137862021-10-31T10:43:00.002+00:002021-11-01T18:35:57.620+00:00Generalised Golden Ratios<p>Back in <i>The Games and Puzzles Journal</i>, #13 p.223 (May 1996), [the first issue of volume 2 which came out 25 years ago] the first puzzle was on "A Question of Proportion". It was about rectangles of various shapes including the standard paper sizes and the golden rectangle. </p><p>The 'Numberphile' videos on YouTube, some by Ben Sparks, have recently been featuring questions of this nature. I also borrowed from our local library the book <i>One to Nine</i> by Andrew Hodges which includes similar material, including the "Plastic" number. </p><p>Towards the end of the book Hodges implies that "Computable" numbers as defined in Alan Turing's famous paper are denumerable, that is they can be matched with the set of natural numbers. </p><p>I hadn't realised this before! </p><p>This also raises the philosophical question of whether any other "real" numbers "really" exist, if they cannot be calculated by a finite set of rules. Such numbers are those that appear in Cantor's diagonal proof of the nondenumerability of the continuum.</p><p>All the usual suspects like root 2, root 3, the golden section, e and pi and Euler's constant gamma are all computable numbers. This just means that they can be computed to any number of decimal places and that the calculation by whatever method always gives the same digit in the nth place for any n. </p><p>They really are real real numbers! </p><p>Going back to my puzzle in the <i>G&PJoutrnal</i> the last question asked was: What shape of rectangle gives a smaller rectangle of the same proportions when a rectangle of m squares by n squares is removed from one end?</p><p>The answer was given in G&PJournal #14 p.245 (December 1996). By applying the same methods as for the golden ratio I found a formula for such a generalised golden ratio, in terms of m and n. </p><p>It now occurs to me that we can denote the larger and smaller golden ratios by capital and lower case Phi as before, but with a suffix 1, and the generalised golden ratios by the same letters with a suffix m/n. This also indicates that these golden ratios all form a denumerable set since they can be placed in one-to-one correspondence with the ratios m/n, which are known to be denumerable.</p><p>The formulas are: Phi suffix m/n = [root (m^2 + 4.n^2) +/- m]/2.n</p><p>For example Large Phi suffix 1/2 = (root 17 + 1)/4 = 1.2807764... </p><p>Small phi suffix 1/2 = (root 17 - 1)/4 = 0.7807764...</p><p>Their difference is 1/2 and their product is 1 (or 0.99999...).</p><p>Some of these generalised ratios work out to be rational. The simplest case is when m=3, n=2 which gives the two Phi with suffix 3/2 as 2 and 1/2, but most of the ratios will be irrational.</p><p>ADDENDUM: It now occurs to me that the ratio m/n of the rectangle removed can in fact be any number not just a rational fraction. And conversely the shape of the whole rectangle can be given by any number x, the rectangle removed being x - 1/x. </p><p><br /></p>George Jellisshttp://www.blogger.com/profile/14912766967103087963noreply@blogger.com0tag:blogger.com,1999:blog-9104469096692192283.post-69623461568766443142021-10-09T10:23:00.002+01:002021-10-09T10:23:55.565+01:00Another Onitiu 18x18 Attempt<p> Here is another attempt at solving the Onitiu problem of constructing a knight tour with 90 degree rotational symmetry on the 18x18 board with the squares in a knight path (the red line). It uses two wazir moves in each quarter, equalling my previous result, making it an Emperor tour.</p><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-tgEA0pi7ETY/YWFfFuTx93I/AAAAAAAAA44/L8HpOHNB4KIJQ5WVvOYaaU54upQuAD-DQCLcBGAsYHQ/s416/Onitiu%2B18%2BQ2.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://1.bp.blogspot.com/-tgEA0pi7ETY/YWFfFuTx93I/AAAAAAAAA44/L8HpOHNB4KIJQ5WVvOYaaU54upQuAD-DQCLcBGAsYHQ/s320/Onitiu%2B18%2BQ2.jpg" width="312" /></a></div><div class="separator" style="clear: both; text-align: center;">Onitiu Problem 18x18 Quaternary Emperor Tour</div><br /><p>The pattern is considerably different from the previous example. So perhaps a solution is possible, if the exactly right configuration can be found. Or maybe two wazir moves per quarter is the best that can be done. Maybe someone could programme a computer to settle it.</p>George Jellisshttp://www.blogger.com/profile/14912766967103087963noreply@blogger.com1tag:blogger.com,1999:blog-9104469096692192283.post-59292784343296779962021-09-22T12:29:00.001+01:002021-09-22T12:32:20.374+01:00Onitiu Problem 30x30 Solution<p> This was the most difficult to solve so far. It shows a knight tour with 90 degree rotational symmetry with the square numbers in a closed knight path (the red line). There are sixteen cells (marked by squares) where the four circuits intersect. The initial and final cells of each circuit are marked by a darker square (001 and 900 in the case of the circuit of square numbers). </p><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-ELClZZ5GOzc/YUsTEO9R2dI/AAAAAAAAA4w/q9AZDOV3PAU4pLDLSgY9uwLFed9QZt2cgCLcBGAsYHQ/s599/Onitiu30x30.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="595" data-original-width="599" height="318" src="https://1.bp.blogspot.com/-ELClZZ5GOzc/YUsTEO9R2dI/AAAAAAAAA4w/q9AZDOV3PAU4pLDLSgY9uwLFed9QZt2cgCLcBGAsYHQ/s320/Onitiu30x30.jpg" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;">Solution to Onitiu Problem 30x30</div><br /><p>The thick black lines mark single-move links between the circuits. The dotted cells mark the links of two or three moves. These short links place considerable restrictions on the way the paths can be arranged.</p><p>Numerologists may notice that the numbers 666 and 216 (6 cubed) occupy diametrically opposite cells. This is because their difference is 450, half of 900. </p><p><br /></p>George Jellisshttp://www.blogger.com/profile/14912766967103087963noreply@blogger.com0tag:blogger.com,1999:blog-9104469096692192283.post-50910874886766684112021-09-17T18:24:00.000+01:002021-09-17T18:24:18.945+01:00Locked Out of Twitter Again<p> Twitter is again asking for a mobile phone number from me to sign in, but there doesn't seem to be any allowance for people who don't have mobile phones. I hope this is not becoming a general thing. It will mean that people without mobile phones are becoming second class citizens. I was also made to pass an "I'm Not A Robot" test, but apparently that was not sufficient! </p><p> </p>George Jellisshttp://www.blogger.com/profile/14912766967103087963noreply@blogger.com1tag:blogger.com,1999:blog-9104469096692192283.post-51692455074940565072021-09-09T23:19:00.000+01:002021-09-09T23:19:28.527+01:00Another Onitiu Solution<p>After a couple of months I have at last solved another of the knight tours with 90 degree rotational symmetry and with the square numbers in a knight circuit (the red line). The type face for the numbers is necessarily vey small. They range from 001 to 676 which is the board size 26x26.</p><p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-mzmsfk8d8t0/YTqH-huol5I/AAAAAAAAA4k/qFotAxdQwDQofSuk9WLR3k15jQP4KgVLgCLcBGAsYHQ/s586/Onitiu26x26.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="586" data-original-width="575" height="320" src="https://1.bp.blogspot.com/-mzmsfk8d8t0/YTqH-huol5I/AAAAAAAAA4k/qFotAxdQwDQofSuk9WLR3k15jQP4KgVLgCLcBGAsYHQ/s320/Onitiu26x26.jpg" width="314" /></a></div><div class="separator" style="clear: both; text-align: center;">Onitiu 26x26 Birotary Solution</div><br /> <p></p>George Jellisshttp://www.blogger.com/profile/14912766967103087963noreply@blogger.com0tag:blogger.com,1999:blog-9104469096692192283.post-70904129344609360152021-08-28T23:49:00.001+01:002021-08-31T21:15:39.532+01:00Enumeration of Grid Points<p>I've been reading all my books that include some Number Theory, and doing some Study of Numbers, which I'm thinking of publishing under the title "Numerology: The Wisdom and Folly of Numbers".</p><p>Here is a little item that is probably not new, but I can't find it in any of my sources. Can anyone locate it in some publication? I've a vague idea I've seen something like it somewhere. </p><p>I call numbers of the form (2^r)x(3^s) "Basals". In other words they are any numbers that exclude prime factors greater than 3. .The sequence runs: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 64, 72, 81, 98, 108, 128, ...</p><p>The corresponding powers of 2 and 3 in the sequence run: (0,0), (1,0), (0,1), (2,0), (1,1), (3,0), (0,2), (2,1), (4,0) and so on. Every pairing occurs and they are listed in a unique order.</p><p>This scheme thus provides an enumeration of the grid cells of an endless board, as partially illustrated in this tour diagram, from (0,0) = 1 to (9,0) = 512. </p><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-XjjLEYlIdVQ/YSq8gLnteEI/AAAAAAAAA4U/vRQQ5cIhA6cx9uT13hqQuV4B6HvSR4bbwCLcBGAsYHQ/s432/612ab760.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="432" data-original-width="432" height="320" src="https://1.bp.blogspot.com/-XjjLEYlIdVQ/YSq8gLnteEI/AAAAAAAAA4U/vRQQ5cIhA6cx9uT13hqQuV4B6HvSR4bbwCLcBGAsYHQ/s320/612ab760.jpg" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;">Enumeration of Coordinate Points</div><br /><p>It is interesting that the path appears never to cross itself. </p><p>Further: This diagram shows the similar result obtained using (2^r)x(5^s) to determine the sequence. This goes up to 5^4 = 625 at (0,4). The moves in the upward direction tend to be knight moves. </p><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-VUcB5e7Bgn0/YS6NpmLKw5I/AAAAAAAAA4c/1i9DSxPtBCQaPP8Xg8yLij7UghhiQUITgCLcBGAsYHQ/s442/612e8bfb.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="432" data-original-width="442" height="313" src="https://1.bp.blogspot.com/-VUcB5e7Bgn0/YS6NpmLKw5I/AAAAAAAAA4c/1i9DSxPtBCQaPP8Xg8yLij7UghhiQUITgCLcBGAsYHQ/s320/612e8bfb.jpg" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;">Enumeration using primes 2 and 5.</div><div><br /></div><br /><p><br /></p><p><br /></p>George Jellisshttp://www.blogger.com/profile/14912766967103087963noreply@blogger.com0tag:blogger.com,1999:blog-9104469096692192283.post-27553460518527369662021-07-20T14:32:00.002+01:002021-08-16T18:20:57.888+01:00Locked out from Twitter<p> I'm unable to log in to Twitter, since I don't have a mobile phone and am unwilling to let them know my landline telephone number, which already gets too many unwanted calls. The same would apply to any other online site. </p><p>A notice on my twitter page, @mayhematics, it says there has been some unusual activity on my account. But the only difference from usual is that I used my old computer to access the site last week, since it was cooler in the back room where that computer is.</p><p>They would be better devoting their resources to stopping people who send nasty messages or fake news. I just try to entertain with my knight tour discoveries and occasional humour and logic.</p><p>UPDATE: 10 August 2021. The requirement for me to enter a mobile phone number has now been removed and I am able to enter the Twitter site by signing in with my user name and password, but I am still unable to post any messages there. A fleeting sign comes up that says my account is suspended. In my Profile all the images have been blanked out. I thought I was making some good contributions to their content, but it seems my efforts are not appreciated.</p><p>Further: After making an "Appeal" on their system I have been reinstated on Twitter. However I find I haven't missed it, and no-one seems to have noticed my absence, so I probably won't be using it much in future. I've found a lot of other things to do, though I don't seem to be making progress with any of my projects at present, which is why I haven't been posting here. </p>George Jellisshttp://www.blogger.com/profile/14912766967103087963noreply@blogger.com2tag:blogger.com,1999:blog-9104469096692192283.post-61580715862158607802021-07-15T09:05:00.001+01:002021-07-20T10:29:43.431+01:00Emperor Solutions to the Onitiu Problem<p>Despite considerable efforts I have not made much further progress on the Quaternary Onitiu problem, that is of constructing knight tours with 90 degree rotational symmetry and with the square numbers in a knight path. These diagrams show the best I have found on the 10x10 and 14x14 boards in the form of Emperor tours which use wazir moves as well as knight moves.</p><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-R_XQ-eKw7-Q/YO_qxU1Gn9I/AAAAAAAAA3c/Za9se6-1TAw2uB8qxAEYrIuL4A6ZWq6iACLcBGAsYHQ/s352/Onitiu%2B10QE.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="352" data-original-width="352" height="320" src="https://1.bp.blogspot.com/-R_XQ-eKw7-Q/YO_qxU1Gn9I/AAAAAAAAA3c/Za9se6-1TAw2uB8qxAEYrIuL4A6ZWq6iACLcBGAsYHQ/s320/Onitiu%2B10QE.jpg" /></a></div><div class="separator" style="clear: both; text-align: center;">10x10 Emperor tour</div><div class="separator" style="clear: both; text-align: center;">with 28 wazir moves</div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-vnxdvSCA328/YO_q4HHCuvI/AAAAAAAAA3g/IKU9R9-6Ss8pFBh5cUT-LUF7X-jwPp68QCLcBGAsYHQ/s480/Onitiu%2B14%2BQE.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="480" data-original-width="480" height="320" src="https://1.bp.blogspot.com/-vnxdvSCA328/YO_q4HHCuvI/AAAAAAAAA3g/IKU9R9-6Ss8pFBh5cUT-LUF7X-jwPp68QCLcBGAsYHQ/s320/Onitiu%2B14%2BQE.jpg" /></a></div><div class="separator" style="clear: both; text-align: center;">14x14 Emperor tour</div><div class="separator" style="clear: both; text-align: center;">with 40 wazir moves</div><p>I am now convinced that a knight tour solution on the 10x10 is impossible, and probably also on the 14x14 board, though I don't have a simple conclusive proof. In the above diagrams the square numbers are shown on the red line. The other coloured lines are rotations of the red line.</p><p>ADDENDUM (20 July 2021)</p><p>I have now solved the same problem on the 18x18 board, using only two wazir moves in each quarter. Can this be modified to make a knight tour solution? </p><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-wLv4vhuvBcY/YPaW_M3zFGI/AAAAAAAAA34/rXoZWfjV_Uchdvaj_hOwJp_IM0cUgnX-gCLcBGAsYHQ/s624/Onitiu%2B18%2BQ.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="624" data-original-width="608" height="320" src="https://1.bp.blogspot.com/-wLv4vhuvBcY/YPaW_M3zFGI/AAAAAAAAA34/rXoZWfjV_Uchdvaj_hOwJp_IM0cUgnX-gCLcBGAsYHQ/s320/Onitiu%2B18%2BQ.jpg" /></a></div><div class="separator" style="clear: both; text-align: center;">18x18 Emperor tour</div><div class="separator" style="clear: both; text-align: center;">with 8 wazir moves</div><div><br /></div>This case is particularly interesting in having eight intersections instead of four. I hope to improve the diagram.<br /><p><br /></p><p><br /></p><p><br /></p><p> </p>George Jellisshttp://www.blogger.com/profile/14912766967103087963noreply@blogger.com0tag:blogger.com,1999:blog-9104469096692192283.post-58429817842824498462021-06-18T15:37:00.002+01:002021-06-22T17:48:09.569+01:00Onitiu Problem on 22x22 Boards<p>These diagrams are the results of a month's struggle to find a knight's tour with 90 degree rotational symmetry with the square numbers in a knight path (the red line). The tour on the circular board was completed first. Both tours use the same central pattern of knight moves. Both boards are of 22x22 = 484 cells. The yellow, blue and green paths are rotations of the red path. The circled cells mark where the coloured paths meet. These are at the points numbered 121, 242, 363, 484.</p><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-NCVILYDnqwM/YMy0k5IsywI/AAAAAAAAA24/I8F3QeTsQo4jmvacxKxB4-rZUdItGeFLACLcBGAsYHQ/s586/Onitiu%2B22%2BRound.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="586" data-original-width="575" height="320" src="https://1.bp.blogspot.com/-NCVILYDnqwM/YMy0k5IsywI/AAAAAAAAA24/I8F3QeTsQo4jmvacxKxB4-rZUdItGeFLACLcBGAsYHQ/s320/Onitiu%2B22%2BRound.jpg" /></a></div><div class="separator" style="clear: both; text-align: center;">Circular Board (radius 13)</div><div class="separator" style="clear: both; text-align: center;">Dots mark {0,13} and {5,12} moves from centre.</div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-hBQh4GIQacM/YMy05rx2AXI/AAAAAAAAA3A/TyUglUoHeeYEoaSuH7ftfc4vTMd-HUGzwCLcBGAsYHQ/s501/Onitiu%2B22%2BSquare.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="501" data-original-width="490" height="320" src="https://1.bp.blogspot.com/-hBQh4GIQacM/YMy05rx2AXI/AAAAAAAAA3A/TyUglUoHeeYEoaSuH7ftfc4vTMd-HUGzwCLcBGAsYHQ/s320/Onitiu%2B22%2BSquare.jpg" /></a></div><div class="separator" style="clear: both; text-align: center;">Square Board 22 by 22</div><div class="separator" style="clear: both; text-align: center;">With square numbers and their rotations noted.</div><div><br /></div>Addendum: <div>Here is a black and white version that shows the overall and its symmetry pattern more clearly.</div><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-XotM3zB-Sbs/YNIUAQlaFnI/AAAAAAAAA3I/FR5p_ScmsoYGBSApLoYY6dPCd_mvlPkLgCLcBGAsYHQ/s501/Onitiu%2B22x22%2BBW.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="501" data-original-width="490" height="320" src="https://1.bp.blogspot.com/-XotM3zB-Sbs/YNIUAQlaFnI/AAAAAAAAA3I/FR5p_ScmsoYGBSApLoYY6dPCd_mvlPkLgCLcBGAsYHQ/s320/Onitiu%2B22x22%2BBW.jpg" /></a></div><div class="separator" style="clear: both; text-align: center;">Black and White version.</div><br /><div><br /><p>These are the only solutions to the quaternary Onitiu problem I have found so far. Whether it is possible on smaller boards (of sides 10, 14, 18) is unknown. </p><p><br /></p></div>George Jellisshttp://www.blogger.com/profile/14912766967103087963noreply@blogger.com0tag:blogger.com,1999:blog-9104469096692192283.post-57382857976436002222021-05-28T18:09:00.002+01:002021-05-28T18:09:47.819+01:00Circular Boards Again<p> These new versions of the previous four tours keep strictly within the circle marked by the black dots. There are no cells whose corners go outside the circle. </p><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-tnSzmxXhRyU/YLEjNBXCXNI/AAAAAAAAA1k/S5oaI8Www8YvmsKh3vVdVYTAVqeu_7QqgCLcBGAsYHQ/s192/Circle%2BR5.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="192" data-original-width="192" src="https://1.bp.blogspot.com/-tnSzmxXhRyU/YLEjNBXCXNI/AAAAAAAAA1k/S5oaI8Www8YvmsKh3vVdVYTAVqeu_7QqgCLcBGAsYHQ/s0/Circle%2BR5.jpg" /></a></div><div class="separator" style="clear: both; text-align: center;">Radius 5</div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-fph059VoDOg/YLEjNJDBuYI/AAAAAAAAA1g/PJW5RKTrhTgioY6l908n5zdSunBd-9mGQCLcBGAsYHQ/s256/Circle%2BRroot-50.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="256" data-original-width="256" src="https://1.bp.blogspot.com/-fph059VoDOg/YLEjNJDBuYI/AAAAAAAAA1g/PJW5RKTrhTgioY6l908n5zdSunBd-9mGQCLcBGAsYHQ/s0/Circle%2BRroot-50.jpg" /></a></div><div class="separator" style="clear: both; text-align: center;">Radius root-50</div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-Ldf_9ZUYFfE/YLEjNHl7CWI/AAAAAAAAA1c/1qWjPeEbK2QjG6oW_dEuTbYNq4BxdIbZACLcBGAsYHQ/s288/Circle%2BRroot-65.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="288" data-original-width="288" src="https://1.bp.blogspot.com/-Ldf_9ZUYFfE/YLEjNHl7CWI/AAAAAAAAA1c/1qWjPeEbK2QjG6oW_dEuTbYNq4BxdIbZACLcBGAsYHQ/s0/Circle%2BRroot-65.jpg" /></a></div><div class="separator" style="clear: both; text-align: center;">Radius root-65</div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-vGnqXnRfKJE/YLEjNlEvR-I/AAAAAAAAA1o/7YZRu1co7lkyeWQ2YDwBJuf_llKCLQM3ACLcBGAsYHQ/s344/CircleRroot85.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="328" data-original-width="344" src="https://1.bp.blogspot.com/-vGnqXnRfKJE/YLEjNlEvR-I/AAAAAAAAA1o/7YZRu1co7lkyeWQ2YDwBJuf_llKCLQM3ACLcBGAsYHQ/s320/CircleRroot85.jpg" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;">Radius root-85</div><br /><p>As before the fourth shows 180 degree rotation, the others 90 degree.</p><p><br /></p>George Jellisshttp://www.blogger.com/profile/14912766967103087963noreply@blogger.com0tag:blogger.com,1999:blog-9104469096692192283.post-58822574226271711782021-05-27T09:19:00.002+01:002021-05-27T09:19:38.135+01:00Smaller Circular Boards<p>Here are four tours on the smallest circular boards, with at least 12 fixed corners on the circle. These show boards with radii 5, root-50, root-65, root-85 corresponding to the smallest double-pattern fixed-distance leapers. The first three tours show 90 degree rotational symmetry but the fourth only has 180 degree rotation, due to the quarter-board containing an even number of cells (64). </p><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-eiD5C2EZWVc/YK9Sr2SisLI/AAAAAAAAA1E/VdNIWZ6vBDc_VDs8SKlDKGfiuI0y5O5jwCLcBGAsYHQ/s192/Radius%2B5.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="192" data-original-width="192" src="https://1.bp.blogspot.com/-eiD5C2EZWVc/YK9Sr2SisLI/AAAAAAAAA1E/VdNIWZ6vBDc_VDs8SKlDKGfiuI0y5O5jwCLcBGAsYHQ/s0/Radius%2B5.jpg" /></a></div><div class="separator" style="clear: both; text-align: center;">Radius 5</div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-k1IhVOvOQb0/YK9SrwVP7yI/AAAAAAAAA08/t6F7pZHlGqgWD1rXzckG4G0zK-FUJXmcwCLcBGAsYHQ/s256/Radius%2Broot-50.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="256" data-original-width="256" src="https://1.bp.blogspot.com/-k1IhVOvOQb0/YK9SrwVP7yI/AAAAAAAAA08/t6F7pZHlGqgWD1rXzckG4G0zK-FUJXmcwCLcBGAsYHQ/s0/Radius%2Broot-50.jpg" /></a></div><div class="separator" style="clear: both; text-align: center;">Radius root-50</div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-y1mSptwlAEM/YK9Srwcf1vI/AAAAAAAAA04/mnQLpRM8iZE9sC3hNLxe6j-RDlc2OadZACLcBGAsYHQ/s288/Radius%2Broot-65.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="288" data-original-width="288" src="https://1.bp.blogspot.com/-y1mSptwlAEM/YK9Srwcf1vI/AAAAAAAAA04/mnQLpRM8iZE9sC3hNLxe6j-RDlc2OadZACLcBGAsYHQ/s0/Radius%2Broot-65.jpg" /></a></div><div class="separator" style="clear: both; text-align: center;">Radius root-65</div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-IGbBNnVUpaw/YK9SsQYYqwI/AAAAAAAAA1A/wUrC1hL4FMIvqKCjZ6ogXX2rAKTM4SpXgCLcBGAsYHQ/s344/Radius%2Broot-85.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="328" data-original-width="344" src="https://1.bp.blogspot.com/-IGbBNnVUpaw/YK9SsQYYqwI/AAAAAAAAA1A/wUrC1hL4FMIvqKCjZ6ogXX2rAKTM4SpXgCLcBGAsYHQ/s320/Radius%2Broot-85.jpg" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;">Radius root-85</div><div><br /></div>There are 17 other two-pattern fixed-distance leapers to consider before the first three-pattern leaper of this type (root 325) is encountered, as used in the large tour previously shown. I doubt if I will get round to constructing examples of all of these! <div><br /></div><div>There is a certain arbitrariness concerning which cells, whose corners go beyond the circle, should be included or not.</div><div><br /><div><br /><p><br /></p><p><br /></p><p><br /></p><p><br /></p></div></div>George Jellisshttp://www.blogger.com/profile/14912766967103087963noreply@blogger.com0tag:blogger.com,1999:blog-9104469096692192283.post-48261466863379538262021-05-24T08:55:00.000+01:002021-05-24T08:55:30.456+01:00Circular Board<p> This is a knight's tour with 90 degree rotational symmetry on a circular board of 964 cells. The 24 black dots on the circle are at exact distance root-325 (approx 18.03) from the centre point. The four corner cells that make possible the birotary symmetry by ensuring the quarter-board has an odd number of cells (241) extend slightly beyond this circle to root-338 (approx 18.38).</p><p><br /></p><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-GDZjSbj3YH8/YKtbpVnVLLI/AAAAAAAAA0w/tKmWMWIeMyAbkWEdjuOGWGqB1vz-S3nQACLcBGAsYHQ/s608/Circular%2BBoard.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="608" data-original-width="608" height="320" src="https://1.bp.blogspot.com/-GDZjSbj3YH8/YKtbpVnVLLI/AAAAAAAAA0w/tKmWMWIeMyAbkWEdjuOGWGqB1vz-S3nQACLcBGAsYHQ/s320/Circular%2BBoard.jpg" /></a></div><br /><p><br /></p><p>The black dots are at points with coordinates {1, 18}, {6, 17} and {10,15} relative to the centre, while the corner points are at {13, 13}. </p><p><br /></p>George Jellisshttp://www.blogger.com/profile/14912766967103087963noreply@blogger.com0tag:blogger.com,1999:blog-9104469096692192283.post-67564890926896541692021-05-23T15:58:00.000+01:002021-05-23T15:58:01.339+01:00Antelope Pettern<p>Here is another to add to the list. I hope I've got it right. It was difficult to be sure a move wasn't missed. </p><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-7uXu_9vq07w/YKptWzOQLWI/AAAAAAAAA0o/rixIW5uCcs83o6vnzIeQZzbIIl2wwLGLwCLcBGAsYHQ/s520/Antelope%2BPattern.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="432" data-original-width="520" src="https://1.bp.blogspot.com/-7uXu_9vq07w/YKptWzOQLWI/AAAAAAAAA0o/rixIW5uCcs83o6vnzIeQZzbIIl2wwLGLwCLcBGAsYHQ/s320/Antelope%2BPattern.jpg" width="320" /></a></div><br /><p><br /></p><p>I suppose this is a sort of "painting by numbers".</p><p><br /></p>George Jellisshttp://www.blogger.com/profile/14912766967103087963noreply@blogger.com0tag:blogger.com,1999:blog-9104469096692192283.post-82343877583384058332021-05-07T23:15:00.003+01:002021-05-07T23:15:34.969+01:00Leaper Move Patterns<p> I was inspired to produce these patterns as a contribution to the #Maydala season on Twitter for mathematics and art. They use rainbow-sequence colouring in place of the numbers of moves from the centre to other cells. </p><p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-L60d1C27BvE/YJW5r12c5NI/AAAAAAAAA0I/0JN3E519qGUnHJWY1qpKblClHV52HIZOgCLcBGAsYHQ/s573/Zebra%2BPattern.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="496" data-original-width="573" src="https://1.bp.blogspot.com/-L60d1C27BvE/YJW5r12c5NI/AAAAAAAAA0I/0JN3E519qGUnHJWY1qpKblClHV52HIZOgCLcBGAsYHQ/s320/Zebra%2BPattern.jpg" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;">Zebra {2,3}</div><p><br /></p><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-YzLhccxSofU/YJW54G63W9I/AAAAAAAAA0Q/QG2FDntb7eQ_zmp4qTUtc5whQeaSb9HhgCLcBGAsYHQ/s520/Giraffe%2BPattern.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="432" data-original-width="520" src="https://1.bp.blogspot.com/-YzLhccxSofU/YJW54G63W9I/AAAAAAAAA0Q/QG2FDntb7eQ_zmp4qTUtc5whQeaSb9HhgCLcBGAsYHQ/s320/Giraffe%2BPattern.jpg" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;">Giraffe {1,4}</div><p><br /></p><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-IIsfT-cjMKk/YJW6BeIY9II/AAAAAAAAA0Y/AAgLlPcmvC0Ep8b7rtbXB4RCmnDodIBiACLcBGAsYHQ/s448/Knight%2BPattern.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="368" data-original-width="448" src="https://1.bp.blogspot.com/-IIsfT-cjMKk/YJW6BeIY9II/AAAAAAAAA0Y/AAgLlPcmvC0Ep8b7rtbXB4RCmnDodIBiACLcBGAsYHQ/s320/Knight%2BPattern.jpg" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;">Knight {1,2}</div><p><br /></p><p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-s8bUayp3GfI/YJW5yphXNJI/AAAAAAAAA0M/SEqo7-GyByMBn6CIiMemfT09gVLRvT9uQCLcBGAsYHQ/s320/Wazir%2BPattern.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="184" data-original-width="320" src="https://1.bp.blogspot.com/-s8bUayp3GfI/YJW5yphXNJI/AAAAAAAAA0M/SEqo7-GyByMBn6CIiMemfT09gVLRvT9uQCLcBGAsYHQ/s0/Wazir%2BPattern.jpg" /></a></div><div class="separator" style="clear: both; text-align: center;">Wazir {0,1}</div><p><br /></p><p> I suppose the Giraffe pattern could be cut off at the 9x9 size, if we stop when the first 8-move cells are reached.</p><p><br /></p>George Jellisshttp://www.blogger.com/profile/14912766967103087963noreply@blogger.com0tag:blogger.com,1999:blog-9104469096692192283.post-62310678026294752432021-05-04T14:25:00.002+01:002021-05-05T07:59:30.237+01:00Dawsonian Tours on Odd Boards<p> It occurred to me that I haven't seen any examples of Dawsonian tours on odd square boards, so I composed a set. Of course closed circuits using all the square numbers are not possible. Other formations have to be considered. Here simple up and down steps.</p><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-Zt62-LjbDdc/YJFK6dlQ5yI/AAAAAAAAAzk/tn082NJT4_Ukt575g8l_Bbu3gvxo8nmiwCLcBGAsYHQ/s181/Dawsonian%2B7x7.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="181" data-original-width="171" src="https://1.bp.blogspot.com/-Zt62-LjbDdc/YJFK6dlQ5yI/AAAAAAAAAzk/tn082NJT4_Ukt575g8l_Bbu3gvxo8nmiwCLcBGAsYHQ/s0/Dawsonian%2B7x7.jpg" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-yZ3Whny4XEQ/YJFK6QTsxII/AAAAAAAAAzg/yio456_ib9AQMxEuUTM5WQWfI38Aqv05QCLcBGAsYHQ/s224/Dawsonian%2B9x9.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="224" data-original-width="213" src="https://1.bp.blogspot.com/-yZ3Whny4XEQ/YJFK6QTsxII/AAAAAAAAAzg/yio456_ib9AQMxEuUTM5WQWfI38Aqv05QCLcBGAsYHQ/s0/Dawsonian%2B9x9.jpg" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-x3Y875ep55Q/YJFK6nVEU8I/AAAAAAAAAzo/YbDXcJNpd40AhsNrDq5j1XHaWBSxQRqXQCLcBGAsYHQ/s266/Dawsonian%2B11x11.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="266" data-original-width="256" src="https://1.bp.blogspot.com/-x3Y875ep55Q/YJFK6nVEU8I/AAAAAAAAAzo/YbDXcJNpd40AhsNrDq5j1XHaWBSxQRqXQCLcBGAsYHQ/s0/Dawsonian%2B11x11.jpg" /></a></div><br /><p>A 5x5 knight solution is not possible but there is a simple wazir tour with the squares in this formation with a uniquely determined path, and also emperor solutions using knight and wazir moves.</p><p> Diagrams:</p><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-izn27qEn1z4/YJJB_T_NMBI/AAAAAAAAAz4/6tJRVGJyWyQ5CGGb1e3Tt7QnxyWatJWfQCLcBGAsYHQ/s139/Dawsonian%2BEmperor.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="139" data-original-width="128" src="https://1.bp.blogspot.com/-izn27qEn1z4/YJJB_T_NMBI/AAAAAAAAAz4/6tJRVGJyWyQ5CGGb1e3Tt7QnxyWatJWfQCLcBGAsYHQ/s0/Dawsonian%2BEmperor.jpg" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-L6WWK3ChpHw/YJJCHF2sKoI/AAAAAAAAAz8/-hgUWwK5UNoOXSkLSZZdXPKRq96tNkW9QCLcBGAsYHQ/s139/Dawsonian%2BWazir.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="139" data-original-width="128" src="https://1.bp.blogspot.com/-L6WWK3ChpHw/YJJCHF2sKoI/AAAAAAAAAz8/-hgUWwK5UNoOXSkLSZZdXPKRq96tNkW9QCLcBGAsYHQ/s0/Dawsonian%2BWazir.jpg" /></a></div><br /><p><br /></p>George Jellisshttp://www.blogger.com/profile/14912766967103087963noreply@blogger.com0