Magic Wizard Problem

(a) 10x10 magic square formed of two orthogonal latin squares. Found by Ernest Tilden Parker 1960. (See Frontispiece of Rouse Ball's Mathematical Recreations 12th and later editions)
(b) Permuted ranks and files so diagonal is 00, 01, ..., 08, 09.

(1) Find a permute that has the minimum number of different move types
between successive cells (00-01, 01-02, ..., 98-99).

Is a Magic Wizard tour possible? (No moves crossing an intermediate cell)
The 49-50 and 79-80 moves would have to be wazir steps {0,1}. Moves like {2,2}, {2,4}, {2,6}, {3,3}, {3,6} crossing other cell centres would be avoided. Maybe keep to moves of type {1,n} and {n,n+1}? i.e. just "off" being lateral or diagonal.

In the first try below the following 13 moves fail:16-17, 21-22, 33-34, 35-36, 51-52, 53-54, 64-65, 67-68, 88-89, 91-92, 93-94, 97-98, 99-00.

Is a Magic Queen tour possible? (all moves lateral or diagonal) Probably not.
Is a Magic Witch tour possible? (all moves crossing an intermediate cell)

Missing Magic Empress Tour

An email from Jaime Gutierrez Salazar 21 Dec 2016 enclosed the following image:

This magic Empress tour (or Magic Two-Knight Tour if you prefer) was apparently included in the collections of magic knight tours published by General Parmentier in the 1890s, but it is missing from the page of such tours on the KTN website. Whether this was in the H. J. R. Murray 1951 manuscript and I missed it out I'm not sure. There are subtle differences in the tours.

There is also another that I reported here in 2015:
http://jeepyjay.blogspot.co.uk/2015/08/magic-empress-tour.html
So I must get round to updating the page.

Note added 19 Jan 2017: I now realise that the above array from Jaime is the reverse numbering of tour O as listed on the Knight's Tour Notes web-page.

Spectacular Trouble

One support arm of my glasses fell off this morning, so I had to take them along to the opticians (now Boots who have taken over Dolland and Aitchison). They couldn't repair it so I have had to pay £25 to leave the glasses with them. Apparently they need to find another frame of the same type.

I asked about re-using one of my old frames and was told it would cost £95 for reading glasses and £250 for varifocals, but apparently it would be cheaper just to buy a new pair! How does this make any sort of sense? It doesn't, as far as I can see, without my glasses.

This is a policy that will result in all unused frames ending up in landfill.

I'm not keen on varifocals, and always had bifocals before the present pair. However it seems they don't make bifocals any more. This doesn't seem like progress to me. Probably they sacked the expert bifocal makers and replaced them by machines.

When I had an eye test recently the optometrician said he would recommend I go in for an operation to remove a cataract in the left eye.  Apparently this means replacing the material in the lens with something else, which can be done in a way that is minimally invasive.

I'm getting grumpy at the way everything seems to be breaking down at present.

Figured Tour with Legendre Numbers

At the August Bank Holiday Rapid-Play at the Hastings Chess Club I somehow did well enough to win a £5 book voucher. I spent it on "Professor Stewart's Incredible Numbers". On page 46 there is an account of numbers that cannot be expressed as a sum of three squares. A-M. Legendre found the formula (4^k)(8n + 1) for all such numbers. Here is a figured tour that I just constructed this afternoon showing all the Legendre Numbers less than 65 on the diagonals.

They consist of the eight odd numbers that are one less than a multiple of eight (that is 8n - 1) and the two even numbers 28 and 60. They occur in pairs that differ by 32 thus allowing a symmetric arrangement in a symmetric knight tour. The reader may like to try constructing a similar tour with the odd numbers in a different sequence or the even numbers on different cells.

Atheism is Not a Religion!

On Friday 2 December I went to London to attend two meetings held at the University College London (UCL) Institute of Archaeology, 31-34 Gordon Square, on Unbelief and Atheism.

The meetings were sponsored by the Templeton Foundation, which automatically makes me distrustful of the speakers, since its purpose is promote supposed scientific study of religion.

The first part was an "SSNB Roundtable: Who Cares about Unbelief?" SSNB stands for "Scientific Study of Nonreligious Belief". The most coherent of the speakers to my way of thinking was Dr Judith Everington from University of Warwick. In the Q&A session at the end I tried to point out that even people with Religious beliefs also have a background Nonreligious Worldview. It is the factual knowledge they have of the real world, that they rely upon to conduct their everyday life. It is what they are left with if  they lose their religion. So there is no great mystery about it.

The second part was the "NSRN Annual Lecture: Is Atheism a Religion?" A lecture, with three speakers. The third speaker, Prof Chris French from Goldsmiths College, London, gave the straight answer No! And was quite amusing, but the other two speakers I found illogical.

The first speaker Jonathan Lansman, from Queens University Belfast, thought the subject was absurd, but mainly because he could not find coherent definitions of Religion and Atheism. He maintained that they are like the concept of Weeds which has no scientific meaning in botany, despite being useful in Gardening. With this sort of vagueness I don't see how it is possible for him to think clearly about anything. He thought however that there were some "natural types" like Water that did have a definite meaning. This type of terminology made me wonder if he might be a Creationist, since they think of animal species in this way.

The second speaker Miguel Farias from Coventry University, seemed to favour the proposition, but his arguments I now only vaguely recall, and bore little relevance to my own experience.

I recognised and spoke to Norman Bacrac from Conway Hall and David Pollock from the Rationalist Association and BHA. Rupert Sheldrake was also pointed out to me in the audience but I didn't get into conversation with him.

Dramatic Cloudscape this Afternoon.

I went to the Post Office in London Road this afternoon to get envelopes and stamps, and post my entry for the Hastings Chess Congress. I've only entered the Christmas and New Year Morning tournaments this year. Hope I will be fit enough at the time.

Took this photo just after crossing the road to the promenade. I was struck by the unusual pastel shades in the clouds.

View From My Window

This was the view from one of my windows the day after the scaffolding was removed.

The fearsome looking spider has since gone. Previously all I could see were planks and poles.

Scaffolding Removed At Last

When I got back from my walk yesterday I was glad to see the scaffolders starting to remove the last layer of scaffolding from round the building. It had been there since March 2015, as reported here on 16 April 2015. Here is how the building looks now:

There remains some rubbish to be removed, but the contractor wants more money to remove it all, and passers by keep adding to the pile. At least I can now see out of the windows. I cleaned them yesterday afternoon.

Saturday Morning Walk

I noticed what looked like a large white tea-caddy on top of the structure on the pier yesterday evening as I walked along the prom. When I went to have a closer look it turned out to be a camera obscura. It was occupied by some giggling children so I didn't get to look inside behind the curtain.

My feet were feeling uncomfortable this morning. I've come to the conclusion it is due to the more flexible shoes I've been wearing, which may be giving my toes too much exercise. At least I hope that's what is the cause. It may also be that they retain too much heat. I had to cut short my visit to Leicester on 10th to 14th September because my feet had had enough of walking around.

This morning I got the urge to go for another walk, so put on an old pair of shoes with a hard sole, though they need a repair to the worn-down heels. I found a new path round the Hastings Museum which comes out at Cornwallis Gardens where the chess club is. The shoes proved satisfactory. I got as far as the top of West Hill, by the Castle, and came down the steep steps to the sea-front.

There weren't many people about at 8 am, but there were quite a lot of runners practicing circuits by the pier as I came back. I'm not tempted to join in. Must now repair the heels, as I had intended for at least a year, having bought the materials, but never got round to the task. A Saturday morning exercise may be a good idea on a regular basis.

Adventure of the Halogen Oven.

The halogen oven that I have had for a number of years broke down on the 4th of October, and there seemed no prospect of getting it repaired, since it is no longer in the Easy Life catalogue.

So I ordered a new one from Tesco Direct. They took the payment from my bank account and claimed the item had been dispatched on 12th of October, but it never arrived, though I stayed in all day waiting for it. According to their delivery system, which seems to be independent from Tesco, they attempted to deliver the goods but they were refused. I can only assume they tried to deliver it to the wrong address. No one contacted me by phone to say the goods were on their way, which is the usual procedure. They didn't offer to try to deliver it again to the right address. At any rate, after the exchange of a couple of emails they did refund the money to my bank account.

So yesterday I ordered a similar oven from the Argos catalogue and went along this morning to pay for and collect it myself. That all went OK. It has proved to be a bit larger than I thought, but I shall try it our this evening. I didn't take up the £9.99 offer of replacement if it breaks down within three years, since surely anything that does not last that long isn't worth replacing.

The Knight's Tour Work

It is time I gave an update on my work in putting my Knight's Tour Notes webpages into book form. This is a project I have been working at for years. The current proposal is to put all the material into a series of eight separate monographs, each of around 100 pages.

The subjects of the studies are roughly as follows. The numbering is provisional. 1. History of Tours. 2. Knight-move Geometry, 3.Small-Board Knight Tours. 4.Symmetry in Knight Tours. 5. Simple Linking of Pseudotours. 6.Shaped Board Tours. 7.Leaper Tours. 8. Magic Tours.

The first six titles are almost entirely about knight tours. The last two include tours using pieces with other moves. The magic tours end with magic squares that use up to four different types of move.

If my health holds out I aim to get this completed by the end of the year. I've given deadlines before and not met them, but this time I'm more confident.

A magic 5 by 7 rectangle tour

This 5 by 7 symmetric magic tour using six move types was composed by me this evening (12 Jun 2016 - our Queen's official 90th birthday) and is the best so far found on this board. The magic constants are 90 and 126 (i.e. 5×18 and 7×18). It uses six different types of move (my previous best was eight). {0,1}{0,2}{1,1}{12}{1,3}{1,4}

This was derived in part from the "arithmic" king tour on this board, but also used the "method of complementary differences" by Charles Planck, as described in Andrews (Magic Squares and Cubes 1917) p.257-262. The example by Planck shown there uses ten move types.

The Tree

It's along while since I posted a photo of the oak tree opposite my front door.

Strong contrasts of black and green in the sunlight yesterday.

Maximal Bergholtian Symmetry

Bergholtian symmetry is centrosymmetry of the type that passes twice through the centre,
and is possible only on boards with one side odd and the other singly even (e.g. 6×7 or 5×10).

When numbered from the ends of the central cross it has the property that diametrally opposite numbers add to a constant sum. This is in contrast to the more common tours with Eulerian symmetry in which opposite numbers have a constant difference.

On other boards it is possible to construct tours which have partial Bergholtian symmetry, not all of the pairs adding to the constant. An example of this was sent to me by Prof D. E. Knuth, which inspired the following examples, which I think show the maximum amount of Bergholtian symmetry on the 8×8 and 10×10 boards.

The dots which, form two rhombs, mark the ends of the symmetric parts. One rhomb is connected in the opposite way to the other and this is the only asymmetry. If one of the rhombs is reversed this results in a tour with Eulerian symmetry.

The geometrical forms:

The arithmetical forms:

Diametrally opposite numbers add to 62
with the exception of the three pairs in bold.

47 52 45 26 05 24 21 28
44 03 48 53 64 27 06 23
51 46 01 04 25 22 29 20
02 43 54 49 30 63 12 07
55 50 31 62 13 08 19 60
42 33 40 37 58 61 16 11
39 56 35 32 09 14 59 18
34 41 38 57 36 17 10 15

crosslinks c6-e7-f5-d4-f3 and b3-d2-c4-e5-g6

03 14 53 16 55 34 37 40
52 17 02 13 42 39 56 35
31 04 15 54 33 36 41 38
18 51 32 01 12 43 62 57
05 30 19 50 61 64 11 44
24 21 26 29 08 47 58 63
27 06 23 20 49 60 45 10
22 25 28 07 46 09 48 59

crosslinks: d5-f4-h3-g5-e4 and d3-b4-a6-c5-e6

Connecting instead 01-48, 00-47 or 51-98, 50-97
results in a tour with Eulerian symmetry

10×10 example. This linkage will not work on the 8×8 board,
since moves through two corners are prevented.

17 22 89 26 45 38 87 34 31 36
90 25 18 01 88 27 44 37 86 33
21 16 23 46 39 48 03 32 35 30
24 91 00 19 02 43 28 57 04 85
15 20 93 40 47 56 49 84 29 06
92 69 14 99 42 51 58 05 78 83
13 94 41 70 55 96 79 50 07 74
68 63 66 95 98 59 52 75 82 77
65 12 61 54 71 10 97 80 73 08
62 67 64 11 60 53 72 09 76 81

Diametrally opposite numbers add to 98 with the exception of the three pairs in bold.
00 underlined indicates 100.

crosslinks: 01-00-99-98-97 and 47-48-49-50-51

Symmetric Tours of Squares and Diamonds

I reported starting a count of these tours a while ago. The result found was 274 tours in all of which 82 are of the double halfboard type. The first figure may still be short, and a further check is needed.

The second figure is definitely correct, checked by an independent method. There are three types of double halfboard tour according as the separation between the two links that connect the halves is 1, 3 or 5 cells. The numbers of these types are respectively 26, 28 and 28 adding to 82.

I'm continuing to work on my book on tours. At present in is in two parts each of about 250 pages. The first part being on History and the second on Theory.
 

Batten Down the Hatches!

It was disappointing to find that my chess grade has only gone up to 90. I was hoping it might get back to 100 as I had put in considerable effort and achieved some good results. It seems it is easier to slide two yards back down the greasy pole than to climb one yard up.

This afternoon I received a copy of a paper I had requested from a journal in Canada only a few days ago. On the other hand a cheque I sent to a company in Kent two weeks ago disappeared in the post, and I had to report the details to the bank to ensure it is not passed for payment.

When I went for an evening walk along the front yesterday evening for exercise and to see the sunset, I happened to see that a shop selling carpets was open, in the parade beneath the Marina building. This reminded me that I needed a mat to go under my chair to protect the fitted carpet from wear. I bought a nice colourful rug, about 1 by 2 metres, for £25. This has brightened up the room considerably.

It was windy out today and apparently the winds are going to get stronger during my 76th birthday on Monday, so it doesn't look as though I will be going out on any trip as I had hoped. Time to batten down the hatches and try to get some work done on my books.

Hastings Chess Congress

The Hastings Chess Congress occupied most of my time from 28 December to 5 January, since I entered all four of the supporting events. Just in the lowest graded sections of course, not the Masters! Scored 3/5 in the Xmas AM event (winning £15 grading prize), but only 1/5 in the Xmas PM event. Then in the Weekend event scored 3/5 again, against stronger players (winning £25 grading prize). Only managed 1.5/4 in the New Year event. 

These results continue my improved play over the past six months, with similar 3/5 results at Thanet in August and Bournemouth at the end of October. I'm also doing well in the internal Hastings Chess Club Rush Cup event. I hope all this will restore my ECF grade to near 100 again when the new gradings are announced later this month.