Catching Up on Knight Tours

 The following are links to various work on Knight Tours that have been sent to me while I was in hospital or recovering.

Dtego Armin Montes is using knight tours in his art works

diego · armí n { conceptual + interdisciplinary

Grayson Blanchard  has developed a site to show knight's tour puzzles

Knight's Tour by gs0n

Ben Schulz sent this knight tour with latin square that is shown on a site by Dan Thomasson and due to someobnne called Paws 

Knight's Tour Latin Squares I - Dan Thomasson

Donald Knuth was able to answer this question a follows:

I investigated the "latin knight's tour problem". It turns out that exactly

10 closed tours do the job on a 6x6. One of them is 00 21 14 45 42 23 15 50 01 22 13 44 02 55 20 43 24 41 51 34 53 30 05 12 54 03 32 11 40 25 33 52 35 04 31 10 and there are seven others obtained by rotation and reflection. The other two come from one of the solutions with 90-degree symmetry, namely the one with "12 slants"! Very pretty. Surprisingly, however, there are NO such tours on an 8x8 board. (At least, that's what my computer tells me, and I used essentially the same program for 8x8 as for 6x6.)

Ben Schultz sent colourful diagrams of the ten cases but I have not been able to reproduce the image here. 

William Walkington wrote back in March to point out that the first 16x16 tour in the Split and Fix section of the KTN website is not in fact magic. Some ranks and files have other totals.

16 by 16 Magic Knight's Tours

This brings me just about up to date.

In Hospital

 On 24 March I went into Leighton Hospital for an operation to remove stones from my bladder. I   expected to be out in a couple of days, but was kept there for two weeks. They successfully lasered one of the stones.

After coming out of hospital I had a period where I could hardly eat without throwing up, but gradually this has improved until I now enjoy my food once again.

I've also had trouble seeing clearly.. This was partly due to being given antihistamine in hospital. This is taking a  long while to clear up. .

The 19 ay date proved to be another pre-op assessmant. A date for the operation has not been set. I have a CT scan scheduled for11 June.

Update: I went into hospital for an operation to remove the second stone on 15 September and came out on the 17th. This seems to have been successful.

On the 22 September I was referred to the Ophthalmology department, and they have diagnosed macular degeneration in my left eye.  It seems there is nothing that can be done to repair this. 

I am having to use a magnifying glass to read anything., including on screen.

Algebraic Numbers

For my Numberlore book  I've been looking into the subject of Algebraic numbers. These are numbers that are the (real) solutions of algebraic equations with integer or rational coefficients. They include all rational numbers and expressions involving nth roots (square roots, cube roots etc), known as radicals. Thee are sufficient to solve all equations up to the fourth degree.

They do not include transcendental numbers like the circular constant, pi, or the exponential constant, e, which require infinite series

There are, however, further algebraic numbers between radicals and transcendentals. Little seems to have been written about them. They are solutions of quintic or higher equations and are known as ultra-radicals. One such is the solution to x^5 - x = 1. which is 1.1673... to four places of decimals. This and similar numbers were studied by a Swedish mathematician, Erland Bring (1736 - 1798). This information is from Wickipedia. I would be interested to know more.

 

In the News

 I found this Polar map projection among my History files. I copied it from somewhere years ago. It shows the location of Greenland, which is in the news, very clearly.

The quality of videos on You Tube seems to have deteriorated considerably lately, due to inferior productions based on Artificial Intelligence, and the automatic generation of mis-spelt captions, which often translate the same word or name in different ways in the same article. A lot of them also have the same didactic style of presentation.

We had a bit of snow on 5th January and since then it has been very cold here in Crewe. However the two named storms didn't reach this far north. Here's a photo I took from my front porch.