Saturday, 9 October 2021

Another Onitiu 18x18 Attempt

 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.

Onitiu Problem 18x18 Quaternary Emperor Tour

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.

Wednesday, 22 September 2021

Onitiu Problem 30x30 Solution

 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). 

Solution to Onitiu Problem 30x30

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.

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.  


Friday, 17 September 2021

Locked Out of Twitter Again

 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! 

 

Thursday, 9 September 2021

Another Onitiu Solution

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.

Onitiu 26x26 Birotary Solution

 

Saturday, 28 August 2021

Enumeration of Grid Points

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".

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. 

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, ...

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.

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. 

Enumeration of Coordinate Points

It is interesting that the path appears never to cross itself. 

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. 

Enumeration using primes 2 and 5.




Tuesday, 20 July 2021

Locked out from Twitter

 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. 

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.

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.

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.

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. 

Thursday, 15 July 2021

Emperor Solutions to the Onitiu Problem

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.

10x10 Emperor tour
with 28 wazir moves

14x14 Emperor tour
with 40 wazir moves

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.

ADDENDUM (20 July 2021)

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? 

18x18 Emperor tour
with 8 wazir moves

This case is particularly interesting in having eight intersections instead of four. I hope to improve the diagram.