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.


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.




  

Friday, 18 June 2021

Onitiu Problem on 22x22 Boards

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.

Circular Board (radius 13)
Dots mark {0,13} and {5,12} moves from centre.

Square Board 22 by 22
With square numbers and their rotations noted.

Addendum: 
Here is a black and white version that shows the overall and its symmetry pattern more clearly.

Black and White version.


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. 


Friday, 28 May 2021

Circular Boards Again

 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. 

Radius 5

Radius root-50

Radius root-65

Radius root-85

As before the fourth shows 180 degree rotation, the others 90 degree.


Thursday, 27 May 2021

Smaller Circular Boards

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). 

Radius 5

Radius root-50

Radius root-65

Radius root-85

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! 

There is a certain arbitrariness concerning which cells, whose corners go beyond the circle, should be included or not.






Monday, 24 May 2021

Circular Board

 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).




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}. 


Sunday, 23 May 2021

Antelope Pettern

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.  



I suppose this is a sort of "painting by numbers".