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.