## 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.

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.