The Onitiu Problem with Birotary Symmetry

 I've spent some time considering whether it might be possible to solve the Onitiu problem of constructing a knight tour with the square numbers in a knight chain but showing birotary symmetry (i.e. unchanged by 90 degree rotation) which is possible on boards of oddly even side (10, 14,18, 22, 26, etc). So far without success, though I can see no simple argument to prove such a tour impossible. 

Here is an Emperor (Knight + Wazir) tour that shows the general idea.

The red line indicates the square numbers. The other colours show its rotations. The black dots in the middle are the positions of 25, 50, 75 and 100 in this tour. It starts with 1 at d5. 

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Earlier today I had my second vaccination dose against the coronavirus.

  

The Onitiu Problem 24x24

 I have now solved the Onitiu Problem on the 24 by 24 board. This is the first case where there are six intersections of the circuit of squares (shown in red) with the circuit of complements (shown in blue). The six square numbers where this occurs are 1, 36, 196, 289, 324, 484, the common difference being 288 which is half of 24x24 = 576.

   Once again the squares and diamonds feature strongly but even so I found this a difficult task.

The Onitiu Problem on Larger Boards

 I have decided to publish my results for the 12x12, 14x14 and 16x16 boards here rather than go to all the trouble of writing the subject up for publication in a journal. I no longer have the patience for that frankly. These were solved in reverse order, the largest 16x16 first because I thought its division into 16 areas 4x4 would make the solution easier by use of the squares and diamonds method in each quarter, and this proved to be the case. 

12x12

16x16

14x14

I'm not sure why Blogger has put these images in the wrong order! 

In the 14x14 case the sequences of squares (red) and the sequence of complements (green) have no points of intersection, but in the 16x16 case there are two (16 and 144 differing by 128), while in the 12x12 case there are four (two pairs, 9 and 81, 49, and 121, with difference of 72).