Friday, 7 May 2021

Leaper Move Patterns

 I was inspired to produce these patterns as a contribution to the #Maydala season on Twitter for mathematics and art. They use rainbow-sequence colouring in place of the numbers of moves from the centre to other cells. 

Zebra {2,3}


Giraffe {1,4}


Knight {1,2}


Wazir {0,1}


 I suppose the Giraffe pattern could be cut off at the 9x9 size, if we stop when the first 8-move cells are reached.


Tuesday, 4 May 2021

Dawsonian Tours on Odd Boards

 It occurred to me that I haven't seen any examples of Dawsonian tours on odd square boards, so I composed a set. Of course closed circuits using all the square numbers are not possible. Other formations have to be considered. Here simple up and down steps.




A 5x5 knight solution is not possible but there is a simple wazir tour with the squares in this formation with a uniquely determined path, and also emperor solutions using knight and wazir moves.

 Diagrams:




Tuesday, 27 April 2021

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. 

***

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

  


Wednesday, 14 April 2021

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.



Tuesday, 6 April 2021

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). 


Wednesday, 31 March 2021

The Onitiu Problem

 In 1939 in Fairy Chess Review the Romanian problemist Valeriu Onitiu solved the difficult problem of constructing a symmetric chessboard knight tour with the squares in a knight circuit. It turned out that there is only one possible solution. 


I have been looking at this problem to see if it is solvable on larger boards. Initially I found it very difficult. However eventually on 27 March I found a solution on the 16x16 board (which can be split up into 4x4 sections which makes the solution by 'squares and diamonds method' feasible). This quickly led to solution on the 14x14 board, then on the 12x12 board, and finally tonight I solved the 10x10 case. I posted a copy on Twitter (misdated 2921). Here is a better presentation using coloured lines to mark the sequence of squares and the sequence of their complements.



I think I will reserve the other solutions for publication elsewhere in some chess or mathematics journal, where I can write it up in more detail. 


Demolition in Progress

 Back in 12 May 2019 and 4 May 2020 I showed photos of the Crewe town centre area where preparations were being made for demolition of many of the old shops including the British Home Stores (BHS) building on the corner of Queen Street and Victoria Street, and also the Clock tower which I had hoped could be saved. Here are some recent photos of the actual demolition in progress. 







Photos taken in January (2), February (1), and March (2) of  2021.

Most of the buildings have now been reduced to piles of rubble.