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.


 

Monday, 15 March 2021

Trees

 Some snaps of trees from recent walks round Crewe. 





The above four are from the Brooklands area between Ford Lane and Broad Street.


This, taken a few days before, is in front of the police station, library, and swimming baths, and on the corner of the grounds of the old bombed-out church.


Saturday, 27 February 2021

Dawsonian 12x12 Tour

 Following on from the 10x10 tours with square numbers in knight chains it suddenly occurred to me that the 12 squares on the 12x12 board could be arranged to show a 3-4-5 triangle.

3-4-5 triangle

It will be seen that my solution makes much use of squares and diamonds in the nine 4x4 areas that the board divides into. It may be improvable in this respect.


Thursday, 25 February 2021

Dawsonian 10 by 10 Tours

 These tours that I have been posting initially on Twitter are part of some work I am doing to put an updated page on Figured Tours on my Knight's Tour Notes pages. I found I had no examples of Dawsonian tours on larger boards. That is tours showing the square numbers in knight paths, preferably symmetric circuits. T. R. Dawson constructed a complete set of such tours on the 8x8 board.

Rectangle 1x4

Hexagon

Octagon

Constructing these tours makes a good puzzle for solving. Probably the larger board makes it a little easier to complete the paths than on the 8x8 board. My procedure is to start with the shorter sections 1-4, 4-9, 9-16 and so on, trying not to leave inaccessible unused cells. Fitting the last two segments 64-81, 81-100 are of course the tricky part of the problem. Sometimes one ends up with two loose ends that do not connect by a knight move.

Addendum: A fourth example to complete the set of symmetric convex polygons.

Rectangle 2x3


Tuesday, 9 February 2021

Tiles and Key Patterns

 In my Knight's Tour Notes on Wazir tours and in various other places I included the following diagram outlining the various ways of forming a frieze with vertical and horizontal moves. 

Wazir Paths

In Twitter I have been following a series of items by Tom Ruen on tiling with a wide range of various shapes. His most recent articles inspired me to put together the following two Greek Key Pattern borders. They are constructed in each case of a single rectangular (metasquare) tile pattern, in two colours and various different orientations.

Key Pattern 1

Key Pattern 2

This would make a good frame for a shaving mirror!