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.


 

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!



Magic King Tours in Colour

 I've been having some exchanges on Twitter with people interested in mathematical art, or art based on geometrical patterns. Some of the examples reminded me of the diagonally magic king tours I studied with Tom Marlow a few years ago. Here is a selection from them with areas coloured in. 

4 magic king tours

This was an exercise just to see what they would look like. Rather gaudy colours. 

This was done on my 81st birthday yesterday.


Tuesday, 2 February 2021

A 9x12 Tour and a Group of 10x11 Tours

 The following 9x12 tour was constructed quickly as a sort of doodle. 

9x12

This design was the basis for the following tours of the 10x11 board, showing the different types of symmetry possible on this board. The Bergholtian tour was the first constructed. The others were derived from this by simple transformations. 
 
10x11 Open

10x11 Eulerian

10x11 Bergholtian

10x11 Axial

The Bergholtian tour passes twice through the centre. The Eulerian tour circles round the centre. The Open tour passes once through the centre. These three all have 180 degree rotational symmetry.

Wednesday, 27 January 2021

An Open Tour 20x21 with Crossing Zigzags

This is a tour I have been trying to construct since before the closed examples previously shown. I have had difficulty connecting it up round the edges without spoiling the central pattern. The darker lines in the first diagram show the parts of the basic zigzag that have been preserved. The other paths across the zigzag can take various forms.

Zigzag pattern


20x21 Open Symmetric Tour

Here is a closed tour using a similar crossing zigzags plan that I constructed the next day. The connections round the edges worked out much more easily than for the open tour. 

Crossing Zigzags

I have emphasised the two moves through the centre point.


Thursday, 14 January 2021

Some 20x21 tours

 Here are some closed knight tours on the 20x21 board for the new year. They all show 180 degree rotational symmetry of Bergholtian type (i.e. passing twice through the centre. Eulerian symmetry (going round the centre) is impossible on this board (or any board doubly even by odd). 

I first show a pattern on which the first tour was based. The spots indicate cells where no knight move is possible. 

20x21 array


20x21 tour


20x21 tour improved


20x21 tour with "Fair Isle" pattern


A Happy New Year, I hope.


Friday, 1 January 2021

More 18x18 Birotary Tours

 Here are some more 18x18 knight tours with 90 degree rotational symmetry. I've also been showing these on Twitter, where they have received some appreciation. 

knots,or diabolos

moths

moths

 The two lower diagrams maybe give an impression of moths around a fire.