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.