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.


Sunday, 27 December 2020

30x30 Ribbon Tour

 This is about the largest board I can draw a tour on, and I've had to reduce the scale as compared with the previous tours. I'm not sure if I've reduced the links, shown in black, to a minimum. This board like the 6x6 and 18x18 allows 90 degree rotational symmetry.




Tuesday, 22 December 2020

12x12 Ribbon Tour

This 12x12 knight tour with 180 degree rotational symmetry completes the set of ribbon tours constructed over the last few days, the others being on boards 18x18 and 24x24. I've also been showing them on Twitter. 



Also shown here is the well known 6x6 tour of similar design. This like the 18x18 tour has 90 degree rotational symmetry.


Monday, 21 December 2020

A 24x24 Ribbon Tour

This tour solves the same task as on the 18x18 case published previously, but proved much more difficult to achieve. Given the diagonal braids shown in blue, and putting in the natural connections in each 6x6 area, similar to the 6x6 tour with the same diagonal pattern, results in a pseudotour of twelve separate circuits (4 of 14 cells, 4 of 20, 4 of 60 and 2 of 100, giving the total 576 = 24x24). 



The task is then to put in the minimum number of links to join the circuits up into a single tour and to ensure that it is a symmetric tour, which on this size of board means 180 degree rotational symmetry (which I call rotary symmetry). This is tricky since some of the circuits are themselves symmetric. 

 

Sunday, 20 December 2020

More on diagonal braids

 It is possible, as shown here, to put an extra set of obtuse diagonal braids between those used in my 18x18 tour, to cover any size area, but they cannot be incorporated in a tour since they leave isolated cells uncovered at intervals of {0,3} moves, as if the whole board is composed of centreless 3x3 boards. All other cells are used. 




Knight's Tour 18x18

This is a tour I constructed yesterday, partly inspired by some work by Robert Bosch on twitter, showing patterns on 24x24 boards. This is a closed tour on the 18x18 board which has 90 degree rotational symmetry (birotary symmetry as I call it). It features complete diagonal braids of oblique angle type. It is based on the familiar 6x6 tour of this type.