Knight's Tours with Direct Binary Symmetry

Tours showing axial symmetry are perhaps unfamiliar, since such symmetry is not possible on the orthodox chessboard. By direct symmetry we mean a pattern having just one axis of symmetry, being unaltered by reflection in that axis, or rotation about that axis. There are two types: Murraian with cells on the axis and Sulian without cells on the axis.


Murraian Symmetry

Closed tours with binary symmetry that have cells on the axis of symmetry I call Murraian since H. J. R. Murray made a special study of this type with diagonal axis in his 1942 manuscript. The axis must contain just two cells and can be lateral or diagonal. A Murraian tour can be regarded (in two ways) as an axisymmetric open tour with two extra moves connecting the ends to a second point on the axis.

LATERAL AXIS

I have just five examples of this little studied type. The 10-cell board tour is the smallest possible (O. E. Vinje 1949). The 12-cell example was found in the study of 12-cell closed tours by Jelliss and Marlow (1995). The 44 and 74 cell examples are by Murray (1942). The 52-cell example (with the cross-shaped hole) is from Zurcher Illustrierte 27 June 1930.



DIAGONAL AXIS

Murray comments on ths type: "The axis is the band of cells lying along the diagonal of a square. Only two cells can be used on the diagonal, and it will be found necessary to omit an even number of cells in addition. If the square contains n cells in a side, the minimum number of cells which must be removed, including those on the diagonal is 2n−4 if n is even and 2n−5 if n is odd. In the diagrams the dexter diagonal is taken as the axis of symmetry."

The smallest boards showing a closed tour with diagonal symmetry are of 12 cells, all of Murraian type with two cells on a diagonal. There are four 4×4 (all with a hole) and ten 5×5 (four with a hole). Only the first 4×4 tour here was shown by Murray in his study of this type of symmetry, the rest were found by Jelliss and Marlow (1995):



All the larger examples that follow are from the study by Murray (1942). One of 20 cells, two of 28 cells, two of 40 cells.



Two of 52 cells, one of 68, one of 84. (He also gave two other diagonal examples of 52 cells on the 8 by 8 board, but they omit cells that result in boards that are not wazir-connected, so are not included here.)



Finally two larger tours of 104 and 124 cells.




Sulian Symmetry

Closed tours with binary symmetry and no cells on the axis I call Sulian (after as-Suli who, circa 900, constructed an 8×8 tour incorporating a section with this type of symmetry). In Sulian symmetry the axis can only be lateral, not diagonal. The number of cells must be of the form 4n + 2, twice an odd number. This type of symmetry seems to me particularly fascinating, but very little work has been done on it until now. There are many other cases to consider besides the examples shown here.

10, 14, 18 and 22 cells

There is one 10-cell example. There are four 14-cell examples on shaped boards within 3×6 and 4×4 frames (the first 4×4 example was given by Murrray), two 18-cell examples within the 4×5 frame (the first example was given by Murray, inverted here), and 11 examples of 22 cells within a 4×6 frame.



26 cells

Here are 21 tours omitting two cells from the 4×7 frame.



And here are 28 tours on a cross-shaped board of 26 cells (Jelliss 2009) in batches of 12 and 16.

A. The 12 with right angles at a2 and a4:



B. The 16 with acute angle at a2. The last tour is the only case with acute angles at both a2 and a4.



30 cells

As-Suli's original axisymmetric tour was on the 4×8 board omitting two cells. The following charts show all possible tours of this type. The axis of symmetry can be the short or long median. There are 45 with axis along the short median. Here are the 15 with missing cells at bg1 (3 cases), cf1 (3 cases, the last of which is Suli's original example) and de1 (9 cases).



Here are the 30 cases that omit cells at the corners (ah1).

A.



B.



There are 47 tours with axis along the long median. Here are the 15 with missing cells in the edges: ad2 (3 tours), ad3 (2 tours), ad4 (10 tours).

C.



Here are the 32 tours with corner cells (ad1) missing, in two batches of 16.

D.



E.



34, 38, 46 and 50 cells

Three examples of 34 cells. The first one, on the 6×6 with 2 indentations, is from Zurcher Illustrierte 23 January 1931, collected by H. J. R. Murray (1942). The next two are on the 6×6 with 2-cell hole. The example with 38 cells is also from Zurcher Illustrierte 8 June 1934. The 6×8 with 2-cell hole (46 cells) is by Bergholt 1916. The 50-cell shape is from Zurcher Illustrierte 13 July 1934. I have joined it to one of the 34-cell tours to give the impression of a robot. The tour looks as if it may have been intended to resemble a human skeleton. Psychologists use patterns with this type of bilateral symmetry in the Rorschach inkblot test, because of its biological significance, which the human eye has evolved to recognise quickly.



62 cells

Here we consider boards formed by omitting two cells from the standard chessboard. First we show Sulian examples for each of the 7 ways of omitting two cells from the edge of the 8×8 board (one case has two examples). The second tour is after Naidu 1922. The fourth is by Kraitchik 1927. The fifth is from Murray 1942. The other five are my own (Jelliss 1996).



Next we show one Sulian example for each of the 7 positions when the two missing cells form one or two holes in the 8×8 board. (The cases with holes a knight move from the corner cells do not of course admit a tour.) The first three are my own work (Jelliss 1996). The fourth is after Hoffmann 1893. The fifth is after a German example of c.1860. The sixth is after Babu 1901. The seventh is from Murray (1942) but has been inverted.



The tours described as "after" Naidu, Babu, Hoffmann and a German example were originally published with the path diverted through the void cells to give either a near-symmetric open tour, or a pseudotour.

66, 70 and 78 cells

These concluding examples on larger boards were all collected by H. J. R Murray from German publications. (a) 66 cell tours from Denken und Raten 25 August 1929 and from Zurcher Illustrierte 28 August 1931. (b) 70 cells in 8x10 from Zurcher Illustrierte 22 December 1933 and from Denken und Raten 25 August 1929. (c) 78 cells in 8x10 from Denken und Raten 25 January 1931.



114 cells

This concluding example is another collected by Murray from Zurcher Illustrierte (7 July 1933), composer unknown.