Closed tours showing axial symmetry are perhaps unfamiliar, since such symmetry is not possible on the orthodox chessboard. By axial symmetry (also known as 'direct binary symmetry') we mean a pattern having just one axis of symmetry, being unaltered by reflection in that axis, or rotation about that axis. There is a separate web-page on symmetric open tours.
There are two types of axial closed tour:
Murraian with cells on the axis, named after H. J. R. Murray who made a special study of this type in his 1942 manuscript. The axis must contain just two cells and can be diagonal or lateral. A Murraian tour can be regarded (in two ways) as an axially symmetric open tour with two extra moves connecting the ends to a second point on the axis.
Sulian without cells on the axis., named after As-Suli who c.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. This type of symmetry seems to me particularly fascinating, but very little work was done on it until my own reported here. Corresponding cells on opposite sides of the axis are of opposite colours if the board is chequered, so the paths of knight moves leading from one to the other must be of an odd number of moves 2×h + 1. The whole tour must therefore cover a board of 4×h - 2 cells, twice an odd number. Sulian tours cannot exist on boards where the number of cells is a multiple of 4, nor on any square boards without holes.
For axial tours on boards of 10, 12, 14 cells see the section on Smallest Boards.
Back to topMurray 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The diagrams here are all from the unpublished 1942 manuscript by H. J. R. Murray, among his papers at the Bodleian Library, Oxford University.
20 cells. Board in 5×5 frame with 5 cells omitted, including one hole.
28 cells. Board 6×6 with 8 cells omitted.
40 cells. Board 7×7 with 9 cells omitted.
52 cells Board 8×8 with 12 cells omitted..
(Murray 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.)
68 cells. Board 9×9 with 13 cells omitted.
84 cells. Board 10×10 with 16 cells omitted.
104 cells. Board 11×11 with 17 cells omitted.
124 cells. Board 12×12 with 20 cells omitted. A final Murraian diagonal example.
44 cells. 7×7 frame with 5 cells omitted. (Murray 1942).
54 cells. 9×9 frame with with 29 cells omitted. This 52-cell cross-pattern example was collected by Murray from Zurcher Illustrierte 27 Jun 1930.
56 cells. 7×9 frame with 7 cells omitted. Example by Murray (1942).
58 cells. 7×9 frame with 5 cells omitted. Example by Murray.
74 cells. 9×9 frame with 7 cells omitted. (Murray 1942).
18 cells. There are two with axial symmetry of Sulian type.
22 cells. There are 11 tours with Sulian symmetry within a 4×6 area. The third is in Murray.
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 16 and 12.
A. 16 with acute angle at a2.
The final tour is the only case with acute angles at both a2 and a4.
B. The 12 with right angles at a2 and a4:
30 cells. 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 30 cases that omit cells at the corners (ah1): First 16:
remaining 14:
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).
There are 47 tours with axis along the long median: 32 tours with corner cells (ad1) missing.
continued
plus 10 with missing cells in the edges at ad4.
plus ad2 (3 tours) and ad3 (2 tours).
34 cells. Sulian symmetry on the 6×6 with 2 cells missing, from Zurcher Illustrierte 23 Jan 1931, collected by H. J. R. Murray, the second is also in Murray's notes. Two with a 2-cell hole (Jelliss).
38 cells. Tour with direct binary symmetry, Sulian type, from Zurcher Illustrierte 8 June 1934.
46 cells. Sulian symmetry on a 6×8 board with 2-cell hole by Ernest Bergholt First Memorandum 24 Feb 1916.
50 cells Sulian symmetry on a 50-cell shape from Zurcher Illustrierte 13 July 1934. The tour looks as if it may have been intended to resemble a human skeleton (the 38-cell tour above may be its head). Psychologists use patterns with this type of bilateral symmetry in the Rorschach ink-blot test, because of its biological significance, which the human eye has evolved to recognise quickly.
Also Sulian tour on board 6×9 minus corners (Jelliss 2018).
62 cells. Here we consider boards formed by omitting two cells from the standard chessboard.
We first show examples with direct binary symmetry (Sulian type) for each of the 7 ways of omitting two cells from the board edge. The first two examples are from Kratchik (1927) and Murray (1942).
The others are by Jelliss (constructed 1996) the first below being after Naidu (1922).
We also show one Sulian example for each of the 7 positions of the two voids forming 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 last is from Murray (1942) diagram inverted, the rest are Jelliss (1996), the third being after Hoffmann (1893), the fifth after a German example (c.1860), the sixth after Babu (1901).
The tours described as 'after' earlier composers were originally published with the path diverted through the void cells to give either a near-symmetric open tour, or a pseudotour.
66 cells. Sulian binary direct symmetry with vertical axis, collected by H. J. R. Murray from Zurcher Illustrierte 28 Aug 1931 and from Denken und Raten 25 Aug 1929 with two holes.
70 cells. Sulian direct binary tours on shaped boards with holes, collected by Murray from Zurcher Illustrierte 22 Dec 1933, and from Denken und Raten 25 Aug 1929.
78 cells. Sulian type on 8×10 board with two holes from Denken und Raten 25 Jan 1931.
114 cells. This is our final Sulian example with vertical axis, another collected by Murray from Zurcher Illustrierte (7 Jul 1933), composer unknown.
