Most tours with quaternary symmetry are of the oblique (rotational) type. Direct quaternary symmetry is less
familiar and its study has been neglected, mainly because it is impossible in tours of rectangular boards.
It requires boards with shaped (non-rectangular) outlines or with holes (or both).
Direct quaternary symmetry occurs in two forms: with diagonal or lateral axes of symmetry.
This type of symmetry is impossible in a knight's tour on a square board since the tour must enter only two cells on each diagonal. If there are to be no holes in the board then the diagonal cells used must be adjacent and the other cells of the board must grow cross-like from this central 2×2 hub.
The following four diagrams show all possible tours with diagonal quaternary symmetry on boards without holes and less than 48 cells. The two on regular cross-shaped boards were found by Euler (1759).
If we allow holes, there are four tours with this type of symmetry within the 5×5 frame (the 16-cell tours shown below). Within the 6×6 frame I find 54 paths with this symmetry, of which 35 are connected tours (including the two above without holes), three examples, one of 20 and two of 28 cells are shown below.
This is evidently a subject in need of further study.
It is easy to fall into the error of supposing that a tour with quaternary symmetry can only be possible on a board with a multiple of 4 cells, since it seems obvious that each quarter of the tour must have the same number of moves, m 1, and of cells, m. This is true in the case of oblique quaternary symmetry and in the case of direct quaternary symmetry with diagonal axes. However in direct quaternary symmetry with lateral axes a quaternary tour may be possible using 4m + 2 cells instead of 4m. This is because of the possibility of Bergholtian symmetry, in which the path crosses itself at the centre of the board, so that each of the two central moves contributes half a move to each quarter of the symmetry, thus making the total 4m + 2 possible. The containing rectangle from which the board is cut must be odd by even.
Drawing the central knight moves horizontally, such tours are in the form of a convoluted figure-of-eight. The mid-point of each loop of the eight must be on the vertical median, and there can be no other cells on this axis, so the board is dumbbell shaped. Apart from the 10-cell tour (given by O. E. Vinje Fairy Chess Review 1949) this type of symmetry does not seem to have been noticed before.
Here are all tours of this type with less than 50 cells and omitting cells only in the middle file. With 3 files the number of rows can only be 4 or 6 since more would leave no move through the corner cells. Tours with 5 files are impossible since on the 5×4 the central moves and the corner moves form a circuit and on boards with more ranks there are again no moves through the corners. On the 7×6 no tour is possible although there is a pseudotour as shown.
The next size boards for direct lateral quaternary symmetry use 50 cells, and three differently proportioned dumbbells are possible. I find 5 tours with frame 7×8, 23 tours 9×6 and 9 tours 13×4.
A. 7×8 case: 5 tours. In the striking first tour the knight's moves are all within the boundary.
B. 9×6 case: (a) 6 tours with straight-through centre. Only the first one is contained within the border.
B. 9×6 case: (b) 9 tours with obtuse angled centre. The first two are within the border.
B. 9×6 case: (c) 8 tours with right angled or acute angled centre. The first one is within the border.
C. 13×4 case: 9 tours. Six are withn the border.
When both sides of the containing rectangle are odd there is necessarily a central hole of at least one cell. Tours of 4n cells with lateral quaternary symmetry are then feasible, formed of four quarter-paths each joining a cell on the vertical median to a cell on the horizontal median, these being the only cells on the medians that are used. (The 3×3 octonary star is a special case of this.) Some quaternary examples are shown here, the smallest using 32 cells.