Closed Knight's Tours with Oblique Binary Symmetry

by G. P. Jelliss, including new work by Michael Creighton

Tours with oblique binary symmetry are invariant to 180 degree rotation, but not to 90 degree rotation, nor to any reflection. This type of symmetry can also be termed 'rotary' or 'diametral'. The assortment of tours shown here is not the result of any particularly systematic study, just an accumulation of various special cases, so there is obviously scope for further work. Many more examples of this type of symmetry will be found in the separate studies of Square and Oblong boards. The open tours previously shown here are now on the new Open Tours page.

Dates given against my name are when the tour was composed (when I remember to record it), against other names the date is that of first known publication. This page was prepared as a chapter for a possible book on knight's tours at the same time as the other pages on shaped boards in this section. However, it was only when I received examples of similar work from Michael Creighton of Liverpool at the end of April 2013 that I decided to publish it here. Ten examples by Mr Creighton are now included. He has a particular interest in using board shapes that fit together to tile a larger area. He has also produced some versions in which the cells are coloured, an example is shown at the end.

Rotary symmetry in closed tours is either Eulerian or Bergholtian. Eulerian symmetry circles round the centre of the board without passing through it. This is the symmetry that is most often seen in knight's tours, since it is the only one possible on the orthodox chessboard. Euler was the first to construct multiple examples of it. Bergholtian symmetry is a type of symmetry in closed tours in which the path crosses itself at the centre of the board.

The frame of the board for Bergholtian symmetry must obviously be of EO type (one side even the other odd) so that the centre is at the mid-edge where two cells meet. The containing rectangle for an Eulerian tour can be of any of the three types: even by even (EE), even by odd (EO), or odd by odd (OO). It should be borne in mind that any of the central cells may be void, that is there may be a hole at the centre of the board, and in fact since an Eulerian tour does not pass through the centre, the central cell in the OO case must be void.

If we consider any cell A in the tour and the corresponding cell A' equidistant on the other side of the centre, then in an Eulerian tour there are two congruent paths A-A' and A'-A connecting these points and together forming a circuit. On EE and OO boards, when chequered, the diametrally opposite points A and A' are of the same colour, so the paths A-A' and A'-A each consist of an even number of knight moves. The number of moves, and cells, in the tour in these cases is thus twice an even number, that is of the form 4n. On EO boards however A and A' are of opposite colour, so the the two paths are of an odd number of moves, and the number of moves and cells in the tour is twice an odd number, that is of the form 4n + 2.

The smallest closed tour on any board is the 8-cell circuit on the 3×3 board, but while this is invariant to 180 degree rotation it also has other symmetries and so appears in the section on Octonary Tours, not here.

10 Cells

On the 10 cell boards examined by O. E. Vinje (see the section on Smallest Knight-Tourable Boards for full details) four tours are symmetric, one each of the Bergholtian, Murraian, Sulian and Eulerian types. The Bergholtian tour, while it has 180 degree rotation, also has axial symmetries. So the first tour with purely oblique binary symmetry is this Eulerian example within a 4×5 frame:

12 Cells

On the 12-cell boards we find a wider range of examples, six Bergholtian (i.e. passing twice through the centre point, one 3×6 and five 4×5) and three Eulerian (circling the centre, two boards in a 4×6 frame, one with two tours):

14 cells

From this point on we give various examples without claiming any completeness in our catalogue. Before writing this item I had diagrams of only three 14-cell tours with oblique binary symmetry, one within the 3×6 frame, and one board within a 4×5 frame (a 3×4 board with a cell added on each longer side) that has two symmetric tours, one Bergholtian and one Eulerian (this board also has two asymmetric tours). The other five shown were constructed fairly quickly, each has only the one tour.

16 cells

No closed tour is possible on the 4×4 board, but moving two corner cells will produce two centro-symmetric boards (with equal numbers of dark and light cells when chequered). One of these has four symmetric tours (shown here). The other has 16 tours but none closed or symmetric (see the section on open tours).

18 cells

The first two examples are tours on the 4×5 board with two corner cells omitted. There are two centrosymmetric closed tours, one Bergholtian and one Eulerian. The other tours were constructed impromptu to provide further examples. The second and third, seventh and eighth examples are related by 'folding out' of pairs of moves.

20 cells

These four 20-cell tours on an oblique quaternary board (Jelliss 1996) at first glance appear to have oblique quaternary symmetry, but each of them has four moves on the central 4×4 squares that are in direct quaternary symmetry instead of oblique; they are examples of "mixed quaternary symmetry".

The next diagram uses a 20-cell board with oblique binary symmetry. An asymmetric tour on this board was given in L'Echiquier. However it also allows a tour with Bergholtian symmetry (Jelliss 2009).

22, 24 and 26 cells

These three assorted tours by Godron and Vatriquant are from L'Echiquier January 1929. No tours are possible on the centreless 3×9 board (27 cells) due to colour mismatch, but by omitting two corner cells the three tours shown become possible (Jelliss 2003).

Here are six tours with Bergholtian symmetry on 24-cell boards three of type 4×7 − 4 and two of type 5×6 − 6. The first tour here was constructed by Ernest Bergholt, 1917 specifically as an example of this type of symmetry on a non-rectangular board. The other five tours are of my own construction (Jelliss 2009). An asymmetric tour on the second board-shape was given in L'Echiquier. These six tours are the only symmetric closed tours possible on these boards.

28 cells

These four tours (on a sort of 'tanned hide' shape) appeared in L'Echiquier December 1928 where they are attributed to four different composers: Tolmatchoff, Marques, Godron and Vatriquant.

30 cells

Of course no tours are possible on the 4×8 with opposite corner cells removed, since there are 16 cells of one colour and 14 of the other. Here are two symmetric tours by S. Vatriquant, L'Echiquier May 1929, one Bergholtian and one Eulerian on the same 30-cell board.

32 cells

The first two diagrams below are symmetric tours on the 6×6 without corners (Jelliss 1999) and Euler (1759). Euler also used the board for an asymmetric tour. Naidu (1922 #S3) gives a symmetric closed tour on the 6×6 board with four holes. The 3×11 centreless board (32 cells) is the smallest odd-by-odd board without centre that admits tours (Jelliss 1999).

This 32-cell stepped board, can be regarded as a 'modal transformation' of the white or black cells on an 8×8 chequered chessboard (as was pointed out by T. R. Dawson). It has just four distinct knight tours with Bergholtian symmetry (Jelliss, 2 September 2009). They correspond exactly to the four camel tours on the chessboard.

36 cells

The first example is from L'Echiquier August 1928. Tours are impossible on the centreless 3×13 board (38 cells) due to colour mismatch, but this can be overcome by omitting two more cells as my examples show.

44 cells

Two more examples from L'Echiquier April 1929. On the centreless 3×15 board there are 14 tours, of which I show four examples; also an impromptu on the centreless 5×9 board.

48 cells

Here are two tours which are my best attempts (Jelliss 29 May 2013) to construct smallish symmetric tours without acute angles. The first has Bergholtian symmetry and the second Eulerian. The board shapes are extremely irregular. (I also found a much smaller 28-move tour, but it is asymmetric on an asymmetric board.) See the 52 cell tour below for a more regular example.

See the section on Oblique Quaternary Symmetry for examples on the centreless 7×7 board; those shown here (all Jelliss 2003) have the simpler (or is it more complex?) binary symmetry.

52 cells

This is another tour without acute angles (see 48-cell tours above). This has mixed quaternary symmetry. It also solves the problem of a longest possible closed knight path on the 8×8 chessboard without acute angles.

56 cells

This is the first example by Michael Creighton (2013). Note that the board shape fits together with the 64 cell example on the right below.

64 cells

The first tour here, by Cognet, L'Echiquier, October 1930 is a symmetric, Bergholtian tour. The board is formed by combining two identical 32-cell boards.
The other two tours are by Michael Creighton (2013). Besides fitting wth copies of themselves, the first board fits with the 96 cell board shown below
and the second with the 56 cell board above.

68 cells

Another tour by Michael Creighton. It pairs with the 92 cell tour shown below.

72 cells

This example (Jelliss 2009) is on a board with octonary symmetry within an even-by-even frame (12×12) but on which a tour with oblique quaternary symmetry is impossible, since 72 = 4×18 and 18 is even. Harikrishna 1871 used it for an asymmetric tour.

80 cells

The first Greek Cross tour here is by Ernest Bergholt Queen 22 January 1916. A handwritten note by Bergholt alongside the diagram on the copy of his article in Murray's collection asserts: "I have since constructed several still more remarkable for their elegance" but no others are given there (but see the 84-cell example on the oblique quaternary page. H. J. R. Murray's example is from his 1942 manuscript "selected from some 60 tours on this board in mixed quaternary symmetry composed by the author". Below these is a tour of the centreless 9×9 board. [It may be noted that quaternary symmetry is also possible on this centreless OO type board, but not on the 80-cell cross which is an EE type board.]

88 cells

These two by Bergholt are in his notes on symmetry sent to Murray in 1917. They give the impression of quaternary symmetry but are in fact binary.

92 cells

These are further examples by Michael Creighton (2013).

96 cells

Another example by Michael Creighton (2013).

100 cells

This 100-cell example by Post is from L'Echiquier November 1929.

108 cells

Another example by Michael Creighton (2013).

112 cells

This tour (Jelliss 2009) is on an octonary board that does not admit quaternary symmetry. It was used for an asymmetric tour by Harikrishna 1871.

116 cells

This 116-cell example (Jelliss 2003) is another in my series of centreless odd-by-odd board tours, this time on the 9×13.

136 and 144 cells

These tours by Michael Creighton (2013), are on boards designed to fit together to form a continuous strip or tile an area.

152 cells

This X-shaped tour by Michael Creighton (2013) fits horizontally with the first 192 cell tour below, and vertically with the 248 cell tour further below.

160 cells

Edward Falkener (1892) gave a collection of 18 tours showing approximate quaternary or octonary symmetry on the four-handed chess board of 160 cells. On close inspection however Falkener's tours mostly deviate from symmetry considerably. Two are pseudotours and thirteen are open. Only his number 3 has true binary symmetry. The tours shown here as "after Falkener" are closely based on his designs and achieve proper binary symmetry.

188 cells

Another tour by Michael Creighton (2013).

192 cells

Three tours by Michael Creighton (2013). The first 192 cell tour fits with the 152 cell tour above. The other two 192 cell tours complement each other.

208 cells

Another example by Michael Creighton (2013).

232 and 248 cells

Another pair of tours on complementary boards by Michael Creighton (2013).

A coloured version of the 248 cell tour. Click on the image to see a larger version. Photo from Michael Creighton.

See also this site on Flickr Quixote2013 which displays photos of Knight's Tour inspired art by Michael Creighton.

336 cells

A 20×20 board with an 8×8 central hole by Michael Creighton (2013). He has also produced tours on even larger shaped boards.