Closed Knight's Tours with Asymmetry on Shaped and Holey Boards

For diagrams of many asymmetric tours on boards of 7 to 12 cells see the study on the Smallest Knight-Tourable Boards, and see the Open Tours study for open examples on larger boards. This new page on asymmetric closed tours is somewhat incomplete but has been awaiting inclusion on the Knight's Tour Notes for a long time; I hope to improve it in due course.

The reasons for studying asymmetric tours are various. In enumerating all the tours possible on a given board, some are usually asymmetric, and generally constitute the majority (however there are exceptions, e.g. the 5×6 board). Also of interest are the problems of what board shapes admit unique tours of given types. It is a paradox that some symmetric boards do not admit symmetric tours. Of course if the board is asymmetric then so must the tour be. A series of tours of these paradoxical types (symmetric boards with only asymmetric tours) were given by various contributors to the chess magazine L'Echiquier in the 1920s and 30s.

Included is a selection of Indian 'pictorial' examples some of which use irregularly shaped boards. In Harikrishna (1871), as reproduced by Iyer (1982) these tours are presented on unchequered chessboards, the tour being given by numbered squares within a line drawing depicting the pictorial subject. In his review of Naidu (1922), and in regard to these pictorial tours, Murray (1930) expressed the view that "These appear to me to be of no importance or interest". However, I find them a welcome antidote to too much of a diet of mathematically perfect symmetry. The Indian tours include a study of tours on boards of 32 cells that retain the biaxal symmetry of the 4×8 by cutting out some squares and affixing them in new positions so that the revised board admits a closed tour, which the rectangular board does not.

20 cells

Asymmetric tour of symmetric board 5×6 – 10, or 4×5 with two corners moved, by Godron and Vatriquant, L'Echiquier January 1929. However this board does admit a symmetric tour with Bergholtian symmetry.

24 cells

These five asymmetric tours on three boards with rotary symmetry are by Godron and Vatriquant. (a) 4×8 frame minus 8 cells, or 4×6 board with corner cells moved, from L'Echiquier July 1928. (b) 5×6 – 6, from L'Echiquier November 1928. (c) 6×6 – 12, from L'Echiquier June 1928 and January 1929; a symmetric tour is also possible, however a symmetric open tour is impossible since the centre point is not centre of a cell or mid-point of an edge.

28 cells

The amusing 'leotard' board has a closed tour, but it must be asymmetric, although there are symmetric pseudotours of two types: one in which both components are symmetric, and one in which the two components are the same and are symmetrically related. The tour, from L'Echiquier February 1929 is attributed to Vatriquant, Post and Deprez. The other tour shown is from Naidu 1922, representing a peacock.

32 cells

This 6×7 - 10 board, or 5×6 board with two attached cells, is another example that has symmetric pseudotours but on which a closed tour must be asymmetric. The tour shown is by Tolmatchoff (L'Echiquier March 1929).

The asymmetric tours on the 6×6 without corners are by Euler (1759) and Naidu (1922). Euler also gave a symmetric tour on this board. The other example is also from Naidu (1922, #S26).

The following tours form part of a systematic study of boards derived from the 4×8 rectangle with some edge cells cut out and reattached elsewhere. This is a natural subject to study since closed tours on the 4×8 are impossible. The first four boards are biaxially symmetric and the asymmetric tours are #21, #30, #45 and #64 from Naidu (1922). The next three boards only have one axis of symmetry, and the tours are #20 and #S51 and #S22 from Naidu the third has two holes.

Two tours from Naidu (1922). The first (his #S13) is a nearly symmetric tour of a board with diagonal axes of symmetry. The second has been rotated 90 degrees.

38 cells

#57 in Harikrishna (1871) representing a horse also occurs, reflected, as #33 in Naidu. Another 38-cell tour is #61 in Naidu (not shown here).

40 cells

These 40-cell boards are typical of a number of Indian tours on irregular boards. The first is #54 in Harikrishna (1871) and #S31 in Naidu (1922) and represents an "installed Shiva-linga". The second is #55 in Harikrishna and represents a bull. The third is #32 in Naidu and represents a yak.

42 cells

#60 in Harikrisha (1871) representing a vase. Two very irregular 42-cell tours in Naidu #15 and #25 represent camel and tiger (not shown here).

44 cells

The 6×8 – 4. A symmetric board that cannot be toured symmetrically by S. Vatriquant L'Echiquier October 1928.

46 cells

Naidu #43 camel.

48 cells

Naidu #46 elephant.

50 cells

#8(S) from Naidu (1922) represents a peacock.

52 cells

Naidu #10 is supposed to represent an elephant.

64 cells

This tour by Kraitchik 1927 is asymmetric on an asymmetric board. (For a tour on a similar symmetric board see the Oblique Binary section). The board is formed by combining two identical 32-cell boards, as indicated by the zigzag join.

72 cells

This is an asymmetric tour, #15 by Harikrishna 1871, said to represent a lake with a platform in the centre. A symmetric tour is possible on this board (see the page on Oblique Binary symmetry) but not oblique quaternary symmetry.

74 cells

This 74 cell example, #25 from Harikrishna 1871, was designed to represent a tree, but nowadays invokes the more ominous mushroom cloud of a nuclear explosion. A similar tour, with two extra squares appears as #35 in Naidu 1922. Harikrishna's #28 representing a prancing horse also uses 74 cells, but on an irregular board with two single-cell holes (not illustrated here).

86 cells

This tour #20 from Harikrishna 1871 is said to represent an aerial chariot. The board has axial symmetry but the tour is asymmetric.

100 cells

An asymmetric closed tour, #23 from Harikrishna 1871, said to represent intertwined serpents. The board is 12×12 minus 44 cells.

108 cells

This diagram is an asymmetric tour #17 from Harikrishna 1871, the nine areas removed are described as ponds. Harikrishna's #21 is also on 108 cells but uses an irregular board with a single cell hole, it is said to represent "an installed Shiva-linga" (not reproduced here).

112 cells

The diagram is an asymmetric tour (144 – 32), #12 by Harikrishna 1871.

128 cells

H. Staeker MS. These closed tours are very nearly symmetric, appearing to show direct quaternary symmetry, but this of course is impossible on a board within an even by even frame and with an even number of cells in each quarter, and on close examination small flaws in the symmetry can be found. These flaws can be rectified but the result will be a pseudotour. This board is of a type used for four-handed chess.