Knight's Tours of Larger Oblong Boards

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« Introduction

Here we continue our study of knight's tours on oblong boards, that is rectangular boards that are not square. Following on from our studies of 3×n and 4×n tours, we consider boards having the smaller side greater than 5. Unlike those earlier studies however this section on larger boards is far less systematic, consisting mainly of a collection of somewhat arbitrary examples.

The only tours of oblong boards constructed during the mediaeval period were those on the 4×8 half-chessboard, in some cases two such tours being joined together to give a full-board tour. Tours of other oblong boards were first considered in Euler's seminal paper (1759). The next after Euler to study small boards was Willis (1821).

For closed tours of course the board must have an even number of cells. Some boards allow open tours but not closed tours. The obvious cases are rectangles odd by odd and boards 4×2n.

« Knight's Tours of Five-Rank Boards

5×6 board

On the 5×6 there are, surprisingly, only three closed tours, two of them having axial symmetry, these were found by Warnsdorf 1858 and Haldeman 1864, the other asymmetric tour found earlier by Euler 1759, combines the two halves of the symmetric tours.

If the central cells c3, d3 are omitted then there is a pseudotour of four circuits (two of 8 cells, two of 6 cells) on the remaining cells. The closed tours delete one move in each of these circuits and connect them with four new connections — two single links and two double links that pass through the central cells. Many of the open tours on this board are also of this form. Others split the circuits up. I give some assorted open tours on this board. The Willis tour is the earliest. The Murray tours show three two-move lines.

5×7 board

I find 288 symmetric open tours (U.Papa, 1920, claimed 260). Classified by separation of end-points they are: {0,2} 18 (4 horizontal c3-e3, and 14 vertical d2-d4); {0,6} 14 (all a3-g3); {2,4} 56 (12 horizontal, b2-g4 or b4-g2, and 44 vertical, c1-e5 or c5-e1); {4,6} 200 (a1-g5 or a5-g1). The corner-to-corner tours comprise much the larger class. They can be subclassified according to the direction of the straight line though the centre cell. If they are all drawn a5-g1 then the central two moves are b2-f4 (32 cases), b4-f2 (28 cases), c1-e5 (64 cases) or c5-e1 (76 cases). Examples of each case are shown. The first diagram is from Papa 1920, the second from Murray 1942, the third Jelliss 1985, the others Jelliss 1997.

Willis 1821 and Jaenisch 1862 gave asymmetric examples.

5×8 board

Murray (1942) writes: "The main interest in this board is that it can be used to build up tours on the chessboard by compartments. We [i.e. Murray, Bergholt and Moore] have found 11 tours in central diametral symmetry on this board." (Central diametral symmetry is Papa's term for bergholtian symmetry.) The 11 diagrams are then given. This agrees with my own enumeration, done independently in 1988. The first tour shown here is due to Bergholt, it is the only one in which the centre moves are cut four times. There are four in which they are cut twice, and in the remaining six they are not cut. The last of these was given by Papa (1922).

There are 22 reentrant symmetric open tour solutions (two from each of the 11 closed solutions). A few non-reentrant examples are shown.

5×9 board

Two example symmetric tours.

5×10 board

This board admits sulian, eulerian and bergholtian symmetry as well as asymmetry.

5×11 board

Four example symmetric tours, and two others by taking the alternative dashed routes:

For other examples of tri-directional tours see boards 3×4k, 6×9 and 7×7.

5×12 board

One example, a closed tour with bergholtian symmetry:

5×20 board

This tour of 100 cells (Jelliss 1988) was composed to mark the centenary of Australia: the central cross symbolising the Southern Cross.

« Knight's Tours of Six-Rank Boards

6×7 board

There are closed tours of all three symmetric types: Eulerian (not enumerated), Bergholtian (19 in all) and Sulian (266 in all); seven of each are shown below. The first to show examples of all three types was Bergholt. I find the patterns on this board particularly attractive and varied. Asymmetry is also possible of course.

A complete set of diagrams of the Bergholtian and Sulian tours are now (20 June 2014) available on a separate page: 6x7 symmetric tours.

Eulerian (the first is by Bergholt):

Bergholtian (the first is by Bergholt himself):

Sulian (the first is by Bergholt). The axis of symmetry in these is horizontal. It is probably more natural for the human eye to perceive such symmetry if the axis is rotated to the vertical.

Asymmetry is also possible of course. This example is by Warnsdorf (Schachzeitung 1858):

Total open symmetric tours unknown, there are 38 reentrant solutions (two from each of the 19 closed solutions of bergholtian type). A few non-reentrant examples:

6×8 board

Eulerian symmetry only. Tours by Haldeman, Bergholt and Jelliss (3).

6×9 board

Sulian examples by Murray and Jelliss. My example joins an asymmetric 3×9 open tour to a copy of itself.

6×10 board

Example tours by Murray and Jelliss.

« Knight's Tours of Larger Oblongs

7×8 board

Two examples, bergholtian symmetry. The first is from British Chess Magazine 1918.

7×9 board

Two examples.

8×9 board

Border method. On the 8×9 board the two concentric braids each consist of two equal strands (of 26 cells on the outer and 10 cells on the inner). So to join them by simple linking requires only four deletions and insertions, but the result is asymmetric as in the first diagram. The symmetric, bergholtian, tour in the second diagram uses six deletions.

8×10 board

Border method. On this board the two concentric braids each consist of four equal strands (of 14 cells on the outer and 6 cells on the inner). So to join them by simple linking requires eight deletions and insertions. The symmetric (Eulerian) tour here achieves this.

8×11 board

Border method. On this board the number of strands in each braid reverts to two, as for the 8×9 board (but of 30 cells in the outer and 14 cells in the inner), so simple linking requires four deletions. The symmetric, bergholtian, tour here uses eight deletions.

8×12 board

The symmetric closed tour is by Murray 1942. The three open tour examples of near biaxial symmetry are by Victor Gorgias from the Dubuque Chess Journal 1871.

9×10 board

Border method. The braids are formed of four equal strands (16 or 14 cells outer, 8 or 6 inner), so eight deletions are necessary for simple linking. The shorter circuits are symmetric, so for a symmetric tour two deletions must be made in each of these, making a minimum of 12 deletions; this is achieved in the example shown. There are two cells in the centre that also have to be joined into the tour, the two moves through these can be treated like a single insertion (Jelliss 1999).

9×11 board

Border method. There are two strands in the braids (48 and 24 outer, 16 and 8 inner) and three central cells to join in. Each strand is in direct quaternary symmetry. Asymmetric and symmetric examples by Jelliss 1999.

10×11 board

Border method. There are four strands in each braid (18 or 16 outer, 10 or 8 inner), two being centrosymmetric, and there are 2×3 central cells to be joined in. Thus twelve deletions at least for a symmetric tour. The example, bergholtian, uses 14 deletions, Jelliss 1999.

« Knight's Tours of Extra Large Oblongs

20×32 board (640 cells)

This tour by Pierre Dehornoy (2003) shows how to construct a tour with most of the knight's moves in two directions. The board can be expanded lengthwise in units of 12 and vertically in units of 6 by duplicating the edge and central components.

(Compare his 16×16 tour which has moves mainly in one direction.)

24×37 board (888 cells)

This tour (Jelliss 1985) which has oblique binary symmetry of bergholtian type, is constructed mainly in the form of a patchwork of areas each of which exhibits one of the eight possible patterns in which an area of board can be covered by straight lines of knight moves, one passing through each cell (see the note on translational symmetry on the Symmetry page). This was made by me for a Christmas/New Year card I sent out in 1985-6.

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