2 for an n ×m board used in this paper. The coordinates of the squares in the first row

2. Definitions and Notations

At the beginning, we would like to introduce the coordinate system shown in Fig.

2 for an n ×m board used in this paper. The coordinates of the squares in the first row

are (0, 0), (0, 1), (0, 2), ..., (0, m-1), that in the second row are (1, 0), (1, 1), (1, 2), ...,

(1, m-1). The rest may be deduced by analogy. So the coordinates of the squares in the

last row are (n-1, 0), (n-1, 1), (n-1, 2), ..., (n-1, m-1). In the later content of this paper,

if we need to point out the location of any particular square, the coordinate system

discussed above will be used.

Fig. 2. The coordinate system used on an n×m board

Closed knight's tour

A knight's tour is said to be closed if the last square visited is also reachable from

the first square by a knight's move (Please refer to Ian Parberry [13]). In other words,

the knight can visit every square exactly once along a closed knight's tour on the

board and then go back to the starting point.

Fig. 3 shows an example of a closed knight's tour. If we start the trip at (0, 0) and

move along the closed knight's tour shown in Fig. 3, finally we will return to the point

of departure at the top-left corner.

Fig. 3. A closed knight's tour on a 6×5 board.

Open knight's tour

It is opposite to the closed knight's tour. A knight's tour that every square on the

board is visited exactly once without getting back to the origin is called an open

knight's tour (Please refer to Ian Parberry [13]).

Fig. 4 demonstrates an open knight's tour that starts at (0, 0) and ends at (4, 0) on

a 5×5 board, so the tour is not reentrant.

Fig. 4. An open knight's tour on a 5 ×5 board.

Corner-missed closed knight's tour

The colors of the visited squares must be black and white interlaced when the

knight moves on the chessboard. If the knight is supposed to visit every square

exactly once on the board and goes back to the starting point, says there exists a

closed knight's tour, the number of black squares must be equal to that of white ones.

While on a board with both of its dimensions n and m odd, the difference in number

between black squares and white squares is one, there doesn't exist any closed knight's

tour on such kind of boards. In this circumstances if we abandon one surplus square,

we may have a chance to find a closed knight's tour on it. The squares in the corners

must belong to the group that has an extra square. Forsaking a corner square, our

algorithm can be put on to find a closed knight's tour for the rest of the squares on an

n × m board if n, m ≥ 5 are both odd. These kinds of knight's tours are called

corner-missed closed knight's tours (It is classified as closed knight's tour in Ian

Parberry [13], but we separate it out in order to avoid confusion). Fig. 5 shows an

example.

Fig. 5. A corner-missed closed knight's tour on a 5 ×5 board

Structured knight's tour

The knight's moves shown in Fig. 6 are necessary for a structured knight's tour. If

a knight's tour contains these moves, it is said to be structured [13].

Fig. 6. Required moves for a structured knight's tour

Although the pairs of moves connected to the squares in the four corners shown

in Fig. 7 are not listed in the requirements of a structured knight's tour, they come into

being naturally. In any knight's tour, each square has to connect to another two

squares. So there must have just two edges attached to it: one for incoming and the

other for outgoing. Because only two squares can be reached by a knight's move from

a square in the corner, corner squares have no other choices.

Fig. 7. The moves formed naturally in the corners.

Stretched knight's tour

This is our own definition. In fact, the stretched knight's tour is a special case of

the open knight's tour, but we add two extra conditions for it:

(1) The starting point and the ending point must be a corner square and an

adjacent square, respectively.

(2) Except for the corner in which the starting point and the ending point are

situated, the other three corners must satisfy the requirements of a structured

knight's tour.

Fig. 8 shows a stretched knight's tour on a 6 × 6 board. In addition to the

characteristics of the open knight's tour, the starting point and the ending point located

at (0, 0) and (0, 1) satisfy the first condition mentioned above and the other three

corners also satisfy the second condition.

Fig. 8. A stretched knight's tour on a 6 ×6 board

Double-loop knight's tour

This is also our own definition. Actually it refers to a pair of closed knight's tours

and each closed knight's tour forms a cycle visiting just half of the squares on the

board and occupies two adjacent squares per column. The union of the pair of closed

knight's tours must satisfy the requirement of a structured knight’s tour.

Fig. 9(a) shows a double-loop knight's tour on a 4×5 board, and Fig. 9(b)

displays only one closed knight's tour that visiting only half of squares. It is obvious

that the closed knight's tour in Fig. 9(b) occupies two adjacent squares in each column

and interlaces them on opposite sides between adjoining columns.

(a) (b)

Fig. 9. An example of double-loop knight's tour: (a) a double-loop knight's tour on a

4 ×5 board (b) only one loop is shown.