Which rectangular chessboards have a knight's tour?

Asked by: Kiara Anderson
Score: 5/5 (70 votes)

The 8 × 8 chessboard can easily be extended to rectangular boards, and in 1991, A. Schwenk characterized all rectangular boards that have a closed knight's tour.

How many knight's tours are there?

There are 26,534,728,821,064 closed directed knight's tours, and the number of undirected ones is half that or 13,267,364,410,532. If you count equivalence classes under rotation and reflection, there are slightly more than 1/8th of that: 1,658,420,855,433.

How do you solve the knight's Tour problem?

The basic idea is this:
  1. For every possible square, initialize a knight there, and then:
  2. Try every valid move from that square.
  3. Once we've hit every single square, we can add to our count.

Is it possible for a knight to move around an 8x8?

If the start position is the cell (1,1), and the destination is (5,1), we can go there with only 2 moves. First at (3,2), and then at (5,1). There are no restricts to be noticed, we can suppose that there is a blank 8x8 chessboard with only a night in it.

34 related questions found

Can a knight access all squares?

Yes, it can

Therefore, the knight can start at any square on the board and finish on the same square, since it just starts at a different point along the cycle.

Can a knight reach bottom from top by visiting all squares?

Since, total number of moves are odd, the journey will start and end on the squares of the opposite color. Since, the squares on the top-left corner and lower-right corner are both coloured the same, hence the journey is impossible.

What is Warnsdorff algorithm?

We have discussed Backtracking Algorithm for solution of Knight's tour. In this post Warnsdorff's heuristic is discussed. Warnsdorff's Rule: We can start from any initial position of the knight on the board. We always move to an adjacent, unvisited square with minimal degree (minimum number of unvisited adjacent).

What is backtracking algorithm?

Backtracking is an algorithmic technique where the goal is to get all solutions to a problem using the brute force approach. It consists of building a set of all the solutions incrementally. Since a problem would have constraints, the solutions that fail to satisfy them will be removed.

What is a closed Knight's Tour?

A closed knight's tour of a chessboard uses legal moves of the knight to visit every square exactly once and return to its starting position. When the chessboard is translated into graph theoretic terms the question is transformed into the existence of a Hamiltonian cycle.

How many knights can be placed on a chessboard without threatening each other?

Since placing 32 knights is possible, 32 is the maximum number of knights that can be placed on a chessboard so no two attack each other.

How many knight's tours exist for the above chessboard?

In addition to netting a total of 140 distinct semimagic knight's tours, the computation demonstrated for the first time that no 8 x 8 magic knight's tour is possible, thus finally laying this long-open problem to rest.

Which is not a backtracking algorithm?

Which of the following is not a backtracking algorithm? Explanation: Knight tour problem, N Queen problem and M coloring problem involve backtracking.

Is a knight's tour possible on a 4x4?

For example, on a 4x4 chessboard a knight's tour is also impossible. In fact, the 5 x 6 and the 3 x 10 chessboards are the smallest rectangular boards that have knight's tours.

How do you implement backtracking?

Backtracking is an algorithmic-technique for solving problems recursively by trying to build a solution incrementally, one piece at a time, removing those solutions that fail to satisfy the constraints of the problem at any point of time (by time, here, is referred to the time elapsed till reaching any level of the ...

How many moves does a knight have to touch every square?

This image shows every square that a Knight can get to in 3 moves, starting at the Red square in the Center.

Is it possible for a knight to tour a chessboard visiting every square exactly once and returning to its initial square?

according to the rules of chess, must visit each square exactly once. If the knight ends on a square that is one knight's move from the beginning square, the tour is closed otherwise it is open tour.

How many moves can a knight make in chess?

Movement. Compared to other chess pieces, the knight's movement is unique: it may move two squares vertically and one square horizontally, or two squares horizontally and one square vertically (with both forming the shape of an L). This way, a knight can have a maximum of 8 moves.

For which NXM values knight can visit each square atleast once?

Moving according to the rules of chess knight must visit each square exactly once. Print the order of each the cell in which they are visited. Following is a chessboard with 8 x 8 cells. Numbers in cells indicate move number of Knight.

What is the minimum number of moves one need to move the knight from it's current position to any given position?

Input 1: A = 8 B = 8 C = 1 D = 1 E = 8 F = 8 Output 1: 6 Explanation 1: The size of the chessboard is 8x8, the knight is initially at (1, 1) and the knight wants to reach position (8, 8). The minimum number of moves required for this is 6.

What is a zugzwang in chess?

Zugzwang is a German word which basically means, "It is your turn to move, and all of your moves are bad!" There is no "pass" or "skip a move" in chess, so sometimes having to move can lose the game! ... Zugzwang is a German word which translates to "compulsion to move."

Can a knight go anywhere?

Whereas other pieces move in straight lines, knights move in an “L-shape”—that is, they can move two squares in any direction vertically followed by one square horizontally, or two squares in any direction horizontally followed by one square vertically.

Which among the chess pieces can move horizontally or vertically?

Rooks move horizontally or vertically any number of squares. They are unable to jump over pieces. Rooks move when the king castles. Bishops move diagonally any number of squares.