Question

The Knight’s Tour Puzzle asks if it is possible to find a sequence of 64 knight’s moves so that a knight on a chessboard visits all the different squares and ends up on the starting point.
a) Formulate the problem in terms of graphs.

b) Can the puzzle be solved?

c) Can you find an explicit solution?

d) What happens on a 3×3 chessboard

This question from Euler and Hamilton Paths and Circuits.

Thanks in advance.

Answer 1

Knight’s Tour Puzzle :-

A knight's tour is a sequence of moves of a knight on a chessboard such that the knight visits every square only once. If the knight ends on a square that is one knight's move from the beginning square so that it could tour the board again immediately, following the same path,Then the tour is closed or it opens again

The knight's tour problem is the mathematical problem of finding a knight's tour. Creating a program to find a knight's tour is a common problem given to computer science students.Variations of the knight's tour problem involve chessboards of different sizes than the usual 8 × 8, as well as irregular (non-rectangular) boards..

The knight's tour problem is an instance of the more general Hamiltonian path problem in graph theory. The problem of finding a closed knight's tour is similarly an instance of the Hamiltonian cycle problem. Unlike the general Hamiltonian path problem, the knight's tour problem can be solved in linear time

Knight's graph showing all possible paths for a knight's tour on a standard 8 × 8 chessboard. The numbers on each node indicate the number of possible moves that can be made from that position.

a):Knight Graph:-

The knight graph is a graph on vertices in which each vertex represents a square in an chessboard, and each edge corresponds to a legal move by a knight (which may only make moves which simultaneously shift one square along one axis and two along the other).

The number of edges in the knight graph is (8 times the triangular numbers), so for , 2, ..., the first few values are 0, 0, 8, 24, 48, 80, 120, ...

Knight graphs are bipartite and therefore are perfect.

The following table summarizes some named graph complements of knight graphs.

-knight graph	-queen graph
-knight graph	-queen graph

The knight graph is implemented in the Wolfram Language as KnightTourGraph[m, n], and precomputed properties are available in using GraphData["Knight", m, n].

Closed formulas for the numbers of -graph cycles of the knight graph are given by for odd and

A knight's path is a sequence of moves by a knight such that each square of the board is visited exactly once. It is therefore a Hamiltonian path on the corresponding knight graph.

The above figures show six knight's paths on an chessboard, all but the first of which are re-entrant. The final path has the additional property that it is a semimagic square with row and column sums of 260 and main diagonal sums of 348 and 168.If the final position of such a path is a knight's move away from the initial position of the knight, the path is called re-entrant or closed and corresponds to a Hamiltonian cycle on the underlying knight graph. It shows that a knight's tour exists on an board iff and is even.

b)There are several ways to find a knight's tour on a given board with a computer. Some of these methods are algorithms while others are heuristics.

Brute-force algorithms

A brute-force search for a knight's tour is impractical on all but the smallest boards.

For example, there are approximately 4×10⁵¹ possible move sequences on an 8 × 8 board and it is well beyond the capacity of modern computers (or networks of computers) to perform operations on such a large set. However, the size of this number is not indicative of the difficulty of the problem, which can be solved "by using human insight and ingenuity ... without much difficulty.

Warnsdorff's rule

It is a heuristic for finding a single knight's tour. The knight is moved so that it always proceeds to the square from which the knight will have the fewest onward moves. When calculating the number of onward moves for each candidate square, we do not count moves that revisit any square already visited. It is possible to have two or more choices for which the number of onward moves is equal;

This rule may also more generally be applied to any graph. In graph-theoretic terms, each move is made to the adjacent vertex with the least degree.Although the Hamiltonian path problem is NP-hard in general, on many graphs that occur in practice this heuristic is able to successfully locate a solution in linear time.

A graphical representation of Warnsdorff's Rule. Each square contains an integer giving the number of moves that the knight could make from that square. In this case, the rule tells us to move to the square with the smallest integer in it, namely 2.

Neural network solutions:-

The knight's tour problem also lends itself to being solved by a neural network implementation.^[23] The network is set up such that every legal knight's move is represented by a neuron, and each neuron is initialized randomly to be either "active" or "inactive" (output of 1 or 0), with 1 implying that the neuron is part of the solution. Each neuron also has a state function which is initialized to 0.

When the network is allowed to run, each neuron can change its state and output based on the states and outputs of its neighbors (those exactly one knight's move away) according to the following transition rules:

where represents discrete intervals of time, is the state of the neuron connecting square to square , is the output of the neuron from to , and is the set of neighbors of the neuron.

Although divergent cases are possible, the network usually eventually converges, which occurs when no neuron changes its state from time to . When the network converges, either the network encodes a knight's tour or a series of two or more independent circuits within the same board.