This time i have decided to write a different Article in Holiday season.I hope you all are having a great Christmas.i wish you all a Great,happy Christmas and Happy new year.

Maby you are a Grand Master level player,A Beginer,intermediate or what ever.Havent you think about mathematical side of the Chess board ???.maby you are or not...lets talk about it in this article...Basically Chess board Has 64 Squares and 32 Pieces .What else???/ we'll See..

See  the picture belove carefully

1. a knight's graph,
   

   Theory

   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. Note however that, unlike the general Hamiltonian path problem, the knight's tour problem can be solved in linear time

    

   Schwenk's Theorem

   For any m × n board with m less than or equal to n, a closed knight's tour is always possible unless one or more of these three conditions are true:

2. m and n are both odd; m and n are not both 1
3. m = 1, 2, or 4; m and n are not both 1
4. m = 3 and n = 4, 6, or 8.
5. 

King's graph

In graph theory, a king's graph is a graph that represents all legal moves of the king chess piece on a chessboard where each vertex represents a square on a chessboard and each edge is a legal move. More specifically, an king's graph is a king's graph of an chessboard.

For a king's graph the total number of vertices is simply nm.

For a king's graph the total number of vertices is simply n² and the total number of edges is (2n − 2)(2n − 1). Additionally, the number of edges for various n is identified as A002943 in the On-Line Encyclopedia of Integer Sequences.

Rook's graph

In graph theory, a rook's graph is a graph that represents all legal moves of the rook chess piece on a chessboard: each vertex represents a square on a chessboard and each edge represents a legal move. Rook's graphs are highly symmetric perfect graphs; they may be characterized in terms of the number of triangles each edge belongs to and by the existence of a 4-cycle connecting each nonadjacent pair of vertices.

Definitions

An n × m rook's graph represents the moves of a rook on an n × m chessboard. Its vertices may be given coordinates (x,y), where 1 ≤ x ≤ n and 1 ≤ y ≤ m. Two vertices (x₁,y₁) and (x₂,y₂) are adjacent if and only if either x₁ = x₂ or y₁ = y₂; that is, if they belong to the same rank or the same file of the chessboard.

For an n × m rook's graph the total number of vertices is simply nm. For an n × n rook's graph the total number of vertices is simply n² and the total number of edges is n³ − n²; in this case the graph is also known as a two-dimensional Hamming graph or a Latin square graph.

The rook's graph for an n × m chessboard may also be defined as the Cartesian product of two complete graphs K_nK_m. The complement graph of a 2 × n rook's graph is a crown graph.

Perfection

A rook's graph can also be viewed as the line graph of a complete bipartite graph K_n,m — that is, it has one vertex for each edge of K_n,m, and two vertices of the rook's graph are adjacent if and only if the corresponding edges of the complete bipartite graph share a common endpoint. In this view, an edge in the complete bipartite graph from the ith vertex on one side of the bipartition to the jth vertex on the other side corresponds to a chessboard square with coordinates (i,j).

Any bipartite graph is a subgraph of a complete bipartite graph, and correspondingly any line graph of a bipartite graph is an induced subgraph of a rook's graph. The line graphs of bipartite graphs are perfect: in them, and in any of their induced subgraphs, the number of colors needed in any vertex coloring is the same as the number of vertices in the largest complete subgraph. Line graphs of bipartite graphs form an important family of perfect graphs: they are one of a small number of families used by Chudnovsky et al. (2006) to characterize the perfect graphs and to show that every graph with no odd hole and no odd antihole is perfect. In particular, rook's graphs are themselves perfect.

Because a rook's graph is perfect, the number of colors needed in any coloring of the graph is just the size of its largest clique. The cliques of a rook's graph are the subsets of its rows and columns, and the largest of these have size max(m,n), so this is also the chromatic number of the graph. An n-coloring of an n×n rook's graph may be interpreted as a Latin square: it describes a way of filling the rows and columns of an n×n grid with n different values in such a way that the same value does not appear twice in any row or column.

An independent set in a rook's graph is a set of vertices, no two of which belong to the same row or column of the graph. Perfect graphs may also be described as the graphs in which, in every induced subgraph, the size of the largest independent set is equal to the number of cliques in a partition of the graph's vertices into a minimum number of cliques. In a rook's graph, the sets of rows or the sets of columns (whichever has fewer sets) form such an optimal partition. The size of the largest independent set in the graph is therefore min(m,n). Every color class in every optimal coloring of a rook's graph is a maximum independent set.

I've never before seen an animated example of the Knight's Tour!

Great post, Viba!

Holy  COW!!!!!!!!!!!!

According to Didelot and d´´Alembert this problen was well known in ancient India, some 2.000 years ago.

Solvyns (1856) says that in Benarés, by that time, the problems was considered by the Brahmans as a misterious knowledge, and a secret its drawing.

The problem was popular in Europe by XVIII and the Scince Academy of Berlin offered a prize to the most intriguing problem of the knight- - -prize which never was awarded.

In 1796 Euler wrote the first serious article about the matter, although there were some previous solutions (Monmort 1708, Moivre 1722). But with Euler a new era started.

Vandermonde, Collini, Warnsdorf, Jaenisch and Roget .also wrote about the problem.

INTERESTING, AMAZING ACTUALLY

Really nice, although I'm not really a math guy, but I enjoyed this one. Thanks for sharing such a high quality chess math with us.

If you are a genius people, give your reasonable answer for this test. "When Cheryl's birthday is? Explain your answer with logic argument, please!

           You've been here 7 days and you've bumped 3 threads that are 5 yrs old. Why?  Show your answer on a separate paper.