Can visit all squares of a chessboard exactly once ?

by Krishna Mahesh

Deevela Murali

Overview

Problem Statement of Real World Problem

Basic Concepts

Formulating Graph Problem

Solution to Graph Problem and Real world Problem

Interesting Facts

Can visit all squares of a chessboard exactly once ?

Our real world Problem is a very simple Question, it is to find if a Knight can legally visit every square on a Chessboard only once and Return to Starting position?

If Knight’s tour Exist ??

If Yes!! when does it happen??

If Not!!! Why??

Topic sounds Interesting rite… so lets not waste time and jump into topic just like the Knight …

Basics:

8 Possible Moves:

• 2UP, 1 left• 2 UP,

1Right• 1UP, 2Left• 1UP, 2Right

• 2Down, 1 left• 2 Down,

1Right• 1Down, 2Left• 1Down,

2Right

The Knight’s Legal Moves: Each move of a Knight comprises 2 directions as specified below.

A Hamiltonian cycle (or Hamiltonian circuit) is a cycle in an undirected graph which visits each vertex exactly once and also returns to the starting vertex. E.g 0,1,2,3,0

Hamilton Circuit

•A path visits each vertex of a graph once and only once. E.g 0,1,2,3

Hamilton Path

Solution to Graph Problem

Formulating Graph Problem

In a chessboard the vertices would represent the squares of the board and the edges would represent a knight’s legal moves between squares results in a graph as shown below, Known as Knight’s graph.

Highlights :Every Vertex represents square on chessboard.

Every edge is one legal move of a Knight.

Special Properties of Knight’s Graph

MxN chessboard has MN vertices. i.e Number of Squares equal to Vertices.

Graph is Undirected.

Edges of equal length, i.e one legal Knight Move.

Knight’s Move is Symmetric. A to B is equal to B to A.

Every move of knight Alternates between black and white. i.e move from white square will always result in an ending position on a black square and vice-versa

Solution to Graph Problem

The Knight’s Tour is a well-known classic problem. The objective is to move a knight, starting from any square on a chessboard, to every other square once. Note that the knight makes only L-shaped moves (two spaces in one direction and one space in a perpendicular direction).

If we observe the Knight's Tour problem turns out to be an instance of the more general Hamiltonian path problem in graph theory.

The problem of getting a closed Knight's Tour is similarly an instance of the Hamiltonian cycle problem.

We have 2 types of solutions, Hamiltonian Path when Knight starts at a vertex and doesn't end at the same vertex, Open Knight tour.

Hamiltonian Cycle where starting vertex and final vertex is the same, thus resulting a Cycle, Closed tour.

The Solution to both Closed and Open tour are same just that start and final vertices are going to same and different correspondingly.

Solution to Graph Problem

Algorithm:We can use simple algorithm to find if Knights tour Exists on the given board or not.

M x N be size of board i.e mn number of squares.

Total number of moves would be A=m*n-1.

Visited be a two dimentional array [M][N] initialized to false, when Knight visits this square it is set to true.

I be integer between 0 & A representing the number of moves, intialially it is 0, and if knight completes the Path it is A.

After Completeting the path if the one of next possible move results to initial position, it is a Closed tour, else it is Open Tour.

Algorithm move(x,y,i)1. if( (x < 0) OR (x ≥ M) OR (y < 0) OR (y ≥ N) )return false

2. if( visited[x, y] = true ) return false

3. if( m = M ) print “A solution has been found”• print “ x, y ” //This starts printing the solution• set visited[x, y] = true• return true

else //This is a valid move, but a tour has not been completed.

• set result = false OR Move( x+2, y+1, m+1)• set result = false OR Move( x+2, y-1, m+1)• set result = false OR Move( x-2, y+1, m+1)• set result = false OR Move( x-2, y-1, m+1)• set result = false OR Move( x+1, y+2, m+1)• set result = false OR Move( x+1, y-2, m+1)• set result = false OR Move( x-1, y+2, m+1)• set result = false OR Move( x-1, y-2, m+1)

4. if( result = true ) //One of the 8 moves above led to a completed tour.

• print “ x, y ”• return true

5. Else //None of the moves from this position led to a successful tour. //Now we must backtrack and try a different path • set visited[x, y] = false //Unvisit this location• return false

Algorithm move(x,y,i) contd..

Hence our Real World Problem is Solved.. Using Knights Graph and Basic Graph Theory.

The Closed Knight’s Tour

The Open Knight’s Tour where start and final vertex are different

http://commons.wikimedia.org/wiki/File:Knight%27s_tour_anim_2.gif#mediaviewer/File:Knight%27s_tour_anim_2.gifhttp://upload.wikimedia.org/wikipedia/commons/1/1e/Turk-knights-tour_anim.svg

Are you finding it hard to believe this would even work... Then take a live look.

How many closed knight’s tours are there on an 8 × 8 board?

How many open knight’s tours are there on an 8 × 8 board?

26,534,728,821,064

19,591,828,170,979,904

19 Quadrillion

1 ??100 ?? 10000??No No No… it is biggerrrrrr

If we have to find all the closed paths every

human has to find 3500 diff pathsPopulation of world 7 billion but no of

paths are 26 trillion

When does this Closed Knight’s Tour occur

In 1991, Allen Schwenk completely determined which rectangular chessboards have a closed knight’s tour.

An m × n chessboard with m <= n has a closed knight’s tour unless one or more of the following three conditions hold:

(a) m and n are both odd;

(b) m = 1, 2 or 4 ;

(c) m = 3 and n = 4, 6 or 8.

(explaination is not given due to

complexity and time for condition c)

Condition (a): m and n are both odd.

Recall that a legal move for a knight whose initial position is a white square will always result in an ending position on a black square and vice-versa.

Hence, every tour representing the legal moves of a knight alternates between black and white squares.

Condition (a): m and n are both odd.

Suppose m and n are both odd.

There will exist an odd number of squares on the board.

The number of black squares will not equal the number of white squares.

No closed tour can exist.

Condition (b): m = 1, 2 or 4.

Suppose m = 1

Suppose m = 2

K

K

Condition (b): m = 1, 2 or 4.

For 4 × n boards we require a more complex coloring of the board than the traditional black and white coloring.

Condition (b): m = 1, 2 or 4. Note that a knight must move from a brown

square to a black square. Likewise, a knight must move from a white square to a green square. Two closed cycles are now forced to exist and no closed tour exists for the 4×n board.

Solomon Golomb

Double and single loop in 4x5 board

Double loop in 4x8 board

5x6 8x3 10x318x18 board

6x6

7x6 7x8

5x8 8x6 12x3

Closed tours of few Boards

30x30

60x60 board

Semi Magic SquaresFirst semi-magic knight’s tour

In each quadrant, the sum of the numbers equals 520 and each of therows and columns adds to 130

The sum of the numbers in each 2x2 section is 130

Even after so many number patterns it is called semi magic because the diagonals did not also addto 260.Total of 140 semi magic knight’s tours are found. But not even a single fully magic knight’s tour in 8x8 is possible

Knight’s tours and cryptography

A cryptotour is a puzzle in which the 64 words or syllables of a verse areprinted on the squares of a chessboard and are to be read in the sequence ofa knight’s tour.

Knight’s tours and 3-D patternsFour polygons formed by 16 knight moves and their tessellation of

the plane

Knight’s tour tessellations can even be used to create beautiful 3-D patterns such as astersphaira hexastersphaira

References..

Jelliss, George. ”Introducing Knight’s Tours.” http://homepages.stayfree.co.uk/gpj/ktn.htm

“Knight’s Tours”- Ben Hill, Kevin Tostado

“Closed Knight Tour on Rectangular Boards”- sciencedirect.com

A. J. Schwenk, Which Rectangular Chessboards have a Knight.s Tour? Mathematics Magazine

Thankyou..

Any Questions