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.
Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.
by Krishna Mahesh DeevelaMurali
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?
Topic sounds Interesting rite… so lets not waste time and jump into topic just like the Knight …
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.
else //This is a valid move, but a tour has not been completed.
4. if( result = true ) //One of the 8 moves above led to a completed tour.
Hence our Real World Problem is Solved.. Using Knights Graph and Basic Graph Theory.
The Open Knight’s Tour where start and final vertex are different
Are you finding it hard to believe this would even work... Then take a live look.
No NoNo… it is biggerrrrrr
If we have to find all the closed paths every
human has to find 3500 diff paths
Population of world 7 billion but no of paths are
(explaination is not given due to
complexity and time for condition c)
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.
Double and single loop in
Double loop in 4x8 board
5x6 8x3 10x3
5x8 8x6 12x3
First semi-magic knight’s tour
In each quadrant, the sum of the numbers equals 520 and each of the
rows 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 add
Total of 140 semi magic knight’s tours are found. But not even a single fully magic knight’s tour in 8x8 is possible
A cryptotour is a puzzle in which the 64 words or syllables of a verse are
printed on the squares of a chessboard and are to be read in the sequence of
a knight’s tour.
Four 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 astersphairahexastersphaira