MGF 1107 – EXPLORATIONS IN MATHEMATICS
LECTURE 18
The Knights Tour
A famous problem that is analogous to finding a Hamilton path involves
taking a chess board of various (square) sizes, and seeing if you can
move a knight around the board, landing on each square exactly once.
In the case of a board that measures 4 x 4 (or smaller), there is no
solution:
In the case of a board that measures 5 x 5 (or any square board where the
dimensions are odd and larger than 5), there exists a Hamilton path, but
no Hamilton circuit:
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In the case of a board that measures 6 x 6 (or any square board where the
dimensions are even and larger than 6), there exists a Hamilton circuit:
The Repetitive Nearest Neighbor Algorithm
The nearest neighbor algorithm that we looked at in Lecture 17 involved
choosing a starting vertex. However the algorithm is so easy to apply
that (assuming the number of vertices is reasonable) we can check each
of the vertices in turn in order to get a better approximation of the
optimal solution.
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 Spring '08
 EVINSON
 Math, Calculus, Hamiltonian path, Travelling salesman problem, Grigori Perelman, nearest neighbor algorithm

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