Find a knight's tour on the boards of the following sizes: (a) 5-by-5 (b) 6 -by-6 (c) 7 -by - 7

Name: Problem 51, Chapter 11: Introductory Combinatorics
Uploaded: 2021-03-28T17:45:51-08:00
Duration: 3 min 57 s
Channel: Carole Wastog
Description: Find a knight's tour on the boards of the following sizes: (a) 5-by-5 (b) 6 -by-6 (c) 7 -by - 7

Find a knight's tour on the boards of the following sizes:...

Key Concepts

Hamiltonian Path in Graph Theory

In the context of the knight’s tour, the chessboard can be represented as a graph where each square is a vertex and each legal knight move is an edge. A knight’s tour then corresponds to finding a Hamiltonian path (or cycle, if the tour is closed) in this graph, which is a path that visits every vertex exactly once.

Knight's Tour

A knight’s tour is a sequence of moves by a knight on a chessboard so that every square is visited exactly once. This problem is a classic example of a Hamiltonian path problem and has intrigued mathematicians and computer scientists due to its complexity and the variety of board sizes, constraints, and solutions that arise from different approaches.

Knight's Move

A knight's move in chess is an L-shaped move, consisting of two steps in one direction and then one step perpendicular to that direction. This unique movement pattern defines the connectivity in the problem, as it determines which board squares are reachable from any given position, ultimately forming the underlying graph structure of the tour.

Backtracking and Heuristic Methods

Due to the combinatorial nature of the knight’s tour, backtracking algorithms combined with heuristic methods, such as Warnsdorff’s rule (which selects the next move based on the fewest onward moves), are commonly used to efficiently search for a valid tour. These techniques help reduce computational complexity and improve the chances of finding a solution on various board sizes.

Transcript

00:01 Before we talk about how to multiply matrices, let's check to make sure that the multiplication is defined.

00:07 In this first matrix, we see that it is a 2x2 matrix, and our second matrix is a 2 by 1.

00:15 For matrix multiplication to be defined, the number of columns in the first matrix has to equal the same number of rows in the second matrix.

00:24 And you can see that this is the case, so this multiplication is defined, and the resulting matrix will be a 2 by 1.

00:32 So let's multiply these two matrices and see what we get.

00:36 So we know our matrix is going to have two rows and one column.

00:43 So the way to determine what goes into those elements is we multiply the row of the first one to the column of the second.

00:51 So this spot you can see here is the first row, first column, first column.

00:57 So we multiply.

00:58 So we would multiply one times two, which is, 2 and then we add that to 6 times negative 7 which is negative 42 and in this element here which is the second row first column second row first column so we would have negative 3 times 2 which is negative 6 plus 5 times negative 7 which is negative 35 and adding those up you see we're left with the matrix of negative 40, negative 41.

01:42 Let's see what happens if we invert these factors.

01:46 In other words, let's see if i can multiply this matrix to negative 7 to the matrix 16, negative 35.

01:56 So again, let's check to make sure that the multiplication is even defined...

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