Chess Engagement Paper

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Introduction
This is filler text for my personal engagement paragraph, where I will be writing about the wonderful game of chess and how it interests me. I will talk about my past history and interactions with chess and why I chose the knight’s tour as my topic for math exploration. I won’t include too much research or calculations here but it should help my case. I will now copy and paste the paragraph I just wrote to make it look longer. This is filler text for my personal engagement paragraph, where I will be writing about the wonderful game of chess and how it interests me. I will talk about my past history and interactions with chess and why I chose the knight’s tour as my topic for math exploration. I won’t include too much research or …show more content…

One can go about constructing a knight’s tour in many possible ways, they usually always fall into two categories: algorithms1 and Heuristics2.
[1] Algorithms
The first algorithm is one of brute force, appropriately named the brute force algorithm, it involves the use of a computer to place the knight at a random or given square and then test for open possibilities for the knight to move in its iconic “L” shape pattern. It will iterate the board until a solution is reached, even with current technology it is extremely impractical on larger boards.

The second algorithm is one that is referred to as divide and conquer, the board is divided into smaller rectangles and the knight’s tour is constructed on those, then the ending square on one rectangle leads to the beginning on another, connecting rectangles making it extremely easy to solve large …show more content…

In this case, when parts of a term are equal to zero, the entire term is equal to zero, because anything multiplied by zero is zero; for example, if a = 0, then a(x-r_1 )(x-r_2 )=0. The same applies to (x-r_1 ) and (x-r_2 ). With this in consideration, it is trivial to determine which n x n boards do not have any edges in its knight graph. (n-2)=0 and (n-1) = 0 if one of these equations is true, the edges in the knight graph will be zero, further simplified by solving for n (2 or 1) which are the solutions for the n x n knight graph which contain zero edges also known as the roots of the parabola.

Explanation of table: for the knight graph on a n x n board n has a relationship with the number of edges and vertices in its graph. The vertex relation is relatively simple as it is the area of the square, n x n, or n2, shown by the second column in the table. The number of edges the graph has is a more difficult relation as it is an exponential one, represented by f(n) = 4(n - 2)(n – 1) and the third column or by the parabola

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