Mathematics of Nine Rooks’ Problem, Essay Example

Rooks’ Polynomial and Potential Configurations

The rook polynomial is a term used in combinatorial mathematics which demonstrates the number variances in the positioning of nine non attacking rooks on a three dimensions 9 x 9 chessboard.  The restriction is that none of the columns or rows may have two rooks.  The chessboard is composed of random collections of the squares possessed by the rectangular chessboard with nine columns and nine rows.  The rook’s polynomial has a coefficient which is demonstrated as x^k.  The coefficient demonstrates the variety of manners where k represents the number of rooks.   In this circumstance, nine rooks can be positioned on a 9 x 9 chessboard without any of the rooks having the capacity of capturing another (Riordan, 2002).

The designation rook polynomial was created by John Riordan.  The use of the rook polynomial is for the enumeration of permutations that have positions which are limited. The nine rooks’ challenge provides r_s with a value.  The value of r_s  is placed in the foreground of the term which has the most elevated order. The determination of the mathematics of nine rooks is an outcome of the production that eight of the rooks which are not threatening each other can be ordered in 40,320 distinct manners or 8! different ways. Considering the nine rooks, the number of configurations increases to 9!  or 362880 different ways (Riordan, 2002).

The majority of the proofs which are applied in determining the nine rooks’ problem utilize one of the following rules:

1. Law of sums: In the event that object A is selected in m manners and object B is selected in n formats which represents alternate ways, A or B may be selected in m+ n
2. Law of product: In the circumstance that object A may be selected in m formats, and subsequently object B may be selected in n formats, A and B have the potential of being selected in this sequence in mn formats (Riordan, 2002).

Nine Rook’s Problem Applications

This model can be generalized in the review of a chessboard.  It becomes apparent that for the problem to become categorized in the class of problems which can be solved.  The number of rooks must be less than the number of columns and rows on the chessboard.  This is the primary condition.  The sequencing of the rooks can be accomplished by two manners.  Initially, the collection of the number of rooks (9) can be selected.  The number of manners that this can be achieved is the number of rows divided by the number of elements.  In a likewise manner, the collection of n files on which to position the rooks can be selected in a number of manners which is equivalent to the number of rows divided by the number of rooks.  .

Notwithstanding, the endeavor is not complete due to the number of rows and columns which have intersections in the number of squares.  In omitting the files, it can be determined that the comprehension of non-threatening rooks that can be arranged is equivalent to.

It can be demonstrated that the number of manners that three rooks can be sequenced without threatening one another is 8!* 8! / 3!*5! * 5! or 18,816 distinct manners.  Considering the restriction that the rooks must not threaten one another, the selection can be made from the number of columns and rows which are available.

These formulas are applicable considering that k represents the number of non-threatening rooks, m represents the number of columns and n represents the number of rows (Allenby & Slomson, 2011).  The squares which represent the permutation on the chessboard are designated as the hit number. The permutation of 9 rooks on the chessboard has one hit. The hit numbers which avoid the condition can be shown as the quantity of permutations with have k exceptions. This quantity is an Eulerian amount.

The Basic Identity

Proof: Enumerate the pairs of (?, H) which fulfill the condition of j as a subset of the hits which pertain to ?. Selecting ? provides the left aspect. Selecting H provides the right aspect, since the selection of j non- threatening rooks has the potential of being amplified to a permutation of [n] in (n-j)! formats.

In the case of a collection of k characteristics being constant , which is correlated to the non- threatening rooks, then the quantity which fulfills all of these characteristics is equivalent to (n- k)!, else the amount is 0. Consequently, the aggregates over all of the collections of k properties of the quantity of permutations which fulfill these characteristics are r_k (n- k)!

The substitution of the row values causes the number of configurations to become 133, 496.  A conventional polynomial can be formulated which monitors all of the number of rooks on a conventional two dimensional chessboard simultaneously.  This category of polynomial is designated the rook’s polynomial in the variable x and is manifested:

The number of combinations of nine rooks which can be made on a nine by nine chessboard is 6.1234455837688608 x 10¹⁰³ positions. In narrowing the number to the number of positions where each of the nine rooks cannot capture another rook on the model, the number is reduced to 133,496 distinct configurations (Velek, 2014).

There are a variety of challenges which are derived from the distinct spheres of mathematics and science which can be applied to the nine rook problem. One of the examples is the following: An organization must put n number of staff members to work on distinct tasks and each of the tasks must be fulfilled by one employee. What is the various numbers of manners that this arrangement can be achieved? The employees can be placed on the orders of the chessboard which has dimensions of 9 x 9.  In the event that an employee is delegated to a task, the rook is placed on the position where the order a traverses the rank b. Considering that each of the tasks is fulfilled by one employee, all of the orders and ranks will be composed of one rook was an outcome of the configuration of nine rooks on the chessboard, considering that the rooks must be positioned in arrangements which do not allow them to capture each other (Bryant, 1992).

The research in the hypothesis of rook placement on a 9 x 9 chessboard is derived from the applications in statistical physics, number theory, group theory, including applied and pure combination geometry. The specific value delegated to the rook polynomial is derived from the usefulness of the generated function perspective.  In addition, the value delegated to the rook polynomial is derived from the premise that the zeros manifested in the digits of the rook polynomial demonstrate characteristics of the coefficient (Bryant, 1992).

Symmetrical Configurations of the Nine Rooks Problem

In order to add greater complexity to the challenge of placing nine rooks on a two dimensional chessboard without threatening one another, the rooks must also be configured in a symmetrical manner on the chessboard. In order to arrange nine rooks on a two dimensional chessboard where one of the rooks is not threatening the other, the initial restriction is that the rooks can only be positioned on the squares which are black. This would cause each of the rooks to be in distinct columns so that one rook cannot capture another rook.  This mathematical challenge is correlated with the enumeration of permutations of characters which have an assumed collection of limitations. The most fundamental of the configurations is when the nine rooks are positioned symmetrically around the chessboard’s center (Allenby & Slomson, 2011).