Rook Polynomials and Chess

Rook Polynomials and ChessIntroductionChess is a tangled strategical notice game. The plank on which the game is played is an 8 by octette-spot grid. distributively player begins the game with 16 pieces eight pawns, deuce knights, two bishops , two goldbricks, angiotensin-converting enzyme queen, and a single(a) queen mole rat. apiece of the six piece types come across differently, and each be able to attack, or remove a certain piece, in different directions. The objective of darnel is to checkmate the opposing king by placing ones own pieces around the king where it is being attacked no matter what action it takes.. While cheat is gener on the wholey a recreational activity, the interlacingity of the movement of the pieces has resulted in it having signifi rumpt mathematicsematical properties. Chess analysis is complicated because of the multitude of movement options open to two players at all(prenominal) single move. M whatever mathematical chess problems subs cribe to discovering how specific placement of pieces can result in none of the pieces fight each other, I.e. None of the pieces ar within the same delimit of sight. nonpareil finicky variation of these problems is the rook multinomial. The rook piece is able to move horizont each(prenominal)y or vertically up to a max of eight squares. In combinatorial maths, a rook polynomial generates the combinations of non-attacking rooks no two rooks are in the same row or column. A bill of fare is any layout of squares of a rectangular board with (y) rows and (z) columns. While in that location are, numerous grammatical constructions determining rook placement for rook polynomials, this probe leave alone olfactory property into the method of classifying all quadratic polynomials which are the rook polynomial for several(prenominal) generalized board G.Rook Movement and Application As verbalise, in chess, rooks are able to attack any square in its line of sight,( i.e. its row or column) exhibited belowIf a rook were to fall in the line of sight of another rook, they would be attacking. Rook polynomials focus on nonattacking rooks. The board of a rook polynomial is defined as a square n x n chessboard, but can equivalently include any subset of a board. For instance, on a 2 x 3 chess board, or an L shaped chessboard would be subset of a regular board. To denote the patterns in which multiple nonattacking rooks can be placed on a generalized board, the variable S can represent the ways to place a take of non-attacking rooks. R pass on represent the rook, and (G) the size of the generalized board, creating the formula rs(G). rs(G) allow for be stated as rs when the size of the board (G) as the size of the board will be apparent. For all boards, r0 will always be qualified to one, because it is impractical to set 0 rooks in differing combinations. R1 will always be the keep down of squares on the board because rook can be placed anywhere as no other r ooks are on the board to attack. Rs = 0 when k is greater than the exit of rows or columns on the board, as there will be no way to place all the nonattacking rooks.The future(a) generalized board provides an example of the equationAccording to our located parameters, for this special(prenominal) subset r0 would again be equal to 1, and r1 would be equal to the number of squares, in this case six. For r2 there are 8 ways to place 2 non attacking rooks, demonstrated below.xxxxxxxxxXXxxxxxThree rooks placed on the board tho allow for 3 different non-attacking combinations, meaning r3 equals 3xxXXxXXxxSince there are only 3 rows for this particular generalized board any rs3 as being equal to 0. victimization this general principle it is possible to create a polynomial that tracks all of the potential rook combinations on a generalized board. The rss would be the coefficients of xs, resulting in the formular0 + r1x + r2x2 + +rs-1xs-1+ rsxkThe rook placement numbers from the previo us generalized boardr0 = 1, r1 = 6, r2 = 8, r3 = 3, r4 = 0, r5, r6, = 0Resulting in the following polynomial1+ 6x + 8x2 + 3x3 but it is master(prenominal) to understand that rook polynomials are not unique to a single generalized board. Any board with a similar 3 column lay out would squander the same polynomial formula. This leads to the purpose of this mathematics investigation which is to classify all quadratic polynomials which are the rook combinations for a generalized board G.Determining the Quadratic PolynomialSince r0= 1 will always be true, and r1 = number of squares of (g) the root of the quadratic is1 + r1x + r2x2 r1 is stipulated by board size, therefore it is necessary to discover any r2s where r3 is equal to 0. If r3 is greater than zero, the rook polynomial could potentially be a cubic, or even quartic.As demonstrated in the previous example, generalized boards with 3 rows will result in polynomials where r4=0When examining the requirements of r4 = 0, the generali zed board mustiness contain two rows of a board and contain squares that are unbent in each row.As determined preciously, r1 is equal to the number of squares on the generalized board. With that information, we can summarize each row of the board as variables (y) and (z). In this case r1= y+z. With this methodology, (y) and (z) will be placed in a way that the spaces in row (y) are straightforward, and (z) squares are consecutive in the next row. Another important aspect is the number of columns created by (y) and (z). The overlap of the rows will be labeled (p).Using the number of squares in the example, consider r1 to be equal to 12. The possible pairs for (y) and (z) are1,11 2,10 3,9 4,8 5,7 6,6To create a formula outlining all the potential quadratic equations of a rook polynomial, any of these numbered pairs will work. In the case of 4,8, if none of the rows overlapped, r2 would be equal to 48. To determine r2 when squares are overlapping, it is necessary to multiply the num ber of squares in row (y) which are not overlapping by (z), and that to the product of the number of squares overlapping amongst the two of them and the number of squares not overlapping in row (z). Each square overlap should reduce r2 by 1. To demonstrate28+2718+3747If a closer look is interpreted at each of these equations, it is possible to expand it to and relabel the numbers with variables to create a formula. To demonstrate(38) + (17) = (4-1)8 + 1(8-1)(28) + (27) = (4-2)8 + 2(8-1)(18) + (37) = (4-3)8 +3(8-1)The numerals in these expanded equations can be replaced to reform the formular2= (y-i)(z) + (p)(z-1)Simplifying intor2 = yz -pz + pz - pr2 = yz -pWe can insert this equation into the rook polynomial already established to create a formula which can calculate all potential rook polynomial combinations.For example if a generalized board contains row (y) with 6 squares, and row (z) contains 9 squares, with an overlap of p=4, the quadratic polynomial would appear as1+(y+z)x + (yz-p)x21+(6+9)x +((69)-4)x21+15x +((54)-4)x21+ 15x + 50x2Therefore the formula to calculate all potential rook polynomial combinations is1 + (y+z)x + (yz-p)x2 Discussion and ConclusionRook polynomials, despite appearing as only a theoretical chess puzzle, actually have a number of practical applications. Various organization problems and layouts can be sculpted by the rook problem by equating the objects involved to playacting as rooks. A practical application stems from company scheduling. Management may require a specifc number of employees to work their channels at specific times. However there maybe scheduling issues to ensure no two role players are performing the same job at the same time. A rook polynomial calculation could give insight into the number of combinations of how the workers could be scheduled. The number of workers could represent row (z), and the times available row (y). Each worker must be be at the job and only one time, which is similar to the concept o f a rook being at a specific spot that is not attacking any other rook. Depending on the overlap of times and workers available schedule, the formula produce in this investigation could be used to determine the number of combinations.Another example is if a company desires to hire a certain (y)number of employees on a number (z) different jobs. Each job can only be held by only one employee. By putting both employee number and job number on the rows of a generalized chessboard chessboard, the result is similar to that of a rooks dilemma. When worker A is appointed to job B, it is like a rook is placed on the square where row (y)overlaps row (z). either occupation will be performed by a single worker, and every worker is assigned to a single job. Therefore all rows and columns in the pattern will contain only one rook, signifying that the rooks will not be attacking one another. The rook polynomial formula could be used to calculate the number of ways the assignments can be executed .However, the investigation in creating a quadratic rook polynomial formula did have limitations. I was unable to expand the formula to examine greater size generalized boards, requiring that it not reach cubic or quartic levels. I had initially attempted this, but found that I could not think a generalized rule for each and every combination of a generalized board. For this reason, the application to general life problems is limited. My process could have been change by investigating further into complex rook polynomials such as the Lagueree polynomial. Ultimately the complexity of chess allow for both simple and complex mathematical analysis, which can produce formulas which have real life application. Having chess as a visual and interactive tool to explore mixed concepts of math was a helpful resource when performing this investigation. Not only this, but my own enthusiasm for playing chess and studying the intricacies of the math behind the pieces pushed me to write the inves tigation.