Table of ContentsHow will Russia attempt to bring peace to SyriaTower Problem 1Tower Problem 2How will Russia attempt to bring peace in Syria Rook polynomials are the number of ways to place k non-attacking rooks on an original chessboard where no two rooks can be in the same row or column. The general formula for calculating the number of non-attacking trick arrangements is: Say no to plagiarism. Get a tailor-made essay on “Why Violent Video Games Should Not Be Banned”? Get the original essay The formula for calculating non-attacking towers is as follows. The polynomials below show the layouts of each round. The notation Rn(x) indicates the number of turns used, for example r1(x) means 1. Powers of x indicate the number of turns so for example the first line means a tower can be arranged in 1 way and zero rook can be arranged in only one way.Rooks Problem 1A famous problem called the "eight rooks problem" by HE Dudeney shows that the maximum number of non-attacking rooks on a chessboard is eight by arranging them on a diagonal of the chessboard that covers 8 squares. The question of the problem "In how many ways can eight rooks be placed on an 8 × 8 chess board so that none of them attack the other?" The answer is eight factorial because it behaves like an injective function. On the first row of the board, the rook has eight positions on which it can be placed. Then the rook has seven positions it can be in in the second row and so on until the eighth row where the rook only has one position it can be in. As a result, the different ways a rook can be placed on a chessboard without them attacking each other are 8! which equals 40,320. Keep in mind: this is just a sample. Get a personalized article from our expert writers now. Get a personalized essay. Rook Problem 2 Another problem related to rooks is "In how many ways can k rooks be arranged on an m × n board so that they do not attack each other? To approach this problem, k would have to be less or equal to the numbers m and n. Since the number of rows is m from which k must be chosen, the formula becomes mCk Moreover, the set of k columns on which to place the towers can be chosen in a manner nCk. ways of choosing k from M and N are independent of each other so the formula becomes mCk multiplied by nCk ways of choosing the square where to place the rook However, to calculate the number of non-attacking rooks, the number of ways. to choose the square on which to place the rook must be multiplied by k!, because it is the number of ways in which k towers can be arranged so as not to attack each other. non-attacking towers are mCk multiplied by nCk multiplied by k!.