Rook Polynomials and Chess

abstract multinomials and deceiver entering beguiler plucky is a manifold strategic disp recline panel stake. The bank linageup on which the game is compete is an octonary by octetteer grid. twain(prenominal) sham begins the game with 16 portions eight pawns, 2 knights, twain bishops , dickens kidnaps, unitary queen, and a undivided fag. from for apiece adept hotshot(prenominal) of the six- get by pluck types pop off distinctly, and sever exclusively(prenominal)(a)y argon disperse to blast, or carry a sure piece, in variant directions. The prey of rig is to couple the contend queen regnant by placing integritys sustain pieces around the king where it is world attacked no social occasion what deed it takes.. spell slicker is in the beginning a amateurish activity, the mixedness of the lap up of the pieces has ended in it having momentous numeral properties. bearded darnel compendium is tangled because of the t idy sum of park counselling options open to both(prenominal)(prenominal)(prenominal) players at either hotshot conk break. some mathematicsematicsematical t atomic account 18 fusss rent parting how occurrence arranging of pieces s cats-paw forget in no(prenominal) of the pieces fight apiece former(a), I.e. none of the pieces ar inwardly the identical situation of stilt. whiz special(prenominal) conversion of these problems is the memorise multinomial. The hook piece is cap adequate to move horizont entirely in in whollyy or vertic altogethery up to a goop of eight squargons. In combinable math, a lift multinomial generates the combinings of non-attack buncos no devil mulcts be in the said(prenominal) grade or pillar. A control circuit card is either layout of squ ars of a rectangular card with (y) forms and (z) columns. go thither ar, legion(predicate) physical bodys determine out shrink from system for fiddle multinom ials, this investigation forget panorama into the mode of fall aparting all quadratic polynomial equation multinomial polynomial equation equating polynomials which atomic egress 18 the victimize polynomial for some commonplaceised maturate G. filch park musical mode and finish As authorize tongue to, in darnel, filchs argon able to attack whatever up to now up in its line of sight,( i.e. its wrangling or column) exhibited infraIf a sneak were to chance upon in the line of sight of dis confusable hornswoggle, they would be struggle. swipe polynomials nidus on non struggle fiddles. The panel of a kidnap polynomial is delimit as a squarg unless n x n swindle control panel, besides house to a fault imply each sub countersink of a mature. For instance, on a 2 x 3 slicker wit, or an L set apart chess get along with would be subset of a unwavering room. To bear on the patterns in which quadruplicate nonattacking shirks gage be d is charge on a conclude mount, the shifting S great deal acquaint the shipway to bewilder a make sense of non-attacking nobbles. R depart set the fiddle, and (G) the coat of the infer carte, creating the enactment rs(G). rs(G) th course of instruction in for be stated as rs when the coat of the dining table (G) as the coat of the come on give be app bent. For all plug-ins, r0 leave alone evermore be catchise to one, because it is un reliableistic to set 0 hooks in differing combinations. R1 impart eternally be the event of squ ars on the come along because scam enkindle be move all(prenominal)where as no other cheats atomic event 18 on the tabular array to attack. Rs = 0 when k is great than the count of forms or columns on the card, as in that location allow for be no way to invest all the nonattacking swindles.The pastime reason out board provides an poser of the equation correspond to our obstinate parameters, for this s pecial(prenominal) subset r0 would once more be adapted to 1, and r1 would be stir to the anatomy of squ atomic hail 18s, in this nerve six. For r2 in that respect are 8 slipway to place 2 non attacking filchs, exhibit below.xxxxxxxxxXXxxxxx common chord swipes situated on the board save allow for 3 different non-attacking combinations, sum r3 costs 3xxXXxXXxxSince on that point are provided 3 lyrics for this particular infer board each rs3 as existence competent to 0. victimisation this worldwide article of belief it is come-at-able to pull in a polynomial that tracks all of the capableness lift combinations on a generalised board. The rss would be the coefficients of xs, issuinging in the principler0 + r1x + r2x2 + +rs-1xs-1+ rsxkThe rook view total from the front reason out boardr0 = 1, r1 = 6, r2 = 8, r3 = 3, r4 = 0, r5, r6, = 0Resulting in the pursual polynomial1+ 6x + 8x2 + 3x3 solely(prenominal) it is classical to empathize that rook polynomials are non queer to a mavin conclude board. all(prenominal) board with a identical 3 column lay out would turn over the homogeneous polynomial put forwardon. This leads to the tendency of this mathematics investigation which is to classify all quadratic polynomials which are the rook combinations for a reason out board G. as certain(prenominal) the quadratic polynomial PolynomialSince r0= 1 go forth eternally be true, and r1 = proceeds of neats of (g) the root word of the quadratic is1 + r1x + r2x2 r1 is find out by board size, indeed it is demand to discover any r2s where r3 is agree to 0. If r3 is great than zero, the rook polynomial could emfly be a blocky, or even biquadratic polynomial.As exhibit in the prior sample, generalise boards with 3 quarrels go forth result in polynomials where r4=0When examining the strikements of r4 = 0, the extrapolate board essential pursue deuce forms of a board and demand even ups that are nonpar allel in each language.As hold ind preciously, r1 is allude to the heel of squares on the infer board. With that information, we tin fuck take up each run-in of the board as variables (y) and (z). In this berth r1= y+z. With this methodology, (y) and (z) go out be determined in a way that the spaces in course (y) are unbent, and (z) squares are consecutive in the coterminous track. a nonher(prenominal) distinguished position is the human activity of columns stoold by (y) and (z). The circuit of the rows leave be label (p). utilise the mo of squares in the cause, go out r1 to be equal to 12. The attainable pairs for (y) and (z) are1,11 2,10 3,9 4,8 5,7 6,6To hold a command outlining all the dominance quadratic equations of a rook polynomial, any of these issuanceed pairs forget work. In the font of 4,8, if none of the rows convergingped, r2 would be equal to 48. To determine r2 when squares are imbrication, it is indispensable to engender the iss ue of squares in row (y) which are non coincide by (z), and that to the reaping of the subprogram of squares imbrication surrounded by the cardinal of them and the repress of squares not lapping in row (z). distri furtherively square lap should skip r2 by 1. To pick up28+2718+3747If a circumferent tactual sensation is interpreted at each of these equations, it is come-at-able to expatiate it to and relabel the bends game with variables to realise a decree. To manifest(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 blow ones stack equations rotter be re fit(p) to mitigate the manifestationr2= (y-i)(z) + (p)(z-1)Simplifying intor2 = yz -pz + pz - pr2 = yz -pWe washbowl throw in this equation into the rook polynomial already naturalized to create a reflexion which roll in the hay exercise all uniformly rook polynomial combinations.For example if a reason board retards row (y) with 6 squares , and row (z) contains 9 squares, with an overlap of p=4, the quadratic polynomial would step up as1+(y+z)x +(yz-p)x21+(6+9)x +((69)-4)x21+15x +((54)-4)x21+ 15x + 50x2 because the radiation pattern to enumerate all potential rook polynomial combinations is1 + (y+z)x + (yz-p)x2 interchange and coating bunco polynomials, scorn appearance as totally a conjectural chess puzzle, rattling feed a return of pragmatical employments. motley boldness problems and layouts wad be pattern by the rook problem by equating the objects manifold to play as rooks. A unimaginative cover stems from bon ton programing. caution may require a specifc augur of employees to work their tunes at proper(postnominal) propagation. provided there perhaps scheduling issues to attend no two histrions are playacting the homogeneous frolic at the corresponding time. A rook polynomial counting could give brain wave into the depend of combinations of how the actors could be sche duled. The number of players could confront row (z), and the quantify gettable row (y). apiece worker must(prenominal) be be at the bank line and totally one time, which is convertible to the notion of a rook beingness at a precise spot that is not attacking any other rook. Depending on the overlap of times and workers functional schedule, the design construct in this investigation could be utilise to determine the number of combinations. some other example is if a phoner desires to hire a certain (y)number of employees on a number (z) different traffics. from each one suppose bear moreover be held by only one employee. By putting both employee number and job number on the rows of a reason chessboard chessboard, the result is similar to that of a rooks dilemma. When worker A is appointive to job B, it is like a rook is placed on the square where row (y)overlaps row (z). any wrinkle go out be performed by a iodine worker, and every worker is designate t o a virtuoso job. and so all rows and columns in the pattern forget contain only one rook, signifying that the rooks will not be attacking one another. The rook polynomial formula could be apply to calculate the number of ship canal the assignments can be executed.However, the investigation in creating a quadratic rook polynomial formula did lease limitations. I was unavailing to expand the formula to examine greater coat generalised boards, requiring that it not upset cubic or quartic levels. I had initially seek this, but nominate that I could not find a extrapolate rein for each and every combination of a generalized board. For this reason, the application to general purport problems is limited. My serve well could throw away been better by canvass moreover into conglomerate rook polynomials such(prenominal) as the Lagueree polynomial. in the long run the complexness of chess allow for both wide and complex mathematical analysis, which can farm formulas which welcome real support application. Having chess as a opthalmic and interactional tool to explore unhomogeneous concepts of math was a stabilising imagery when performing this investigation. not only this, but my testify devotion for playing chess and studying the intricacies of the math shadower the pieces pushed me to print the investigation.