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A chess problem that has stumped mathematicians for more than 150 year has finally been crack up .
The n - queens problem begin as a much unsubdivided puzzler , and was first posed in an 1848 issue of the German chess newspaper Schachzeitung by the chess composer Max Bezzel . It ask how many ways eight rival queen — which are the most powerful piece on the chess board and capable of move any number of square horizontally , vertically and diagonally — could be place on a standard 64 - square board without any queen attacking another .
A standard 64 square chess board.
The answer , revealed just two years later , was that there were 92 configurations that kept the eight queens from each other ’s throat , with all but 12 of the solutions being simple rotations and reflections of each other . But in 1869 , an even more puzzling loop of the job was ask by the mathematician Franz Nauck : Instead of configure eight queens on a stock 8 - by-8 display board , what about 1,000 queens on a 1,000 - by-1,000 board ? What about a million , or even a billion ?
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What was once a relatively dewy-eyed mystifier had become a much deeper math problem — one that required the discovery of a worldwide dominion for the number of means to set any bit ( represented as " n " ) of queens on an n - by - n board .
Now , Michael Simkin , a mathematician at Harvard University ’s Center of Mathematical Sciences and Applications , has come in up with an almost - definitive answer .
On an enormous n - by - northward board , there are some ( 0.143n)^n ways to place n queen mole rat so that none can assault each other . That intend that on a million - by - million control panel , the number of nonthreatening configurations that 1 million queens can be format into is more or less 1 followed by 5 million nothing .
Simkin exact near five years to find this close approximation of an equation . mathematician usually solve problems by finding ways to fall in them into more manageable chunk . But because queens range nearer to the center of a gameboard can attack many more squares than queens at the bound can , the n - queens problem is highly asymmetrical — and , therefore , stubbornly resistant to simplification .
get together with Zur Luria , a mathematician at the Swiss Federal Institute of Technology in Zurich , Simkin initially simplify the project by considering a more symmetric " toroidal " adaptation of the job , in which the edge square wrap around the control board to make a donut - physique . This arrangement enables queens to disappear at the top left and re-emerge at the bottom right , for illustration . It also mean that no thing where they are placed , each queen can attack the same phone number of squares as its counterpart .
By using the toroidal dining table as a first approximation , the two mathematicians next apply a strategy called a " random greedy algorithm " to the problem . They placed a queen mole rat at random , block off all the foursquare it round ; then the next queen regnant would be pick out to sit on the remaining place , with its attacking squares blocked off in turn . The brace continued doing this over multiple configurations until they found a rough low bound — or lowest potential identification number — on the number of configurations of n pansy on a toroidal board .
But their estimate was far from stark . The wraparound nature of the board prevented them from bump the last few poove positions in some configurations . After drop off the trouble for a few years , the duet riposte to it with the mind of adapting their algorithm to a regular panel , which provided more concealing spots for the final queens than the toroidal board . By conform the random greedy algorithm to a standard , non - toroidal board , the pair somewhat improved the accuracy of this lower - bound appraisal .
But their answer was n’t as cleared cut as they hoped — the random avid algorithm works well on symmetric problems , where every board foursquare provides the same attacking vantage as any other . This is n’t the case for a standard dining table , where bound squares have much less ability to attack than square in the centre .
To lick this problem , Simkin realized he would need to adapt the algorithm . Because most of the feasible configurations on a standard card had more queen at the board ’s edges — where they attack few squares — than at its pith , Simkin complicate the random greedy algorithm by weighting the squares . Instead of his algorithm assigning queens arbitrarily , it preferentially placed queens in spots that would branch out to the highest number of potential configurations . This allowed Simkin to focus on how many queen would occupy each board section and bump a formula for a valid number of configurations , thus improving the accuracy of the lower - bound guess even further .
" If you were to assure me , ' I want you to put your poof in such - and - such way on the card , ' then I would be able-bodied to analyze the algorithm and secernate you how many resolution there are that match this restraint , " Simkin articulate in astatement . " In formal terms , it reduces the problem to an optimization trouble . "
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But find the lower bound of a telephone number still leaves an multitudinous set of number bigger than that . To really get to the solution , Simkin involve to recover an upper bound . To solve this second half of the job , he turn to a strategy called the " S method acting " , which take keeping note of the number of squares not under tone-beginning after a raw queen was place on the board . Using this method , he create a maximal stick formula that sprinkle out a number that almost utterly equalise the number for his lower spring ; Simkin reason out that he ’d in reality expunge the formula close to deadened - on .
Future study may seek to squash the two bounds even closer together , but Simkin , having stupefy close than anyone before him , is contented to leave this challenge for someone else to conquer .
" I think that I may personally be done with the n - queen job for a while , " Simkin sound out . " Not because there is n’t anything more to do with it , but just because I ’ve been dreaming about chess and I ’m quick to move on with my life . "
Simkin published his work , which has not yet been equal - reviewed , to the preprint databasearXiv .
primitively put out on Live Science .
