On one hand it appears to be an easy game with 64 black or white squares with the game has 16 pieces each and two players vying to take on.
Explore a little more the game can provide incredible possibilities that pose questions to chess theorists and mathematicians that remain unsolved for decades , or hundreds of years.
In the summer of 2021, one of these challenges was solved at least to a certain point. Mathematical scientist Michael Simkin, from Harvard University in Massachusetts has put his thoughts to the n-queens issue that has baffled scientists since it was first thought of in the 1840s.
If you’re a chess player and know the rules, you will realize queens are the strongest part of the game, capable of move the entire board anywhere. The n-queens issue asks with a specific amount of queens (n) what combinations are possible when the queens are spaced from each other that none could take one of the other queens?
If you want to count eight queens for an 8×8 board, it is actually 92. However, many of them are reflected or rotated versions of only 12 basic solutions.
But what happens to 1000 queens on a board with 1,000 squares divided by 1,000? What happens to one million queens? Simkin’s best guess to solve the issue would be (0.143n) n The amount of queens divided by 0.143 which is then increased to the power of n.
What do you have isn’t the exact answer but it’s near as you can achieve today. If you have a million queens the number is an amount with five million digits which is why we can’t duplicate the exact number for you here.
It took nearly five years to Simkin to devise the equation. There were various methods and approaches employed, as well as a few obstacles in the way of finding a solution. The mathematician eventually was able calculate the lower bounds as well as the upper bounds for possible solutions using various techniques and discovered that they match.
“If you were to tell me I want you to put your queens in such-and-such way on the board, then I would be able to analyze the algorithm and tell you how many solutions there are that match this constraint,” Simkin says. Simkin.
“In formal terms, it reduces the problem to an optimization problem.”
In the beginning, Simkin and colleague Zur Luria from Zur Luria at the Swiss Federal Institute of Technology Zurich collaborated on a variant of the n-queens issue, also that is known as the torodial modular problem. In this particular case the diagonals are wrapped around the board which means that a queen can move across the right side of a board , only to appear on the leftside, for instance.
The queens have an asymmetric attack, however this isn’t the way a typical chessboard operates: a queen that is located in the corner does not have as many possibilities of attacking angles like a queen located in the middle.
In the end, the two’s work on the toroidal equation stagnated (although they published some findings) however, Simkin was able to adapt the results of his work into the final solution.
As boards grow larger with more queens grows it is evident that in the most allowed configurations the queens tend to be concentrated on the edges and edges of the table, and there are smaller numbers of Queens in the middle which is where they are vulnerable to attacks. This allows for a better targeted strategy.
In the realm of theory the future, a more precise solution to the n-queen problem is possible, but Simkin has brought us closer to a solution than we’ve ever been before. Simkin is happy to give the challenge to someone else to research further.
“I think that I may personally be done with the n-queens problem for a while, not because there isn’t anything more to do with it but just because I’ve been dreaming about chess and I’m ready to move on with my life,” Simkin adds. Simkin.