Harvard Mathematician Solves 150-Year-Old Chess Problem
A Different Kind of Queen’s Gambit
Harvard mathematician greatly settles 150-year-old chess problem involving most potent piece on board.
In the game of chess, the queen is considered the most powerful piece on the board because of its ability to move in any direction along the rank, file, or diagonal it is placed on. This makes the queen a versatile and dangerous piece that can control a large portion of the board and attack multiple targets at once. However, it’s important to note that the king is still the most important piece on the board, as the object of the game is to checkmate the opponent’s king.
Consider this queen’s gambit: If you put 8 of them on a typical board of 8 squares by 8 squares, how many ways could they be set up to ensure that none could assault the other? It turns out there are 92. However, suppose you position an even bigger number of queens on a chessboard of the same relative dimension, claim, 1,000 queens on a 1,000-by-1,000 square chessboard, and even a million queens on a likewise sized board?
The n-queens chess problem
The initial version of the n-queens mathematical problem emerged in a German chess magazine in 1848 as the “eight-queens problem.” The correct solution emerged a couple of years later on. Then in 1869, the broader version of the problem emerged and stayed unanswered till late last year, when a Harvard mathematician gave a practically definitive solution.
Michael Simkin, a postdoctoral fellow at the Center of Mathematical Sciences and Applications, calculated that there are approximately (0.143 n) n manners the queens can be positioned, so none are attacking each other on gigantic n-by-n chessboards.
Simkin’s last equation does not offer the exact solution. However, instead simply states this figure is as near to the actual number as you can get right presently. The 0.143 figure, which stands for an average level of uncertainty in the variable’s possible result, is multiplied by whatever n is and afterward increased to the power of n to obtain the solution.
For instance, on the very large chessboard with one million queens, 0.143 would be multiplied by one million, resulting in approximately 143,000. After that, that figure would certainly be increased to the power of one million, meaning it is multiplied by itself one million times. The final solution is a figure with 5 million digits.
Simkin claims that he personally is a dreadful chess player but seeks to improve his game. “I presume mathematics is much more forgiving.”
A potential solution arises.
Simkin could come up with the equation by comprehending the underlying pattern for just how multitudes of queens would have to be distributed on these enormous chessboards– whether they would be concentrated in the middle or on the edges– and then using popular mathematical techniques and algorithms.
” Simkin’s statement is related to the field of optimization in computer science and mathematics. When given a specific constraint, such as the placement of the queens on the chessboard, an algorithm can be used to find the optimal solution that satisfies the constraint. In this case, the algorithm would need to find a way to place the queens on the board such that no two queens are attacking each other (i.e., there are no two queens on the same rank, file, or diagonal).“
By concentrating on the spaces that have the greater possibilities of being occupied, Simkin figured out the number of queens that would remain in each area of the board as well as thought of a formula to get a valid amount of arrangements. The calculations led to what is known as the lower bound– the minimum number of plausible configurations.
As soon as he had that number, Simkin, after that, used a strategy called the entropy method to locate the upper bound, which is the greatest number of plausible configurations.
Simkin discovered that the lower bound solution virtually completely matches the top bound answer. Simply put, it showed that the exact solution is sandwiched somewhere between both bounds in a reasonably small mathematical space.
Simkin’s intrigue by the puzzle
Simkin has been tackling the n-queens problem for almost five years. He says that he actually is a horrible chess player but is looking to upgrade his game. “I still take pleasure in the difficulty of playing. However, I guess, math is a lot more forgiving,” claimed Simkin. He ended up being intrigued by the problem due to how he could apply innovations from the field of mathematics he operates in called combinatorics. This field concentrates on counting and selection issues and arrangements.
Handling the problem has been a trial of patience as well as resilience. 4 years ago, as a Ph.D. student at the Hebrew University of Jerusalem, he went to see the mathematician and chess wiz Zur Luria at the Swiss Federal Institute of Technology in Zurich. The pair worked together and established brand-new techniques to access a solution. In the end, after 2 years of work, they just developed a better lower-bound figure and also recognized they were missing something.
Laying the n-queens problem to rest
Simkin finished his Ph.D. in 2020 as well as relocated to Boston to begin working at Harvard. The problem was constantly at the back of his mind. He came back to it when he recognized he had to start focusing on spaces the queens would be as opposed to giving equal weight to every space.
Even though it is in theory possible to obtain a little bit closer to a much more exact solution, Simkin, for now, is happy to let someone else come to it.
“I believe that I may personally be finished with the n-queens problem for a while, not since there is not anything more to do with it. However, just because I have been fantasizing about chess, and I am all set to carry on with my life,” he said.
Read another source: The number of n-queens configurations
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