October 5, 2022

Harvard Mathematician Resolves 150-Year-Old Chess Problem

Now consider this queens gambit: If you put eight of them on a standard board of 8 squares by eight squares, how numerous methods could they be set up so that none could assault the other? Turns out there are 92. But what if you position an even larger variety of queens on a chessboard of the same relative size, state, 1,000 queens on a 1,000-by-1,000 square chessboard, and even a million queens on a similarly sized board?
The initial variation of the n-queens mathematical problem initially appeared in a German chess magazine in 1848 as the eight-queens issue, and the correct response emerged a couple of years later. In 1869, the more extensive variation of the problem emerged and remained unanswered up until late last year, when a Harvard mathematician provided a practically definitive response.
Michael Simkin, a postdoctoral fellow at the Center of Mathematical Sciences and Applications, computed that there are about (0.143 n) n methods the queens can be put so none are assaulting each other on huge n-by-n chessboards.
Simkins last formula doesnt offer the exact answer however instead simply states this figure is as near to the actual number as you can solve now. The 0.143 figure, which represents a typical level of unpredictability in the variables possible outcome, is increased by whatever n is and then raised to the power of n to get the answer.
On the exceptionally large chessboard with one million queens, for instance, 0.143 would be increased by one million, coming out to about 143,000. That figure would then be raised to the power of one million, indicating its multiplied by itself one million times. The last answer is a figure with five million digits.
Simkin states that he personally is a dreadful chess gamer but is seeking to improve his video game. “I guess, math is more flexible.”
Simkin had the ability to create the formula by understanding the underlying pattern for how large numbers of queens would need to be distributed on these massive chessboards– whether they d be concentrated in the center or on the edges– and then using widely known mathematical strategies and algorithms.
” If you were to inform me I desire you to put your queens in such-and-such way on the board, then I would be able to evaluate the algorithm and tell you how numerous options there are that match this restraint,” Simkin said. “In official terms, it minimizes the issue to an optimization problem.”
By concentrating on the areas that have the higher opportunities of being occupied, Simkin determined how numerous queens would remain in each section of the board and came up with a formula for to get a legitimate variety of configurations. The computations resulted in whats known as the lower bound– the minimum variety of possible setups.
As soon as he had that number, Simkin then used a technique referred to as the entropy technique to find the upper bound, which is the highest variety of possible configurations.
Simkin found the lower bound answer almost perfectly matches the upper bound answer. Put simply, it revealed that the specific response is sandwiched someplace in between the two bounds in a reasonably little mathematical area.
Simkin has actually been working on the n-queens problem for nearly 5 years. He says that he personally is a terrible chess gamer but is seeking to improve his video game. “I still enjoy the challenge of playing, but, I guess, mathematics is more forgiving,” stated Simkin, who became thinking about the issue since of how he could use developments from the field of mathematics he works in called combinatorics, which focuses on counting and issues of selection and arrangements.
Working on the issue has actually been a test of perseverance and strength. 4 years ago as a Ph.D. trainee at Hebrew University of Jerusalem, he visited mathematician and chess wiz Zur Luria at the Swiss Federal Institute of Technology in Zurich. The pair collaborated and established new strategies to get at an answer. In the end, after two years of work, they just came up with a better lower bound figure and knew they were missing out on something.
Simkin finished his Ph.D. in 2020 and relocated to Boston to begin working at Harvard. The problem was constantly at the back of his mind, and he returned to it when he realized he had to start focusing on spaces the queens would be rather than giving equal weight to each area.
Although its theoretically possible to get a bit closer to an even more specific response, Simkin in the meantime is happy to let another person concerned it.
” I think that I may personally be finished with the n-queens problem for a while, not because there isnt anything more to do with it however simply because of Ive been dreaming about chess and Im all set to move on with my life,” he said.
Reference: “The number of n-queens configurations” by Michael Simkin, 19 August 2021, Mathematics > > Combinatorics.arXiv:2107.13460.

A Different Kind of Queens Gambit
Harvard mathematician mainly fixes 150-year-old chess problem including most effective piece on board.
The queen is the most effective piece on the chessboard. Unlike any other (including the king), it can move any number of squares vertically, horizontally, or diagonally.

Now consider this queens gambit: If you put 8 of them on a basic board of 8 squares by eight squares, how lots of ways could they be arranged so that none could attack the other? What if you put an even larger number of queens on a chessboard of the very same relative size, say, 1,000 queens on a 1,000-by-1,000 square chessboard, or even a million queens on a likewise sized board?
On the extremely large chessboard with one million queens, for example, 0.143 would be increased by one million, coming out to about 143,000. Simkin has actually been working on the n-queens issue for practically five years. “I still enjoy the difficulty of playing, however, I think, math is more forgiving,” said Simkin, who became interested in the issue because of how he could use breakthroughs from the field of math he works in called combinatorics, which focuses on counting and issues of choice and plans.

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