If you have a few chesss live at home, try this: Arrange eight queens in the yard so that no one rebels. If you can win once, will you get another chance? The third part? How many are there?
This problem is over 150 years old. This is the very first question of a mathematical question called n-testate the problem whose answer is Michael Simkin, a graduate of Harvard University’s Center for Mathematical Science and Applications, was introduced in a paper written in July. Instead of placing eight queens on an 8-by-8 chessboard (where there are 92 different layouts to work), the problem asks how many ways to set it up n queens on ni amn board. These could be 23 queens on a 23-by-23 board or 1,000 on a 1,000-by-1,000 board, or any queens on the same board.
“It’s easy to explain to everyone,” he said Erika Roldán, a colleague of Marie Skłodowska-Curie at the Technical University of Munich and the Swiss Federal Institute of Technology Lausanne.
Simkin confirmed that on the large chess courts with many queens, there are approximately (0.143n)n layouts. As a result, on a million-by-million board, the number of ways to make 1 million non-threatening queens is followed by zero to five million.
The first issue of the 8-by-8 chessboard first appeared in the German chess magazine in 1848. By 1869, n-queens the problem had followed. Since then, mathematics has had little effect on n-smots. Although previous researchers used computer experiments to predict Simkin’s results, he was the first to confirm this.
“They did this harder than anyone else before,” he said Sean Eberhard, a postdoctoral fellow at the University of Cambridge.
One obstacle to deal with n-testing the problem is that there are no obvious ways to simplify it. Even on a small board, the amount of a queen can be huge. On a larger board, the amount of calculations that take place is staggering. In this way mathematicians often expect to find a specific type, or structure, that allows them to divide the numbers into smaller, easier-to-manage pieces. But the file for n-testing the problem does not seem to exist.
“One of the most obvious things about a crisis is that, without much thought, it seems that there is no plan,” Eberhard said.
This is due to the fact that not all of the content in the group is created the same way.
To find out why, just think of your design for the eight queens. If you place your first queen near the center, she will be able to fight any position in a straight line, in a column, or in two very long rows. This leaves 27 unlimited spaces for your next queen. But if you put your first queen next to the board instead, it only threatens 21 places, since the fit is short. In other words, the middle ground and side squares are different — and as a result, the team does not have the same shape that would make the problem easier.
This shortcoming is why when Simkin went to see mathematician Zur Luria at the Swiss Federal Institute of Technology Zurich to help with the problem four years ago, he first dealt with “toroidal” n-prevent the problem. In the modified version, the chess board “wraps around” its side like a torus: When you drop to the right, you also get to the left.
The toroidal problem seems simple due to its similarity. Unlike the top board, all the diagonals are the same, and each queen can compete in the same position: 27.
Simkin and Luria attempted to create a design on the toroidal board using two layers. At each step, they placed the queen at random, and chose any similar location as long as they could be found. Then he closed all the places that he could destroy. Depending on the number of options available for each image, they expect to count fewer – fewer on the layout. Their approach has been called the unchanging greed of greed, and it has been used to solve many other problems in the affiliates.
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