Answer to A Math is a 150 Year Chess Problem

If you have how many sets of chess at home, try the following exercise: Arrange eight queens on a board so that none of them will attack. If you succeed once, will you find a second arrangement? Third? How many are there?

This challenge is over 150 years old. This is the first version of a math question called n-provides problem whose solution Michael Simkin, a postdoctoral associate at Harvard University’s Center of Mathematical Science and Applications, zero sa sa in a paper posted in July. Instead of placing the eight queens on a standard 8-by-8 chessboard (where there are 92 different entrances available), the problem asks how many ways can be placed. n queens of a n-by-n board. It could be 23 queens on a 23-by-23 board-or 1,000 on a 1,000-by-1,000 board, or any number of queens on a board of the corresponding size.

“It’s very easy to explain to anyone,” he said Erika Roldán, with Marie Skłodowska-Curie of the Technical University of Munich and the Swiss Federal Institute of Technology of Lausanne.

Simkin confirms that for many chessboards with multiple queens, there is an estimated (0.143n)n trusts. Thus, on a one million board, the number of ways to prepare 1 million non -threatening queens is almost 1 followed by about 5 million zeros.

The original problem of the 8-by-8 chessboard first appeared in a German chess magazine in 1848. By 1869, the n-next problem follows. Since then, mathematicians have created a drop in the results of n-resurrection Even if past researchers used computer simulations to determine the outcome Simkin found, he was the first to prove it.

“He’s done it a lot smarter than it’s ever been,” he said Sean Eberhard, a postdoctoral fellow at the University of Cambridge.

An obstacle to solve nThe problem with -queens is that there are no clear ways to simplify it. Even on a small board, the number of potential arrangements of queens can be large. On a much larger board, the amount of calculation is shocking. In this case, mathematicians always hope to find some underlying pattern, or structure, that will allow them to break down calculations into smaller pieces that are easier to control. But the n-we see the problem of the problem as nothing.

“One of the things that is famous about the problem is that, even if it’s not thought out very well, it doesn’t have any structure,” Eberhard said.

This stems from the fact that not all blanks on the board are created equal.

To find out why, also imagine doing your own healing of eight queens. If you place your first queen near the center, it can attack any space in its row, on its column, or below two of the highest diagonals on the board. That leaves 27 spaces with no limit for your next queen. But if you put your first queen on the edge of the board instead, it threatens only 21 spaces, because the corresponding diagonals are shorter. That is, the middle and side frames are different – and as a result, the board does not have a symmetrical structure that can make the problem much simpler.

This lack of structure is why, when Simkin visited mathematician Zur Luria at the Swiss Federal Institute of Technology Zurich to work on the problem four years ago, they first solved the more symmetrical “toroidal” n-providing problem to problem. In this modified version, the chess board “wraps” itself around the edges like a torus: If you fall to the right, you will also appear on the left.

The toroidal problem seems simpler because of its symmetry. Unlike the classic slate, all diagonals are the same length, and each queen can attack the same number of spaces: 27.

Simkin and Luria tried building toroidal board trusts using a two -part recipe. At each step, they place a queen with no choice, choosing any space with equal possibilities as long as it is available. They then blocked all the spaces it could attack. By tracking how many options they have at each step, they hope to calculate a lower limit – an absolute minimum for the number of adjustments. Their strategy is called a random greedy algorithm, and it is used to solve many other problems in the area of combinations.

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