A mathematician answers a 150-year-old chess problem

If you have Set a few chess at home, try the following exercise: Arrange eight queens on a board so that none of them attack each other. Once you succeed, can you find a second solution? Third? How many are there?

This challenge is more than 150 years old. This is called the oldest version of a mathematical question n-The Queens problem whose solution has been nullified in a paper posted in July by Michael Simkin, postdoctoral fellow at Harvard University’s Center of Mathematical Sciences and Applications. Instead of having eight queens on a standard 8-by-8 chessboard (where 92 different configurations work), the problem asks how many ways there are n Queen on a n-By-n Board It can be 23 queens on a 23-by-23 board অথবা or 1,000-by-1000 on a 1,000-board, or any queen on a board of the corresponding size.

“It’s very easy to explain to someone,” said Rica Roldon, a colleague of Marie Skodovska-Kuri at the Technical University of Munich and Lausanne at the Swiss Federal Institute of Technology.

Simkin has proven that a huge number of queens have about (0.143) for a huge chess boardn)n Configuration Thus, on a million-by-a-million board, the number of ways to arrange 1 million non-threatening queens is almost 5 million after 1.

The main problem with the 8-by-8 chess board was first published in 1848 in a German chess magazine. By 1869, n-Queens followed the problem. Since then, mathematicians have developed a strategy of results n– Queens. Although recent researchers have used computer simulations to estimate Simkin’s results, he was the first to prove it.

Shawn Eberhard, a postdoctoral fellow at Cambridge University, said, “He basically did it much more sharply than anyone has done before.”

One obstacle to solution n-Queens The problem is that there is no obvious way to make it easier. Even on a relatively small board, the number of possible arrangements for queens can be huge. On a larger board, the amount of calculations involved is staggering. In these situations, mathematicians often hope to find some underlying pattern or structure, which allows them to divide their calculations into smaller pieces that are easier to handle. But n– Queens doesn’t seem to have a problem.

“One of the issues that is significant is that, at least without thinking too hard about it, there doesn’t seem to be any structure,” Eberhard said.

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

To see why, imagine creating your own eight-queen configuration again. If you place your first queen near the center, it will be able to attack anywhere in her row, in her column, or at two of the longest corners of the board. This goes beyond the 27 space limit for your next queen. But if you put your first queen next to the board, it only threatens 21 spaces, because the relevant corners are small. In other words, the center and side squares are distinct – and as a result, the board lacks a symmetrical structure that can make the problem easier.

Why this framework is lacking, when Simkin visited Jur Luria, a mathematician at the Swiss Federal Institute of Technology Zurich, four years ago to collaborate on the problem, they initially tackled a more symmetrical “troidal” n– Queens problem. In this modified version, the chess board “wraps” around itself at the edge like a terrace: if you fall to the right, you reappear on the left.

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

Simkin and Luria tried to create a configuration on a toroidal board using two-part recipes. At each step, they randomly placed a queen, selecting a place with equal probability as long as it was available. They then shut down all the places they could attack. Looking at how many options they had at each step, they expected to calculate a lower threshold – an absolute minimum for the number of configurations. Their technique is called random greedy algorithm and it has been used to solve many other problems in the field of combinatorics.

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