A Harvard scientist solves a mathematical chess problem posed 150 years ago

A Harvard scientist has managed to solve a mathematical chess problem posed 150 years ago. This is the challenge of the n-queens or the eight queens, which has worried specialists since its original approach, in 1848, by the German chess player Max Bezzel.

The problem is to place eight queens on the chessboard without threatening each other. Given that the queens are the most powerful figure on the board and can threaten any piece on the same rank, file or diagonal, the problem asks how many arrangements are possible so that the queens are far enough apart so that they do not attack each other.

Although the original problem was solved a couple of years after it was raised, a broader version emerged in 1869, which remained unanswered until August of last year, when Michael Simkin, a postdoctoral fellow at the Harvard Center for Mathematical Sciences and Applications, provided an almost definitive answer.

The solution

Simkin calculated that there are some (0.143n)n ways to arrange the queens so that none attack each other on giant n-by-n chessboards. Simkin’s final equation does not provide the exact answer, but merely says that this figure is as close to the real number as can be obtained at this time.

On an extremely large chessboard with a million queens, for example, 0.143 would be multiplied by a million, resulting in 143,000. That number would be raised to the power of a million, that is, it would be multiplied by itself a million times. The final answer is a number with five million digits.

“If you told me I want you to place your queens in such and such a way on the board, then I could go through the algorithm and tell you how many solutions there are that meet this constraint,” Simkin said last Friday. “In formal terms, it reduces the problem to an optimization problem,” he added.

As boards get larger and the number of queens increases, research shows that in most allowed configurations, queens tend to congregate on the sides of the board, with fewer queens in the middle, where they are exposed. to attacks. Now the mathematician passes the baton to others to further study this problem.

“I think personally I can get rid of the n-queens problem for a while,” says Simkin. His article on the solution to this mathematical chess problem can be found on the arXiv preprint server.

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