In the chess game, the queen is the most powerful piece. It has the unique ability to move an unlimited number of squares in any direction. It moves horizontally, vertically, or diagonally. For example, how many ways could you arrange eight queens on a typical 8-by-8 square board to ensure that they couldn’t attack each other? There are, in fact, 92 of them. A 1K-by-1K-square-chessboard may have 1K different queens on it. But how about a board with 1,000,000 different royal queens? Would the result be the same?

The original form of the n-queens mathematics problem initially appeared in a German chess journal in 1848 as the eight-queens issue. The right answer came a couple of years later. It wasn’t until late last year that a Harvard mathematician came up with an almost certain answer to the issue.

Postdoctoral fellow Michael Simkin at the Center for Mathematical Sciences and Applications found that there are about n ways(0.143n) the queens may be arranged on enormous n-by-n chessboards so that none of them are attacking each other.

Simkin’s final equation doesn’t give a precise answer. It does say that this number is the closest you can come to the real number right now. The result is obtained by multiplying the 0.143 figure. It reflects an average degree of uncertainty in the probable outcome of the variable, by whatever n is.

For example, Chess with one million queens would multiply 0.143 by one million. It results in a total number of 14,33,000. The number would then be increased to the millionth power, implying that it has been multiplied one million times. In the end, the solution is a five million digit number.

By understanding the fundamental pattern for how massive numbers of queens had to be placed on these large chessboards — whether they would be concentrated in the center or on the periphery — Simkin was able to develop the equation.

In that case, Simkin remarked, “I can examine the algorithm and tell you the different solutions that fit this constraint.” Simkin used this example to illustrate his point. Formally speaking, the problem is reduced to an optimization problem.

It was Simkin’s job to calculate how many queens were likely to be in each part of the board to come up with a formula for viable setups. The lower bound – the smallest possible number of configurations — was the outcome of the computations.

Using a procedure known as the entropy method, Simkin was able to determine the upper bound, which is the maximum number of configurations.

When Simkin looked at the lower bound, he discovered that it was nearly identical to the upper bound response. That is to say, it demonstrated that the precise solution lies somewhere between the two boundaries in a reasonably tiny mathematical area.

N-queens research has taken Simkin nearly five years to complete. It is clear by the demeanor of his actions, however, that he is not a good chess player. But I believe arithmetic is more forgiving,” Simkin said. Simkin became most interested in the topic. Because of how he might use advancements from his school of maths called combinatorics. it is focused on counting and choosing configurations of objects.

It has been a challenge to persevere in the face of this obstacle. He visited Zur Luria in Zurich at the Swiss Federal Institute of Technology four years ago as a Ph.D. student at the Hebrew University of Jerusalem. The two worked together to come up with innovative methods for solving the problem. A better lower limit number emerged after two years of effort, but they realized they were missing something.

Harvard hired Simkin as a postdoctoral fellow in 2020 after he completed his Ph.D. and relocated to Boston. The issue was always at the back of his mind. He came back to it when he recognized that he had to start focusing on the places where the queens would be rather than giving equal weight to each location.

There is potential room for improvement. But for the time being Simkin is willing for someone else to come up with an even more precise solution.

This is the last time I will be dealing with the n-queens problem for a while, not because there is nothing else to do, but because I have been dreaming about chess and I am ready to move on with my life, “he added.