A Harvard Mathematician Has Basically Solved an Epic, 150-Year-Old Chess Problem

chess is a simple game, with 64 individual black or white squares, 16 pieces per side, and two competitors trying to win.

The game offers incredibly complex possibilities, posing challenges to chess theorists and mathematicians that can go undiscovered for decades or even centuries.

One of the challenges was finally solved in July of 2021. The n-queens problem, which has puzzled experts since it was first imagined in the 1840s, was put to Michael Simkin from Harvard University.

The queen is the most powerful piece on the board and can move any number of squares in any direction. The n-queens problem is about how many arrangements can be made if the queens are too far apart.

The answer is 92 for eight queens on a standard 8 x 8 board, although most of these are not fundamental solutions.

There is a board with 1,000 x 1,000 squares. What about a million people? The number of queens is raised to the power of n by Simkin.

It is not the precise answer, but it is as close as you can get right now. We won't reproduce it for you here because the figure comes out as a number with five million digits after it.

It took Simkin almost five years to come up with the equation with a variety of approaches and techniques used. The mathematician was able to calculate the lower and upper bounds using different methods, finding that they almost matched.

If you told me you wanted to put your queens in such a way, I would be able to analyze the solution and tell you how many matches there are.

It reduces the problem to an optimal problem.

Simkin and Luria collaborated on a variation of the n-queens problem known as the torodial or modular problem. In this one, the diagonals wrap around the board so a queen can move from the right edge to the left.

The queen in the corner of the board doesn't have as many angles of attack as the queen in the center.

The pair's work on the toroidal problem stopped, but Simkin adapted some of the fruits of that labor into his final solution.

As the boards get bigger and the number of queens increases, the research shows that in most allowable configurations the queens congregate along the sides of the board, with fewer queens in the middle, where they are exposed to attack. A more weighted approach is enabled by this knowledge.

Simkin has gotten us closer than ever before, and he is happy to pass the challenge on to someone else to study further.

I think that I may be done with the n-queens problem for a while, not because there isn't anything more to do with it, but just because I've been dreaming about chess and I'm ready to move on with it.

Simkin's paper on the solution can be found on the preprint server arXiv.