How many queens can fit in a chess cake?

Take a chessboard (8 x 8 squares) and eight queens. Can you put those pieces together without two of them bumping into each other? Two queens can hit each other if they are in the same row, column or diameter. Yes, answer: One of the possible methods is shown here in the top right of the illustration.

Now imagine that a chessboard is made of elastic rubber. Make a chessboard tube by gluing two opposite sides together. Then glue the two rounded ends together to create a donut shape.

Then it turned out that there was nothing left of the solution on the flat chessboard. Just look at the two tars indicated: On the donut, the tar goes along. Two queens on one diagonal – this is not allowed.

Format as a distinctive feature

Mathematicians have been dealing with math on the chessboard since the 19th century, but it’s only in recent years that real progress has been made. On September 16, Candida Bottle and Peter Kivach, a mathematician from the University of Oxford, published a massive 161-page document. Back to print arXiv. In it they demonstrated a number of results, culminating in a formula for the number of solutions on large donut-shaped chessboards.

The starting point is always a square chessboard, not necessarily in 8×8 format, but in general n×n, which is fixed to a donut cake. The central question: How many methods are there? n To place the queens in a way that no other queen of families can? So it could be a 17 queen on a 17 x 17 donut board, 2021 a queen on a 2021 x 2021 board, or any number.

Exact answers are only available for some small and specific values ​​of n. For example, there are 4,524 solutions if n= 13 and 1,957,725,000 if n= 25. Mathematicians look for patterns and hope to find a general formula for plates of any size.

Unlike a flat chessboard, the donut board has a great deal of symmetry. While on a regular board, it is important that the queen is in the middle or somewhere on the edge, on a donut-shaped board, that each square is even. After all, the lengths of the diagonals on the flat plate are different, but on the donut, the rows, columns, and diagonals are all exactly the same length. This symmetry makes the donut board an interesting mathematical object.

Hungarian mathematician George Polia, author of the famous book published in several languages how to solve it, proven in 1918 when possible n Queens on a donut sized plate n×n For location: This only works if n Not divisible by 2 or 3. On an 8 x 8 flat plate there are no less than 92 solutions, but it follows from Bolea’s theorem that none of them hold if the plate takes the form of a round cake: after all, 8 is divisible by 2.

Pólya proved that there are solutions if n Not a multiple of 2 or 3, but he couldn’t answer the question about the number. How many solutions are there if n A thousand and one? Or two million and five? Is it reasonable to say about it in one form? This question has now been answered by Bowtell and Keevash. At least, a good approximation equation gives: for large values ​​of n, not divisible by 2 or 3, there are approximately (0.0498n)ⁿ Solutions.

Although the formula does not give an exact answer, it becomes clear how fast the number of possible formations is growing n gets bigger. In an email, Bowtell explains: “The growth is ‘very exponential’, because the variable n It occurs in both the base and the exponent. Except for a limited number of cases, our formula gives any value for n A very accurate approximation of the number of solutions.”

jogging outdoors

Bowtell began working on the queen problem nearly four years ago – it was one of the main topics of her doctoral research under Kivash. Many avenues for a possible solution turned out to be dead ends. Bautil sometimes felt he had to give up the whole problem. Nothing works. In those moments, running was a way to break through the wall: “Running outdoors created a space in my head to look at the problem differently.”

Finally, the puzzle pieces fell into place. Bowtell and Keevash made extensive use of so-called graph theory, a branch of mathematics with applications in, among other things, decision theory and computer science. Simply put, a graph is a network of nodes that may or may not be connected to each other. They formulated the problem in terms of “links” in such a network. A “link” is a set of connected line segments without common nodes.

The queen problem is part of a variety of combinatorial problems, as mathematicians search for “associative” generalities and formulas: the approximations that get better. n Increases. Bowtell: “These types of counting problems are generally very difficult. Therefore, our methods can be useful in answering similar questions.”