Harvard mathematician solves 150-year-old chess problem


Known as the Eight Queens Puzzle, it was first mentioned in a German chess magazine in 1848 by a certain Max Friedrich William Bezzel. The problem is to place eight queens on a chessboard in such a way that no two queens threaten each other. The problem is fairly easy to solve – even a complete beginner should be able to construct a position that satisfies the requirement.

But how many solutions are there to the problem? Below are twelve fundamentally different solutions (Source: Wiki). Most positions have eight variants, which you can get by rotating and mirroring them 90, 180 or 270°. However, one of the basic solutions, the last one shown below, is identical to its own 180° rotation and has only four variants.

So it turns out that the total number of different solutions is 92, as was soon finally established. But then the question arose: How many different ways could queens be placed on larger boards? How many ways are there to place n queens on an nxn board in such a way that no queen attacks another queen?

Turns out there’s no known formula to figure this out. On a 9×9 board there are 352 different ways, on a 10x 10 board it is 724. The largest board for which an exact solution has been worked out is the 27×27 board. There are exactly 234,907,967,154,122,528 different ways to place the 27 queens so that none attacks the other. Working it out was a very arduous task.

How about larger numbers, eg 1000 queens on a 1000 x 1000 board? Or a million queens on a million square board? It was a problem that intrigued mathematicians. In 2021, Michael Simkin found a way to calculate the result for very large numbers of n. He has been working on the problem for almost five years, applying breakthroughs in combinatorics, which focus on counting and selection and arrangement problems. He calculated that it was approximately (0.143n)n Ways the queens can be placed on giant n by n chessboards. That last equation doesn’t give the exact number, but it gets as close to the actual result as no one can currently get. You can read about it in more detail in this Harward Gazette story. There we learn what the formula means:

For example, on the extremely large chessboard with one million queens, 0.143 would be multiplied by one million, giving about 143,000. That number would then be raised to the power of a million, meaning it’s multiplied by itself a million times. The final answer is a number with five million digits.

A little tip for our readers: Don’t try to list all of these positions. It would take too long.

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