Maths - The Queens Problem

A chess-based question with many fascinating permutations

Jonny Griffiths

In Elizabeth II's Diamond Jubilee year, why not pose your young mathematicians a royal question to mark the event? For example, how many queens can they place on a 60 by 60 chessboard so that no queen can be taken by any other?

The answer cannot be more than 60, since the Pigeonhole Principle says that if you post n + 1 letters into n pigeonholes, there must be at least two in one pigeonhole. Therefore, if we have 61 or more queens, there must be at least two queens in a row, which would mean they could take each other. It is, in fact possible to place 60 queens on the board. (In general, given an n by n board, you can always place n queens on it, as long as n = 4 or bigger). Taking the more familiar case of an 8 by 8 board, one possible arrangement is shown in Figure 1 (see left).

A really tough question now: in how many fundamentally different ways (if rotations and reflections are the same) can this be done? This is a sequence that gets very big extremely quickly:

There is no known formula for this sequence. Thankfully, there are some far easier questions that are accessible to everyone. What is the maximum number of squares that a queen can threaten on an 8 by 8 board? The minimum? Enlighten students about the nCr button on their calculators. How many ways can you put 8 queens on an 8 by 8 board? The answer is "64 choose 8" or 64C8 (counting rotations and reflections as different), which is 4,426,165,368. Of these, just 92 are solutions to the problem. So if we place 8 queens at random on the board, what is the probability that we will hit on a solution? About 1 in 50 million.

The Queens Problem represents a neat challenge to the computer programmer. The University of Utah has a delightful applet at http:bit.lypWQptp that shows you how to set about finding a solution for any n.

Finally, a related question: on an n by n board, what is the minimum number of queens needed so that every square is either occupied or threatened? It is best to start with smaller boards; for the 8 by 8 board, the answer is 5 (see Figure 2, left).

Jonny Griffiths teaches maths at a sixth-form college.

What else?

For more chess-themed maths, try an activity from MrBartonMaths. What is the fewest number of moves a knight can make to get from one corner to the opposite corner on a chess board with 100 squares?

Or make maths merry with Owen Elton's Diamond Jubilee Arithmetic Game.

Find all links and resources at


In honour of the Jubilee, maths teachers discuss a problem-solving activity built around the idea of a street party.

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Jonny Griffiths

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