Maths - The Queens Problem

18th May 2012 at 01:00
A chess-based question with many fascinating permutations

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 www.tes.co.ukresources035

IN THE FORUMS

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

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