Maths - The Queens Problem

25th May 2012 at 01:00

What it's all about

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, writes Jonny Griffiths.

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 above.

A tough question now: in how many fundamentally different ways (if rotations and reflections are the same) can this be done?

There is no known formula for this sequence but there are some easier questions that are accessible to everyone, such as `What is the maximum number of squares that a queen can threaten on an 8 by 8 board?'

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.

What else?

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

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