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Square Corners

Age 7 to 11 Challenge Level:

Square Corners

On each of these grids the counters lie at the four corners of a square:


What is the greatest number of counters you can place on the grid without four of them lying at the corners of a square?

You might like to use this printable sheet.

This problem is taken from 'Mathematical Challenges for Able Pupils Key Stages 1 and 2', published by DfES.

Why do this problem?

This problem challenges pupils' understanding of the properties of squares in the sense that squares do not necessarily have to be orientated so that their sides are horizontal and vertical. It is a good context in which to encourage children to find a systematic approach.

Possible approach

You could begin by playing the game Seeing Squares a few times as a class. This will provoke discussion amongst pupils about what makes a square a square, and you may wish to address the misconception that tilted squares are called 'diamonds'.

Introduce the problem, perhaps by using the grid on the interactive whiteboard and ask children to begin to work on it in pairs. They could use the large grid with counters and/or this sheet of smaller grids. Remind them to check for squares! After a suitable length of time, share results so far amongst the whole group. What is the largest number of counters so far? Ask a pair or pairs of learners to come up and recreate their arrangement of counters on the screen so that everyone can check there aren't any squares.

At this point, challenge the class to come up with a way of working that will ensure the largest number of counters is definitely found. How will they know that all arrangements have been tested? Some children may suggest a system based on, for example, placing counters on the grid from top left to bottom right and each time checking that a square has not been made. You could model the beginnings of a strategy and then give the class more time to investigate the problem. It may be that you split the group up to investigate different 'families' of arrangements.

In the plenary, you could discuss the solutions and what makes one arrangement of counters different to another. Will rotations and reflections be considered different or the same?

Here is a printable version of the problem.

Key questions

How will you know that you have definitely found the largest number of counters?
Are you sure there aren't any squares on your grid?

Possible extension

Children could investigate larger grids and see whether there is a pattern to the number of counters by looking at smaller grids too. Is it possible to predict the largest number of counters in any size grid?

You might like to use this interactive to check solutions to larger grid sizes.

Possible support

Some pupils could start with a three by three grid.