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Prison Cells

There are seventy eight prisoners in a square cell block of twelve cells. There is one prisoner in one of the cells, two in another cell, three in another, four in another and so on up to twelve prisoners in one of the cells.

The clever prison warder made it easy to check if the prisoners were all there by arranging them so there were twenty five along each wall of the prison block. How did he do it?

(There's more than one solution - send yours in - it might be different to everyone else's!)

Here is an interactive game with which to experiment.


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Why do this problem?

It is possible to solve this problem by trial and improvement but most children (and adults) find this frustrating and lengthy. The value of the problem lies in realising that it is worth doing some logical thinking to reduce the possibilities to a manageable number.

Possible approach

Offer the question and give the children some time to 'get into' it. If they haven't thought of it, you could suggest scraps of paper with the numbers $1-12$ which can be rearranged easily, rather than recording and rubbing out each time. You may see some other ways of recording which you can share with the class too.

Bring the children together and ask how they have started the problem. What can they learn from listening to each other's ideas? Some children will tease out the discrepancy between adding up all the numbers to $12$ ($78$) and adding four lots of $25$ together. Others may focus on the number of odds and evens needed for any one side (one or three) and use that as their starting point. Ask the children if they think there are lots of different solutions and confirm that there are so that everyone feels it is worth continuing even after someone else has found an answer!

Allow plenty of time for investigating and encourage the children to write their solutions and display them somewhere for everyone else to see them. I wonder how many different ones there are ...?

Key questions

What have we got to find out?
What do we know?
What shall we try first?

Possible extension

Children who find one solution quickly could be encouraged to find another one by rearranging some of the numbers in their own solution, rather than beginning afresh. In doing so they are beginning to generalise, an important mathematical skill. If you ask the children to record each solution on a separate piece of paper, then by moving and rearranging them they can see there are 'families' of solutions.

Possible support

Children who find this difficult could be given the grid with the corners filled in so that they start at a different place but end up with a complete solution, as everyone else.