The following puzzle is borrowed from a 1965 edition of "Alan Goode's Mind Bender" column, and reproduced here with the author's permission.

The puzzle is this:
At each of the four corners of a square room (the length of a wall being L metres) is a dog. At a specified moment, each dog sets off at the same speed chasing its neighbour. At any time during the ensuing pursuit, each dog is always travelling directly towards the dog it is chasing. How far does each dog travel before they all meet? 
So here's your activity  think about the puzzle yourself, and work out your answer....
For convenience we'll discuss this puzzle as shown in the diagram, although the answer is the same the other way, as long as all the dogs chase the same way. Also note that this is not a word game, don't look for a trick in the words used, just think about the problem described.
If you're finding it hard to visualize, why not try it out? Get four people, arrange them at the corners of a square, and label them round the square A, B, C and D. On a given count, everyone take one footstep towards the next person, so A steps towards B, B towards C, C towards D and D towards A. Then everyone take a second step, and so on, until you all meet. Ideal pubtime fun. It will at least give you an idea of what's happening, before you get to the pen and paper.
It helps to think about the symmetry of the problem first of all. There are four dogs, which are identical, and all their motions must also then follow the symmetry. According to the author, "It sounds difficult  it isn't!", but I'd say it's not all that easy either.
Once you have your answer (and not before!), see if you agree with the solution. Also, please feel free to submit your thoughts on this problem via email to mail@activityworkshop.net.