Take a square piece of paper about the size of four pennies laid side-to-side, and divide it into sixteen penny-sized squares. Then take seven (small) paperclips and two pennies, and lay out the pennies so that it isn't possible to fit all the paperclips onto the whole square without placing a paperclip diagonally. Each paperclip should take up two squares. There's at least three or four ways to do this. Now - what do all the ways share in common? Is there a way to predict what will work and what won't? Could you have predicted what will work in advance?
If two kids have to divide a cake evenly between the two of them, what's the fairest way? Clearly, the first has to divide the cake and the second chooses which piece he wants, because that way the first always gets the smaller of the two pieces and thus will always divide evenly.
Now, what's the fairest way to divide the cake into three pieces? You can PM me if you want me to tell you if your answer is correct -- this puzzle, while not too challenging, is too fun to give away to others.