First, we see this example. Suppose we have a set of 6 elements, we can get 3 subsets of it, each of which has 2 elements, but no two sets overlap. But if our set has 5 elements, we want to get 3 subsets of it each of which has 2 elements . Then two of them need to overlap on 1 element. Now generally suppose we have a set of #a elements and we want 3 subsets of it each of which has #a' elements. We also want each two of them have the same but least # in common. Could you get the relationship between a and a'? How many does any two of them in common? How many does three of them has in common? For example, if a=3a', the we can have no pair of the three subsets overlap.
You might try the problem for dividing a set into two sets first, and explore the possibilities. For working with three sets, I like to visualize the set as evenly distributed around a circle, with the subsets covering certain arcs of the circle. Perhaps this visualization will help you.
Gerhard "Ask Me About System Design" Paseman, 2010.04.11
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