Sunday, February 2, 2014

Solution to area minimization problem

The problem in my last post is a special case of the following more general problem: What is the minimum combined area of N disks, if the sum of their radii is fixed?

Denoting the radius of the kth disk as r_k, the combined area is proportional to r_1^2 + r_2^2 + ... + r_N^2. By the inequality of geometric and arithmetic means, this sum takes its minimum value when the r_k are all equal. It follows that in the problem in my last post, the minimum area is reached when all three disks are the same size.

If you solved the disk problem more straightforwardly yourself by using algebra and/or calculus, then you probably didn't find the problem very hard. But the more general version of the problem as stated above is even more painless to solve. Surprisingly, it happens pretty often in mathematics that a more general version of a problem is the easier one. Polya talks about this phenomenon in his famous book on problem solving (this link should take you directly to the relevant section beginning on page 108). Polya gives several examples of hard or messy problems, along with generalizations of them, noting in each case that "the second problem is more general than the first, and, nevertheless, much easier than the first. In fact, our main achievement in solving the first problem was to invent the second problem." (Italics in the original.)

Of course, generalization doesn't always work. As Polya also notes, often the best angle of attack is to do the opposite and specialize.

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