>George, John-
>
>Beautiful!!!
>
>I am humbled by both yours and John's intellects. I came up with my own, but it wasn't quite as simple. I approached it completely differently, from a more mathematical side. Here is what I came up with:

You shouldn't be. Your's is as beautiful as any of the others. In looking at it, it occurred to me that it might better in some respects. Certainly, I think that the mathematical approach might be more in keeping with the nature of the problem.