I was playing with Hailstorm a few years back (and why that proof took 30 pages I'll never know!). I realized that it was actually a base 4 problem rather than a binary problem. As soon as I did that, I was able to "map" all real numbers to their "primary" node...1, 5, 21, 85, etc. I wanted to prove to a mathematician friend that any power of 4 minus 1 was evenly divisible by 3. And I couldn't remember enough of my high school geometry to prove it. So I came up with the following: Assume "n" to be the base of the numbering system. (One great thing about this...it doesn't matter how big "n" is). If "n" is the base of the system, then n to the x power is n followed by x zeroes. Now assume "m" to be one less than "n". Subtract 1 from n to the x and the result will be a string of x m's...obviously divisible by m. My friend was not overly amused, and proceeded to show me the traditional proof...which I promptly forgot. I just wonder if others use this sort of slightly unusual solution to some of the basic theorems and axioms of geometry. That seems to be a lot of what I'm seeing on this site.