Tuesday, December 16, 2008
I'm in the complete graphs, solvin' yur problemz
So this is how math works. You get a problem, you fiddle with it, you hope something you know fits, though you never know for sure when something will fit. Because the problem was made up out of whole cloth, sometimes from a real life situation, sometimes as an extension of another math problem, you can't even be sure there is an answer.
The problem a friend asked me to solve was about 36 people put at tables of six each, which I wrote about on Sunday. I haven't solved it. I have solved 9 people at tables of 3 each, 16 people at tables of 4 each, 25 people at tables of 5 each, and oddly enough, 49 people at tables of 7 each.
I may have just gotten lucky with 16 people split into groups of 4. But the prime numbers, not lucky. As a mathematician, and when you prove something, you know when you get it, and it feels really good.
For any prime p no matter how big, you can split p squared into p tables of p each over p+1 evenings so that everybody in the group will sit at a table with everyone else exactly once. It may not work for non-primes, but it will always work for primes.
I'm solvin' yur problemz, 'cuz that's how I roll.