About my custom signature, £åÕ³. This stands for a problem in number theory that dates back to dynastic Egypt. Contributions to its solution were made by such notables as Fibonacci and Sylvester (the latter being the founder of the American Mathematical Society), but a completely satisfactory solution had to await the arrival of computers and computation-intensive techniques (modesty forbids me saying more). I can't really state the problem here in it's full generality, justifying the "sum of the products" notation åÕ, but its most elementary instance can be given in the form of a riddle:
Ben Cartright died, having lived a full and happy life, and he left his estate to his three sons. Included in the estate were 17 horses. The instructions in the will were that the estate be divided in these proportions: one-half to Adam, the oldest son; one-third to Hoss, the second son; and one-ninth to Little Joe, the youngest.
Well, you see the problem. You can't take half of 17 horses. For that matter, you can't take one-third of 17 horses either, or one-ninth. So they called upon the circuit-riding judge to resolve the matter.
The judge's solution was this: She put her horse into the corral with the others, making 18 in all. Then she took half of the eighteen horses (9) and gave them to Adam; she took a third of the eighteen horses (6) and gave them to Hoss; and one-ninth of the eighteen horses (2) and gave them to Little Joe. So everyone was happy, having received exactly the number of horses he had coming to him.
The judge then got back on her horse and rode away.
The riddle is -- wait a minute, didn't the judge give out her horse, too, as one of the 18?