Well, lets say you have 3 elements (1,2,3) with 2 being 0. So since 2 is 0, 1+2=1=2+1, 2+2=2, 3+2=3=2+3. With me so far ? So what's 1+3 ? Well, it cannot be 1 b

Well, lets say you have 3 elements (1,2,3) with 2 being 0. So since 2 is 0, 1+2=1=2+1, 2+2=2, 3+2=3=2+3. With me so far ? So what's 1+3 ? Well, it cannot be 1 because you could then cancel the 1's and 3 would become 0, but 3 isn't 0, its 2, because we just said so. 1+3 is obviously not 3, because by the same reasoning of cancellation, you'd end up with 1 equals 0, which it ain't cause 2 is 0. So by the weird magic of Niels Abel, 1+3 must indeed be 2. Because it just can't be anything else, yeah ? You only have 3 elements to play with, and 1+3 better be one of those. Similarly, you can work it out that 1+1=3, 3+1=2 and 3+3 =1. That completes your Cayley table and you can collect your paycheck and go home to wife and kids, because with 3 elements your Cayley has 9 slots. But when you have 10,001 employees, your Cayley will have 100 million slots. Unless you have a rule to fill those slots, you won't be able to fill the Cayley. "Adding employees" is simply a rule to fill the Cayley. HTH :)