I was busy over the weekend, and only had time to get started on this yesterday. I was relieved to find the grid fill much easier than some recent numerical puzzles and was amazed by the final result, like most others judging by other posts in this thread. Thanks, Elap, a delightfully constructed puzzle. Just one slight worry, concerning the minimum value of N. Nothing in the preamble or hint says that N cannot by zero or negative. To a non-mathematician, the notion of raising a number to a zero or negative power could be hard to fathom, although mathematicians use the equation (x^m)*(x^n)=x^(m+n) to give definitions: x^0=1 and X^-n=1/(x^n). Suing this one could argue that the property is true for N=0, which it would be for any set of six positive integers. I wasn't sure whether or not this would affect the "correctness" of an solution submitted that ignore this.