Minimum number of guesses on sum and product required to find two numbers. I have a series of numbers 1 to N. A system randomly picks up two numbers and computes their sum and product. I have to guess the sum and product, The system will tell if the sum and product are larger or smaller. What is the optimal strategy to find the numbers in minimal number of guesses.

The following strategy reduces the number of options by a factor of at least three at each stage.
I refer to a possible (sum,product) as a $point$.
Let $P$ be a product where at most 1/3 of the possible points have product greater than $P$ and at most 2/3 of the possible points have product less than $P$.
Of those points with product less than $P$, let $S$ be the median of the sums.
The system will compare the true sum and product to S and P, and will reduce the set of possible points to a set no more than 1/3 what it was.
This method takes at most $\log_3{N\choose2}$ guesses.
