Among pairs of numbers whose sum is 16, 8*8=64 is the greatest possible product. However, if we allow for a sum of 17 then there are 3 distinct ways of achieving a higher product with two positive integers: 9*8, 10*7, and 11*6 are all greater than 64.

Among pairs of numbers whose sum is 2n, n*n=n^{2} is the greatest possible product. However, if we allow for a sum of 2n+1 then there are 2012 distinct ways of achieving a higher product with two positive integers.

Find the minimum value of n.

(In reply to

re: A very tricky problem (spoiler) by Charlie)

Charlie:

Thanks for catching this. I missed a digit in my multiplicand. Or maybe it was my multiplier. This is now fixed in my original post.

Actually, I also had a more serious error. I had missed the first pair for non-integral n, namely (n + .5) * (n+ .5). This changed my calculations, and now I have the minimum n as a non-integer (different solution from Broll's) and the maximum n as an integer.

A much more interesting problem than I expected. Thanks, Jer.

Steve

*Edited on ***May 29, 2012, 3:54 pm**