Each of P, Q, R, S and (P+S), with P < Q < R < S, is a non leading zero 10-digit base ten positive integer containing each of the digits from 0 to 9 exactly once. It is known that R is the arithmetic mean of P and S, and Q is the geometric mean of P and S.

Determine the minimum value of P and the maximum value of S.

(In reply to re: Basic thoughts by brianjn)

Yes, I know how you feel Brianjn - it's easy enough to write a program that checks all the possibilities, but will we still be alive when the answer appears?

I too have ruled out the possibility of examining all pairs of P and S values - too long. However, I think there's a chance of pinning the value of R down. After all, P+S=2R, and only a fraction of the 10! pandigitals have a non zero first digit and can be doubled to give P+S as a pandigital.

Apart from the fact that they're all divisible by 9, I know very little else about the mathematical properties of pandigitals. However, I do believe that if all common factors are removed from P, Q, R and S they still obey the AM, GM relationships and I think they may then all be odd numbers with some interesting Pythagorean connections which may allow the R values to be generated.

So that's where I am. It's great fun, but there's probably a very simple approach that we're missing.

Posted by Harry on 2009-07-23 10:55:05