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.

There are two values here on which I want to focus, P and S.

IF I make the assumption that P must have the lowest PD value and (P+S) is allowed the highest then the following results:

P         1023456789
Q         3010108204
R         4938271605
S         8853086421
P+S       9876543210

In the above P, R and (P+S) are pandigital.

To run a computer program on my computer (and QuickBasic) does not seem feasible.

I could set up two For/If..EndIf/Next loops, one to generate incrementing PD values for P and the other to generate decrementing values of S which start at the value in my above table.

Since P and S would be PD, one would, as one option, convert Q, R and (P+S) to Strings and test for duplication.

The difficulty with this process however is that every P value generated (up to that time) needs to be tested against every S value generated (up to that time) until all conditions are met.

If those conditions are met very early I see no comfortable computer resolution from my resources.

That leads me to believe that there is a more direct, analytical method to achieving the goal.

Hint please?

Posted by brianjn on 2009-07-22 01:35:42