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        riddles >> hard >> Sum Of Squares
        
(Message started by: K_Sengupta on Nov 25^th, 2005, 10:27pm)

Title: Sum Of Squares
Post by K_Sengupta on Nov 25^th, 2005, 10:27pm

Considering the actual magnitude corresponding to a string of Roman Numerals as P, let us define N as ; N = (P + 1000^B – 1) / (Q+S), where Q is the sum of the numerical magnitude of the place of letters in the English Alphabet which appear in the string (excluding bars) , S is the total number of letters or numerals in the string (excluding bars ) while B is the maximum number of bars corresponding to a numeral in a given string.
(For example, the magnitude corresponding to L-bar-bar X-bar is (51*10^6 + 10^4 – 1)/38, since B=2 ).
If appearance of a maximum of three (3) Bars in any Roman Numeral inclusive of a string is under consideration, determine the total number of strings of S Roman Numerals (not counting the bars) such that for each string, N is a whole number which is expressible as the sum of two distinct squares whenever 1=< S =< 50 .
Furthermore, if N is a T-gonal number for 3=< T=<5 and the sum of digits of N is a triangular number , what would be the total number of strings?
In addition, does a string satisfying all the other conditions of the problem exist, such that the sum of squares of the digits of N is a perfect cube?