Strange String Manipulation

Time Limit : 8 sec, Memory Limit : 65536 KB

Problem A: Strange String Manipulation

A linear congruential generator produces a series R(⋅) of pseudo-random numbers by the following for- mulas:

where S, A, C, and M are all parameters. In this problem, 0 ≤ S, A, C ≤ 15 and M = 256.

Now suppose we have some input string I(⋅), where each character in the string is an integer between 0 and (M - 1). Then, using the pseudo-random number series R(⋅), we obtain another string O(⋅) as the output by the following formula:

Your task is to write a program that shows the parameters S, A, and C such that the information entropy of the output string O(⋅) is minimized. Here, the information entropy H is given by the following formula:

where N is the length of the string and #(x) is the number of occurences of the alphabet x.


The input consists of multiple datasets. Each dataset has the following format:

I(1) I(2) ... I(N)

N does not exceed 256.

The last dataset is followed by a line containing one zero. This line is not a part of any dataset and should not be processed.


For each dataset, print in a line the values of the three parameters S, A, and C separated by a single space. If more than one solution gives the same minimum entropy, choose the solution with the smallest S, A, and then C.

Sample Input

5 4 3 2 1
7 7 7 7 7
186 8 42 24 154 40 10 56 122 72

Output for the Sample Input

0 1 1
0 0 0
8 7 14

Source: ACM-ICPC Japan Alumni Group Summer Camp 2009 , Day 3, Tokyo, Japan, 2009-09-20