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1995 East-Central Regionals
ACM International Collegiate programming Contest

sponsored by Microsoft

Problem A - Pi

Professor Robert A. J. Matthews of the Applied Mathematics and Computer Science Department at the University of Aston in Birmingham, England has recently described howthe positions of stars across the night skymay be used to deduce a surprisingly accurate value of pi.This result followed from the application of certain theorems in number theory.

Here, we don'thave the night sky, but can use the same theoretical basis to form an estimate for pi:

Given any pair of whole numbers chosen from a large, random collection of numbers, the probability that the twonumbers have nocommon factor other than one (1) is

For example, using the small collection of numbers: 2, 3, 4, 5, 6; there are 10 pairs that can be formed: (2,3), (2,4), etc. Six of the 10 pairs: (2,3), (2,5), (3,4), (3,5), (4,5) and (5,6) have nocommon factor other than one. Using the ratio of the counts as the probability we have:

In this problem, you'll receive a series of data sets. Each data set contains a set of pseudo-random positive integers. For each data set, find the portion of the pairs which may be formed that have no common factor other than one (1), and use the method illustrated above toobtain an estimate for pi. Report this estimate for each data set.

Input

The input consists of a series of data sets. The first line of each data set contains a positive integer value, N, greater than one (1) and less than 50. There is one positive integer per line for the next N lines that constitute the set for which the pairs are to be examined. These integers are each greater than 0 and less than 32768. Each integer of the input stream has its first digit as the first character on the input line. The set size designator, N, will be zero to indicate the end of data.

Output

A line with a single real value is to be emitted for each input data set encountered. This value is the estimate for pi for the data set. The default real/float output format should be used. Answers must be within 0.1% of the correct answer. For some data sets, it may be impossible to estimate a value for pi.This occurs when there are no pairs without common factors. In these cases, emit the single-line message: 
No estimate for this data set.
exactly, starting with the first character, "N", as the first character on the line.

Sample Input

5
2
3 
4 
5 
6 
2 
13 
39 
0

Output for the Sample Output

3.162
No estimate for this data set.

Judge's Data

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Last updated November 6, 1997