Problem:
Problem B: Lagrange's Four-Square Theorem

The fact that any positive integer has a representation as the sum of at most four positive
squares (i.e. squares of positive integers) is known as Lagrange’s Four-Square Theorem. The
first published proof of the theorem was given by Joseph-Louis Lagrange in 1770. Your mission
however is not to explain the original proof nor to discover a new proof but to show that the
theorem holds for some specific numbers by counting how many such possible representations
there are.

For a given positive integer n, you should report the number of all representations of n as the
sum of at most four positive squares. The order of addition does not matter, e.g. you should
consider 42 + 32 and 32 + 42 are the same representation.

For example, let’s check the case of 25. This integer has just three representations 12 +22 +22 +42 ,
32 + 42 , and 52 . Thus you should report 3 in this case. Be careful not to count 42 + 32 and
32 + 42 separately.

Input

The input is composed of at most 255 lines, each containing a single positive integer less than
215 , followed by a line containing a single zero. The last line is not a part of the input data.

Output

The output should be composed of lines, each containing a single integer. No other characters
should appear in the output.

The output integer corresponding to the input integer n is the number of all representations
of n as the sum of at most four positive squares.

Sample Input

1
25
2003
211
20007
0

Output for the Sample Input

1
3
48
7
738

Solution in C# (judge verdict: Accepted)

using System;
using System.IO;
using System.Collections;
using System.Collections.Generic;
using System.Text;
using System.Linq;

class mainClass
{
	static Scanner sc;
	public static void Main(string[] args)
	{
		sc = new Scanner();
		pre();
		while (solve()) {}
	}
	static List<int> n = new List<int>();
	static void pre()
	{
		for (int i = 0; i < (1 << 16); i++)
		{
			int sq = (int)Math.Sqrt(i);
			if (sq * sq == i) n.Add(i);
		}
		n.Reverse();
		return;
	}
	static bool solve()
	{
		int N = sc.nextInt();
		if (N == 0) return false;
		int cnt = 0;
		for (int i = 0; i < n.Count; i++)
		{
			if (N - n[i] > n[i] + n[i] + n[i]) continue;
			if (N - n[i] < 0) continue;
			for (int j = i; j < n.Count; j++)
			{
				if (N - n[i] - n[j] < 0) continue;
				if (N - n[i] - n[j] > n[j] + n[j]) continue;
				for (int k = j; k < n.Count; k++)
				{
					if (N - n[i] - n[j] - n[k] < 0) continue;
					if (N - n[i] - n[j] - n[k] > n[k]) continue;
					for (int l = k; l < n.Count; l++)
					{
						if (N - n[i] - n[j] - n[k] - n[l] == 0) cnt++;
					}
				}
			}
		}
		Console.WriteLine(cnt);
		return true;
	}
}

public class Scanner
{
	public Scanner() { }
	public string next()
	{
		return Console.ReadLine();
	}
	public int nextInt()
	{
		return int.Parse(next());
	}
	public double nextDouble()
	{
		return double.Parse(next());
	}
	public long nextLong()
	{
		return long.Parse(next());
	}
	public string[] nextArray()
	{
		return next().Split(' ');
	}
	public int[] nextIntArray()
	{
		return Array.ConvertAll(nextArray(), e => int.Parse(e));
	}
	public long[] nextLongArray()
	{
		return Array.ConvertAll(nextArray(), e => long.Parse(e));
	}
	public double[] nextDoubleArray()
	{
		return Array.ConvertAll(nextArray(), e => double.Parse(e));
	}
}