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 JAVA (judge verdict: Accepted)

import java.io.BufferedReader;
import java.io.IOException;
import java.io.InputStreamReader;

import static java.lang.Integer.parseInt;

/**
 * Lagrange's Four-Square Theorem
 */
public class Main {

	public static void main(String[] args) throws IOException {
		BufferedReader br = new BufferedReader(new InputStreamReader(System.in));
		String line;
		String[] words;

		//2^15 = 32768
		//sqrt(32768) = 181.01...
		int[] vs = new int[182];
		int[][] dp = new int[32769][4];

		for (int i = 1; i < 182; i++) {
			vs[i] = i * i;
			dp[vs[i]][0] = 1;
		}

		for (int i = 181; i >= 1; i--) {
			for (int j = 1; j < 4; j++) {
				for (int k = vs[i]; k < 32769; k++) {
					if (dp[k][j - 1] > 0 && k + vs[i] < 32769) {
						dp[k + vs[i]][j] += dp[k][j - 1];
					}
				}
			}
		}

		while ((line = br.readLine()) != null && !line.isEmpty()) {

			int n = parseInt(line);
			if (n == 0) break;

			int sum = 0;
			for (int i = 0; i < 4; i++) {
				sum += dp[n][i];
			}
			System.out.println(sum);

		} //end while
	}//end main
}