六角幻方

把 1 2 3 ... 19 共19个整数排列成六角形状，如下：

    * * *
   * * * *
  * * * * *
   * * * * 
    * * *

    要求每个直线上的数字之和必须相等。共有15条直线哦！

    再给点线索吧！我们预先填好了2个数字，第一行的头两个数字是：15 13，参见图【p1.png】，黄色一行为所求。

    请你填写出中间一行的5个数字。数字间用空格分开。

    这是一行用空格分开的整数，请通过浏览器提交答案，不要填写任何多余的内容（比如说明性的文字等）

<img src="https://img-blog.youkuaiyun.com/20150315202424193?watermark/2/text/aHR0cDovL2Jsb2cuY3Nkbi5uZXQvU3Ryb25nTWFuVHdv/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70/gravity/Center" alt="" />
public class A {
	int[][] am = new int[15][19];
	int[] bm = new int[15];
	int[] col = new int[19];

	int[][] line = { { 0, 1, 2 }, { 3, 4, 5, 6 }, { 7, 8, 9, 10, 11 },
			{ 12, 13, 14, 15 }, { 16, 17, 18 }, { 0, 3, 7 }, { 1, 4, 8, 12 },
			{ 2, 5, 9, 13, 16 }, { 6, 10, 14, 17 }, { 11, 15, 18 },
			{ 2, 6, 11 }, { 1, 5, 10, 15 }, { 0, 4, 9, 14, 18 },
			{ 3, 8, 13, 17 }, { 7, 12, 16 } };

	void init() {
		for (int i = 0; i < line.length; i++) {
			for (int j = 0; j < line[i].length; j++) {
				am[i][line[i][j]] = 1;
			}
		}

		for (int i = 0; i < bm.length; i++)
			bm[i] = 38;

		for (int i = 0; i < col.length; i++)
			col[i] = i;
	}

	void switch_row(int a, int b) {
		if (a == b)
			return;

		for (int j = 0; j < am[a].length; j++) {
			int tmp = am[a][j];
			am[a][j] = am[b][j];
			am[b][j] = tmp;
		}

		int tmp = bm[a];
		bm[a] = bm[b];
		bm[b] = tmp;
	}

	int gcd(int a, int b) {
		if (b == 0)
			return a;
		return gcd(b, a % b);
	}

	// k列，用k行消i行
	void xiao_yuan(int k, int i) {
		int u = am[k][k];
		int v = am[i][k];
		if (u == 0 || v == 0)
			return;

		int g = gcd(Math.abs(u), Math.abs(v));

		for (int j = 0; j < am[i].length; j++)
			am[i][j] = am[i][j] * u / g - am[k][j] * v / g;

		bm[i] = bm[i] * u / g - bm[k] * v / g;
	}

	void try_switch_col(int k) {
		int w = k;
		for (int j = k; j < am[k].length; j++) {
			if (am[k][j] != 0) {
				w = j;
				break;
			}
		}

		if (w == k)
			return;

		for (int i = 0; i < am.length; i++) {
			int tmp = am[i][k];
			am[i][k] = am[i][w];
			am[i][w] = tmp;

			tmp = col[k];
			col[k] = col[w];
			col[w] = tmp;
		}
	}

	void gauss() {
		for (int j = 0; j < am.length; j++) {
			for (int i = j; i < am.length; i++) {
				if (am[i][j] != 0) {
					switch_row(i, j);
					break;
				}
			}

			try_switch_col(j);

			for (int k = 0; k < am.length; k++) {
				if (k != j)
					xiao_yuan(j, k);
			}
		}
	}

	void show() {
		for (int i = 0; i < am.length; i++) {
			for (int j = 0; j < am[i].length; j++)
				System.out.print(String.format("%3d", am[i][j]));
			System.out.print(String.format("%6d", bm[i]));
			System.out.println();
		}
		System.out.println("----");
		for (int i = 0; i < 19; i++)
			System.out.print(col[i] + " ");
		System.out.println();
	}

	void test(int[] x) {
		for (int k = 11; k >= 0; k--) {
			int t1 = 0;
			for (int i = 12; i < 19; i++)
				t1 += am[k][i] * x[i];
			int t = (bm[k] - t1) / am[k][k];

			if (t < 1 || t > 19)
				return; // 非法值
			for (int i = k + 1; i < 19; i++)
				if (t == x[i])
					return; // 重复值

			x[k] = t;
		}

		System.out.println("find: ");
		int[] y = new int[19];
		for (int i = 0; i < 19; i++)
			y[col[i]] = x[i];
		for (int i = 0; i < 19; i++)
			System.out.print(y[i] + " ");

		// for(int i=0; i<19; i++) System.out.print(x[i] + " ");
		System.out.println();
	}

	// k: 当前要填充的位置
	void f(int[] x, int k) {
		if (k < 12) {
			test(x);
			return;
		}

		L1: for (int n = 1; n <= 19; n++) {
			for (int i = k + 1; i < 19; i++)
				if (x[i] == n)
					continue L1;
			x[k] = n;
			f(x, k - 1);

			if (k == 18)
				System.out.println(".. " + n);
		}
	}

	void search() {
		int[] x = new int[19];

		f(x, 18);
	}

	public static void main(String[] args) {
		A a = new A();
		a.init();

		a.gauss();
		a.show();

		a.search();
	}
}