所有可能幻方解法使用回溯

幻方 指N^2 格子放入所有1-N^2 的数子 使得格子水平列、垂直行或对角线数字和相同

求得所有可能 

使用回溯法

public class Magiques {
    int max, sumNum, n;
    public List<String> desQuassez(int n) {
        List<String> res = new ArrayList<>();
        if (n < 1) return res;
        int[][] board = new int[n][n];
        this.n = n;
        max =  n * n;
        sumNum = (max + 1) * n / 2;
        boolean[] allUsed = new boolean[n * n];
        int[] colSum = new int[n];
        int[] rowSum = new int[n];
        int forwardSum = 0;
        int inverseSum = 0;
        recurse(res, 0, board, allUsed, colSum, rowSum, forwardSum, inverseSum);
        return res;
    }
    /**
     * 递归求解
     * @param res 结果
     * @param fixN 第N个格子
     * @param board 填充数字的格子
     * @param allUsed 使用过的数字  allUsed[i] = true 指 第i个数字已被使用
     * @param colSum 列的和
     * @param rowSum 行的和
     * @param forward 正斜线的和
     * @param inverseSum 逆斜线的和
     */
    private void recurse(List<String> res,int fixN, int[][] board, boolean[] allUsed, int[] colSum, int[] rowSum, int forwardSum, int inverseSum) {
        int row = fixN / n;
        int col = fixN % n;
        if (col > 0 && row == n - 1 && colSum[col - 1] != sumNum) return;
        if (fixN >= n && col == 0 && rowSum[row - 1] != sumNum) return;
        if (row == n - 1 && col > 0 && inverseSum != sumNum) return;
        if (fixN == max) {
            if (forwardSum != sumNum) return;
            res.add(Arrays.deepToString(board));
            return;
        }
        for (int i = 1; i <= max; i++) {
            if (allUsed[i - 1]) continue;
            allUsed[i - 1] = true;
            board[row][col] = i;
            colSum[col] += i;
            rowSum[row] += i;
            if (col == row) forwardSum += i;
            if (col + row == n - 1) inverseSum += i;
            recurse(res, fixN + 1, board, allUsed, colSum, rowSum, forwardSum, inverseSum);
            if (col == row) forwardSum -= i;
            if (col == n - row - 1) inverseSum -= i;
            colSum[col] -= i;
            rowSum[row] -= i;
            board[row][col] = 0;
            allUsed[i - 1] = false;
        }
    }
    public static void main(String[] args) {
        Magiques testInstance = new Magiques();
        List<String> res = testInstance.desQuassez(4);
        res.stream().forEach(o -> System.out.println(o));
        System.out.println(res.size());
    }
}