Problem:
Score : 900 points

Problem Statement

AtCoDeer the deer has N square tiles. The tiles are numbered 1 through N, and the number given to each tile is written on one side of the tile. Also, each corner of each tile is painted in one of the 1000 colors, which are represented by the integers 0 between 999. The top-left, top-right, bottom-right and bottom-left corner of the tile with the number i are painted in color C_{i,0}, C_{i,1}, C_{i,2} and C_{i,3}, respectively, when seen in the direction of the number written on the tile (See Figure 1).

Figure 1: The correspondence between the colors of a tile and the input

AtCoDeer is constructing a cube using six of these tiles, under the following conditions:

For each tile, the side with the number must face outward.

For each vertex of the cube, the three corners of the tiles that forms it must all be painted in the same color.

Help him by finding the number of the different cubes that can be constructed under the conditions.
Since each tile has a number written on it, two cubes are considered different if the set of the used tiles are different, or the tiles are used in different directions, even if the formation of the colors are the same. (Each tile can be used in one of the four directions, obtained by 90° rotations.) Two cubes are considered the same only if rotating one in the three dimensional space can obtain an exact copy of the other, including the directions of the tiles.

Figure 2: The four directions of a tile

Constraints

6≦N≦400

0≦C_{i,j}≦999 (1≦i≦N , 0≦j≦3)

Input

The input is given from Standard Input in the following format:

N
C_{1,0} C_{1,1} C_{1,2} C_{1,3}
C_{2,0} C_{2,1} C_{2,2} C_{2,3}
:
C_{N,0} C_{N,1} C_{N,2} C_{N,3}

Output

Print the number of the different cubes that can be constructed under the conditions.

Sample Input 1

6
0 1 2 3
0 4 6 1
1 6 7 2
2 7 5 3
6 4 5 7
4 0 3 5

Sample Output 1

1

The cube below can be constructed.

Sample Input 2

8
0 0 0 0
0 0 1 1
0 1 0 1
0 1 1 0
1 0 0 1
1 0 1 0
1 1 0 0
1 1 1 1

Sample Output 2

144

Sample Input 3

6
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0

Sample Output 3

122880

Solution in C# (Mono 4.6.2.0) (judge verdict: Accepted)

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

// (づ°ω°)づﾐe★゜・。。・゜゜・。。・゜☆゜・。。・゜゜・。。・゜
public class Solver
{
    long Fun(int a, int b, int c, int d)
    {
        return Math.Min(Math.Min(1000000000L * a + 1000000 * b + 1000 * c + d, 1000000000L * b + 1000000 * c + 1000 * d + a),
            Math.Min(1000000000L * c + 1000000 * d + 1000 * a + b, 1000000000L * d + 1000000 * a + 1000 * b + c));
    }

    public void Solve()
    {
        int n = ReadInt();
        var a = ReadIntMatrix(n);

        //n = 400;
        //a = new int[n][];
        //for (int i = 0; i < n; i++)
        //    a[i] = new int[4];

        var idx = a.Select(aa => Fun(aa[0], aa[1], aa[2], aa[3])).Distinct().ToArray();
        Array.Sort(idx);
        var b = a.Select(aa => Array.BinarySearch(idx, Fun(aa[0], aa[1], aa[2], aa[3]))).ToArray();

        int m = idx.Length;
        var cnt = new int[m];
        for (int i = n - 5; i < n; i++)
            cnt[b[i]]++;

        long ans = 0;
        Action<List<int[]>, int, int> add = (list, v1, v2) => 
        {
            int size = list.Count;
            if (size > 0 && list[size - 1][0] == v1 && list[size - 1][1] == v2)
                list[size - 1][2]++;
            else
                list.Add(new [] { v1, v2, 1 });
        };
        for (int i = n - 6; i >= 0; i--)
        {
            Func<int, int, int, int, long> fun = (t1, t2, b1, b2) =>
            {
                int back = Array.BinarySearch(idx, Fun(t2, t1, b2, b1));
                if (back < 0)
                    return 0L;
                int right = Array.BinarySearch(idx, Fun(a[i][1], t2, b1, a[i][2]));
                if (right < 0)
                    return 0L;
                int left = Array.BinarySearch(idx, Fun(t1, a[i][0], a[i][3], b2));
                if (left < 0)
                    return 0L;
                long ret;
                if (left == right)
                {
                    if (left == back)
                        ret = 1L * cnt[left] * (cnt[left] - 1) * (cnt[left] - 2);
                    else
                        ret = 1L * cnt[left] * (cnt[left] - 1) * cnt[back];
                }
                else
                {
                    if (left == back)
                        ret = 1L * cnt[left] * (cnt[left] - 1) * cnt[right];
                    else if (right == back)
                        ret = 1L * cnt[right] * (cnt[right] - 1) * cnt[left];
                    else
                        ret = 1L * cnt[left] * cnt[right] * cnt[back];
                }
                if (t2 == b2 && t1 == b1)
                {
                    if (t2 == t1)
                        ret *= 4;
                    else
                        ret *= 2;
                }
                if (a[i][1] == b1 && t2 == a[i][2])
                {
                    if (b1 == t2)
                        ret *= 4;
                    else
                        ret *= 2;
                }
                if (t1 == a[i][3] && a[i][0] == b2)
                {
                    if (t1 == b2)
                        ret *= 4;
                    else
                        ret *= 2;
                }
                return ret;
            };

            for (int j = i + 1; j < n; j++)
            {
                var to = new List<int[]>();
                if (a[i][0] == a[j][1] && a[i][1] == a[j][0])
                    add(to, a[j][2], a[j][3]);
                if (a[i][0] == a[j][2] && a[i][1] == a[j][1])
                    add(to, a[j][3], a[j][0]);
                if (a[i][0] == a[j][3] && a[i][1] == a[j][2])
                    add(to, a[j][0], a[j][1]);
                if (a[i][0] == a[j][0] && a[i][1] == a[j][3])
                    add(to, a[j][1], a[j][2]);
                if (to.Count > 0)
                {
                    cnt[b[j]]--;
                    for (int k = i + 1; k < n; k++)
                        if (j != k)
                        {
                            cnt[b[k]]--;
                            if (a[i][2] == a[i][3] && a[k][0] == a[i][2] && a[k][0] == a[k][1] && a[k][0] == a[k][2] && a[k][0] == a[k][3])
                            {
                                foreach (var t in to)
                                    ans += 4 * t[2] * fun(t[0], t[1], a[i][2], a[i][2]);
                            }
                            else
                            {
                                if (a[i][3] == a[k][0] && a[i][2] == a[k][1])
                                    foreach (var t in to)
                                        ans += t[2] * fun(t[0], t[1], a[k][2], a[k][3]);
                                if (a[i][3] == a[k][1] && a[i][2] == a[k][2])
                                    foreach (var t in to)
                                        ans += t[2] * fun(t[0], t[1], a[k][3], a[k][0]);
                                if (a[i][3] == a[k][2] && a[i][2] == a[k][3])
                                    foreach (var t in to)
                                        ans += t[2] * fun(t[0], t[1], a[k][0], a[k][1]);
                                if (a[i][3] == a[k][3] && a[i][2] == a[k][0])
                                    foreach (var t in to)
                                        ans += t[2] * fun(t[0], t[1], a[k][1], a[k][2]);
                            }
                            cnt[b[k]]++;
                        }
                    cnt[b[j]]++;
                }
            }

            cnt[b[i]]++;
        }

        Write(ans);
    }

    #region Main

    protected static TextReader reader;
    protected static TextWriter writer;
    static void Main()
    {
#if DEBUG
        reader = new StreamReader("..\\..\\input.txt");
        //reader = new StreamReader(Console.OpenStandardInput());
        writer = Console.Out;
        //writer = new StreamWriter("..\\..\\output.txt");
#else
        reader = new StreamReader(Console.OpenStandardInput());
        writer = new StreamWriter(Console.OpenStandardOutput());
        //reader = new StreamReader("input.txt");
        //writer = new StreamWriter("output.txt");
#endif
        try
        {
            new Solver().Solve();
            //var thread = new Thread(new Solver().Solve, 1024 * 1024 * 128);
            //thread.Start();
            //thread.Join();
        }
        catch (Exception ex)
        {
#if DEBUG
            Console.WriteLine(ex);
#else
            throw;
#endif
        }
        reader.Close();
        writer.Close();
    }

    #endregion

    #region Read / Write
    private static Queue<string> currentLineTokens = new Queue<string>();
    private static string[] ReadAndSplitLine() { return reader.ReadLine().Split(new[] { ' ', '\t', }, StringSplitOptions.RemoveEmptyEntries); }
    public static string ReadToken() { while (currentLineTokens.Count == 0)currentLineTokens = new Queue<string>(ReadAndSplitLine()); return currentLineTokens.Dequeue(); }
    public static int ReadInt() { return int.Parse(ReadToken()); }
    public static long ReadLong() { return long.Parse(ReadToken()); }
    public static double ReadDouble() { return double.Parse(ReadToken(), CultureInfo.InvariantCulture); }
    public static int[] ReadIntArray() { return ReadAndSplitLine().Select(int.Parse).ToArray(); }
    public static long[] ReadLongArray() { return ReadAndSplitLine().Select(long.Parse).ToArray(); }
    public static double[] ReadDoubleArray() { return ReadAndSplitLine().Select(s => double.Parse(s, CultureInfo.InvariantCulture)).ToArray(); }
    public static int[][] ReadIntMatrix(int numberOfRows) { int[][] matrix = new int[numberOfRows][]; for (int i = 0; i < numberOfRows; i++)matrix[i] = ReadIntArray(); return matrix; }
    public static int[][] ReadAndTransposeIntMatrix(int numberOfRows)
    {
        int[][] matrix = ReadIntMatrix(numberOfRows); int[][] ret = new int[matrix[0].Length][];
        for (int i = 0; i < ret.Length; i++) { ret[i] = new int[numberOfRows]; for (int j = 0; j < numberOfRows; j++)ret[i][j] = matrix[j][i]; } return ret;
    }
    public static string[] ReadLines(int quantity) { string[] lines = new string[quantity]; for (int i = 0; i < quantity; i++)lines[i] = reader.ReadLine().Trim(); return lines; }
    public static void WriteArray<T>(IEnumerable<T> array) { writer.WriteLine(string.Join(" ", array)); }
    public static void Write(params object[] array) { WriteArray(array); }
    public static void WriteLines<T>(IEnumerable<T> array) { foreach (var a in array)writer.WriteLine(a); }
    private class SDictionary<TKey, TValue> : Dictionary<TKey, TValue>
    {
        public new TValue this[TKey key]
        {
            get { return ContainsKey(key) ? base[key] : default(TValue); }
            set { base[key] = value; }
        }
    }
    private static T[] Init<T>(int size) where T : new() { var ret = new T[size]; for (int i = 0; i < size; i++)ret[i] = new T(); return ret; }
    #endregion
}