Problem:
Score : 900 points

Problem Statement

Let us consider a grid of squares with N rows and N columns. You want to put some domino pieces on this grid.
Each domino piece covers two squares that have a common side. Each square can be covered by at most one piece.

For each row of the grid, let's define its quality as the number of domino pieces that cover at least one square in this row.
We define the quality of each column similarly.

Find a way to put at least one domino piece on the grid so that the quality of every row is equal to the quality of every column,
or determine that such a placement doesn't exist.

Constraints

2 \le N \le 1000

Input

Input is given from Standard Input in the following format:

N

Output

If the required domino placement doesn't exist, print a single integer -1.

Otherwise, output your placement as N strings of N characters each.
If a square is not covered, the corresponding character must be . (a dot).
Otherwise, it must contain a lowercase English letter.
Squares covered by the same domino piece must contain the same letter.
If two squares have a common side but belong to different pieces, they must contain different letters.

Sample Input 1

6

Sample Output 1

aabb..
b..zz.
ba....
.a..aa
..a..b
..a..b

The quality of every row and every column is 2.

Sample Input 2

2

Sample Output 2

-1

Solution in C# (Mono 4.6.2.0) (judge verdict: Wrong Answer)

using System;
using System.Collections.Generic;
using System.Linq;
using System.IO;
using System.Threading;
using System.Text;
using System.Diagnostics;
using static util;
using P = pair<int, int>;

class Program {
    static void Main(string[] args) {
        var sw = new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false };
        var solver = new Solver(sw);
        // var t = new Thread(solver.solve, 1 << 26); // 64 MB
        // t.Start();
        // t.Join();
        solver.solve();
        sw.Flush();
    }
}

class Solver {
    StreamWriter sw;
    Scan sc;
    void Prt(string a) => sw.WriteLine(a);
    void Prt<T>(IEnumerable<T> a) => Prt(string.Join(" ", a));
    void Prt(params object[] a) => Prt(string.Join(" ", a));
    public Solver(StreamWriter sw) {
        this.sw = sw;
        this.sc = new Scan();
    }

    public void solve() {
        int n = sc.Int;
        var ans = new char[n][];
        for (int i = 0; i < n; i++)
        {
            ans[i] = new char[n];
            for (int j = 0; j < n; j++)
            {
                ans[i][j] = '.';
            }
        }
        if (n % 3 == 0) {
            int k = 0;
            for (int j = 0; j < n / 3; j++)
            {
                ans[k][k] = 'a';
                ans[k][k + 1] = 'a';
                ans[k + 1][k + 2] = 'a';
                ans[k + 2][k + 2] = 'a';
                k += 3;
            }
            foreach (var item in ans)
            {
                Prt(new string(item));
            }
            return;
        }
        for (int i = 0; i < 9; i++)
        {
            if (n - i * 4 >= 0 && (n - i * 4) % 6 == 0) {
                int c4 = i;
                int c6 = (n - c4 * 4) / 6;
                int k = 0;
                for (int j = 0; j < c6; j++)
                {
                    ans[k][k] = 'a';
                    ans[k][k + 1] = 'a';
                    ans[k + 1][k + 2] = 'a';
                    ans[k + 2][k + 2] = 'a';

                    ans[k + 3][k] = 'c';
                    ans[k + 3][k + 1] = 'c';
                    ans[k + 3 + 1][k] = 'd';
                    ans[k + 3 + 2][k] = 'd';
                    ans[k + 3 + 2][k + 1] = 'c';
                    ans[k + 3 + 2][k + 2] = 'c';
                    ans[k + 3][k + 2] = 'd';
                    ans[k + 3 + 1][k + 2] = 'd';

                    ans[k][k + 3] = 'c';
                    ans[k][k + 3 + 1] = 'c';
                    ans[k + 1][k + 3] = 'd';
                    ans[k + 2][k + 3] = 'd';
                    ans[k + 2][k + 3 + 1] = 'c';
                    ans[k + 2][k + 3 + 2] = 'c';
                    ans[k][k + 3 + 2] = 'd';
                    ans[k + 1][k + 3 + 2] = 'd';

                    ans[k + 3][k + 3] = 'a';
                    ans[k + 3][k + 3 + 1] = 'a';
                    ans[k + 3 + 1][k + 3 + 2] = 'a';
                    ans[k + 3 + 2][k + 3 + 2] = 'a';

                    k += 6;
                }
                for (int j = 0; j < c4; j++)
                {
                    ans[k][k] = 'a';
                    ans[k][k + 1] = 'a';
                    ans[k + 1][k] = 'b';
                    ans[k + 1][k + 1] = 'b';

                    ans[k + 2][k] = 'c';
                    ans[k + 2][k + 1] = 'd';
                    ans[k + 2 + 1][k] = 'c';
                    ans[k + 2 + 1][k + 1] = 'd';

                    ans[k][k + 2] = 'c';
                    ans[k][k + 2 + 1] = 'd';
                    ans[k + 1][k + 2] = 'c';
                    ans[k + 1][k + 2 + 1] = 'd';

                    ans[k + 2][k + 2] = 'a';
                    ans[k + 2][k + 2 + 1] = 'a';
                    ans[k + 2 + 1][k + 2] = 'b';
                    ans[k + 2 + 1][k + 2 + 1] = 'b';

                    k += 4;
                }
                foreach (var item in ans)
                {
                    Prt(new string(item));
                }
                return;
            }
            if (n - i * 4 >= 0 && (n - i * 4) % 9 == 0) {
                int c4 = i;
                int c9 = (n - c4 * 4) / 9;
                int k = 0;
                for (int j = 0; j < c9; j++)
                {
                    for (int p = 0; p < 3; p++)
                    {
                        for (int q = 0; q < 3; q++)
                        {
                            ans[k + p * 3][k + q * 3] = 'a';
                            ans[k + p * 3][k + q * 3 + 1] = 'a';
                            ans[k + p * 3 + 1][k + q * 3 + 2] = 'a';
                            ans[k + p * 3 + 2][k + q * 3 + 2] = 'a';
                        }
                    }

                    k += 9;
                }
                for (int j = 0; j < c4; j++)
                {
                    ans[k][k] = 'a';
                    ans[k][k + 1] = 'a';
                    ans[k + 1][k] = 'b';
                    ans[k + 1][k + 1] = 'b';

                    ans[k + 2][k] = 'c';
                    ans[k + 2][k + 1] = 'd';
                    ans[k + 2 + 1][k] = 'c';
                    ans[k + 2 + 1][k + 1] = 'd';

                    ans[k][k + 2] = 'c';
                    ans[k][k + 2 + 1] = 'd';
                    ans[k + 1][k + 2] = 'c';
                    ans[k + 1][k + 2 + 1] = 'd';

                    ans[k + 2][k + 2] = 'a';
                    ans[k + 2][k + 2 + 1] = 'a';
                    ans[k + 2 + 1][k + 2] = 'b';
                    ans[k + 2 + 1][k + 2 + 1] = 'b';

                    k += 4;
                }
                foreach (var item in ans)
                {
                    Prt(new string(item));
                }
                return;
            }
        }
        Prt(-1);
    }
}

class pair<T, U> : IComparable<pair<T, U>> {
    public T v1;
    public U v2;
    public pair() : this(default(T), default(U)) {}
    public pair(T v1, U v2) { this.v1 = v1; this.v2 = v2; }
    public int CompareTo(pair<T, U> a) {
        int c = Comparer<T>.Default.Compare(v1, a.v1);
        return c != 0 ? c : Comparer<U>.Default.Compare(v2, a.v2);
    }
    public override string ToString() => v1 + " " + v2;
    public void Deconstruct(out T a, out U b) { a = v1; b = v2; }
}
static class util {
    public static readonly int M = 1000000007;
    // public static readonly int M = 998244353;
    public static readonly long LM = 1L << 60;
    public static readonly double eps = 1e-11;
    public static void DBG(string a) => Console.Error.WriteLine(a);
    public static void DBG<T>(IEnumerable<T> a) => DBG(string.Join(" ", a));
    public static void DBG(params object[] a) => DBG(string.Join(" ", a));
    public static void Assert(bool cond) { if (!cond) throw new Exception(); }
    public static pair<T, U> make_pair<T, U>(T v1, U v2) => new pair<T, U>(v1, v2);
    public static int CompareList<T>(IList<T> a, IList<T> b) where T : IComparable<T> {
        for (int i = 0; i < a.Count && i < b.Count; i++)
            if (a[i].CompareTo(b[i]) != 0) return a[i].CompareTo(b[i]);
        return a.Count.CompareTo(b.Count);
    }
    public static bool inside(int i, int j, int h, int w) => i >= 0 && i < h && j >= 0 && j < w;
    static readonly int[] dd = { 0, 1, 0, -1 };
    // static readonly string dstring = "RDLU";
    public static P[] adjacents(int i, int j)
        => Enumerable.Range(0, dd.Length).Select(k => new P(i + dd[k], j + dd[k ^ 1])).ToArray();
    public static P[] adjacents(int i, int j, int h, int w)
        => Enumerable.Range(0, dd.Length).Select(k => new P(i + dd[k], j + dd[k ^ 1]))
                                         .Where(p => inside(p.v1, p.v2, h, w)).ToArray();
    public static P[] adjacents(this P p) => adjacents(p.v1, p.v2);
    public static P[] adjacents(this P p, int h, int w) => adjacents(p.v1, p.v2, h, w);
    public static Dictionary<T, int> compress<T>(this IEnumerable<T> a)
        => a.Distinct().OrderBy(v => v).Select((v, i) => new { v, i }).ToDictionary(p => p.v, p => p.i);
    public static Dictionary<T, int> compress<T>(params IEnumerable<T>[] a) => compress(a.Aggregate(Enumerable.Union));
    public static T[] inv<T>(this Dictionary<T, int> dic) {
        var res = new T[dic.Count];
        foreach (var item in dic) res[item.Value] = item.Key;
        return res;
    }
    public static void swap<T>(ref T a, ref T b) where T : struct { var t = a; a = b; b = t; }
    public static void swap<T>(this IList<T> a, int i, int j) where T : struct { var t = a[i]; a[i] = a[j]; a[j] = t; }
    public static T[] copy<T>(this IList<T> a) {
        var ret = new T[a.Count];
        for (int i = 0; i < a.Count; i++) ret[i] = a[i];
        return ret;
    }
}

class Scan {
    StreamReader sr;
    public Scan() { sr = new StreamReader(Console.OpenStandardInput()); }
    public Scan(string path) { sr = new StreamReader(path); }
    public int Int => int.Parse(Str);
    public long Long => long.Parse(Str);
    public double Double => double.Parse(Str);
    public string Str => sr.ReadLine().Trim();
    public pair<T, U> Pair<T, U>() {
        T a; U b;
        Multi(out a, out b);
        return new pair<T, U>(a, b);
    }
    public P P => Pair<int, int>();
    public int[] IntArr => StrArr.Select(int.Parse).ToArray();
    public long[] LongArr => StrArr.Select(long.Parse).ToArray();
    public double[] DoubleArr => StrArr.Select(double.Parse).ToArray();
    public string[] StrArr => Str.Split(new[]{' '}, StringSplitOptions.RemoveEmptyEntries);
    bool eq<T, U>() => typeof(T).Equals(typeof(U));
    T ct<T, U>(U a) => (T)Convert.ChangeType(a, typeof(T));
    T cv<T>(string s) => eq<T, int>()    ? ct<T, int>(int.Parse(s))
                       : eq<T, long>()   ? ct<T, long>(long.Parse(s))
                       : eq<T, double>() ? ct<T, double>(double.Parse(s))
                       : eq<T, char>()   ? ct<T, char>(s[0])
                                         : ct<T, string>(s);
    public void Multi<T>(out T a) => a = cv<T>(Str);
    public void Multi<T, U>(out T a, out U b)
    { var ar = StrArr; a = cv<T>(ar[0]); b = cv<U>(ar[1]); }
    public void Multi<T, U, V>(out T a, out U b, out V c)
    { var ar = StrArr; a = cv<T>(ar[0]); b = cv<U>(ar[1]); c = cv<V>(ar[2]); }
    public void Multi<T, U, V, W>(out T a, out U b, out V c, out W d)
    { var ar = StrArr; a = cv<T>(ar[0]); b = cv<U>(ar[1]); c = cv<V>(ar[2]); d = cv<W>(ar[3]); }
    public void Multi<T, U, V, W, X>(out T a, out U b, out V c, out W d, out X e)
    { var ar = StrArr; a = cv<T>(ar[0]); b = cv<U>(ar[1]); c = cv<V>(ar[2]); d = cv<W>(ar[3]); e = cv<X>(ar[4]); }
}