Problem:
Score : 600 points

Problem Statement

We have a tree with N vertices. The vertices are numbered 1 to N, and the i-th edge connects Vertex a_i and Vertex b_i.

Takahashi loves the number 3. He is seeking a permutation p_1, p_2, \ldots , p_N of integers from 1 to N satisfying the following condition:

For every pair of vertices (i, j), if the distance between Vertex i and Vertex j is 3, the sum or product of p_i and p_j is a multiple of 3.

Here the distance between Vertex i and Vertex j is the number of edges contained in the shortest path from Vertex i to Vertex j.

Help Takahashi by finding a permutation that satisfies the condition.

Constraints

2\leq N\leq 2\times 10^5

1\leq a_i, b_i \leq N

The given graph is a tree.

Input

Input is given from Standard Input in the following format:

N
a_1 b_1
a_2 b_2
\vdots
a_{N-1} b_{N-1}

Output

If no permutation satisfies the condition, print -1.

Otherwise, print a permutation satisfying the condition, with space in between.
If there are multiple solutions, you can print any of them.

Sample Input 1

5
1 2
1 3
3 4
3 5

Sample Output 1

1 2 5 4 3

The distance between two vertices is 3 for the two pairs (2, 4) and (2, 5).

p_2 + p_4 = 6

p_2\times p_5 = 6

Thus, this permutation satisfies the condition.

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.Runtime.CompilerServices;
using System.Text;
using System.Diagnostics;
using System.Numerics;
using static System.Console;
using static System.Convert;
using static System.Math;
using static Template;
using Pi = Pair<int, int>;

class Solver
{
    int N,ok;
    List<int>[] g;
    int[][] ct;
    int[] par,res;
    Pi[] ps;
    List<int>[][] s;
    public void Solve(Scanner sc)
    {
        N = sc.Int;
        g = Create(N, () => new List<int>());
        ok = (N + 1) / 3;
        for (var i = 0; i < N-1; i++)
        {
            int a, b;
            sc.Make(out a, out b);
            a--; b--;
            g[a].Add(b); g[b].Add(a);
        }
        s = new List<int>[N][];
        ct = Create(N, () => new int[4]);
        par = new int[N]; 
        ps = new Pi[N];
        res = new int[N];
        dfs(0, -1);
        for (int i = 0; i < N; i++)
        {
            var c = ct[i][3];
            var f = par[i];
            if (f == -1) goto END;
            c += ct[f][2] - ct[i][1];
            var ff = par[f];
            if (ff == -1) goto END;
            c += ct[ff][1] - ct[f][0];
            if (par[ff] != -1) c++;
        END:;
            ps[i] = new Pi(c, i);
        }
        Array.Sort(ps, (a, b) => b.v1 - a.v1);

        var v = 3;
        for (int i = 0; i < N / 3; i++, v += 3)
            res[ps[i].v2] = v;
        /*int ca = 1, cb = 2,t=1;
        var q = new Queue<int>();
        foreach(var i in ps.Skip(N/3))
            if (res[i.v2] == 0)
            {
                q.Enqueue(i.v2);
                if (t == 1) { if (ca > N) Fail(-1); res[i.v2] = ca; ca += 3; }
                else { if (cb > N) Fail(-1); res[i.v2] = cb; cb += 3; }
                t ^= 1;
                while (q.Any())
                {
                    var p = q.Dequeue();
                    foreach(var e in s[p][3])
                        if (res[e] == 0)
                        {
                            if (t == 1) { if (ca > N) Fail(-1); res[e] = ca; ca += 3; }
                            else { if (cb > N) Fail(-1); res[e] = cb; cb += 3; }
                            
                            t ^= 1;
                            q.Enqueue(e);
                        }
                }
            }*/
        c(0, -1);
        Console.WriteLine(string.Join(" ", res));

    }
    int t = 0, ca = 1, cb = 2;
    void c(int i,int p)
    {
        if (res[i] == 0)
        {
            if (t == 1) { if (ca > N) Fail(-1); res[i] = ca; ca += 3; }
            else { if (cb > N) Fail(-1); res[i] = cb; cb += 3; }
            t ^= 1;
        }
        foreach (var e in g[i]) if (e != p) c(e, i);
    }
    void dfs(int i,int p)
    {
        //s[i] = Create(4, () => new List<int>());
        par[i] = p;
        //s[i][0].Add(i);
        ct[i][0]++;
        foreach(var e in g[i])
            if (e != p)
            {
                dfs(e, i);
                for (int k = 0; k < 3; k++)
                {
                    /*if (s[i][k + 1].Count < s[e][k].Count) swap(ref s[i][k + 1], ref s[e][k]);
                    foreach (var v in s[e][k])
                        s[i][k + 1].Add(v);*/
                    ct[i][k + 1] += ct[e][k];
                }
            }
    }

}

#region Template
public static class Template
{
    static void Main(string[] args)
    {
        Console.SetOut(new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false });
        new Solver().Solve(new Scanner());
        Console.Out.Flush();
    }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static bool chmin<T>(ref T a, T b) where T : IComparable<T> { if (a.CompareTo(b) == 1) { a = b; return true; } return false; }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static bool chmax<T>(ref T a, T b) where T : IComparable<T> { if (a.CompareTo(b) == -1) { a = b; return true; } return false; }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static void swap<T>(ref T a, ref T b) { var t = b; b = a; a = t; }
    public static T[] Shuffle<T>(this IList<T> A) { T[] rt = A.ToArray(); Random rnd = new Random(); for (int i = rt.Length - 1; i >= 1; i--) swap(ref rt[i], ref rt[rnd.Next(i + 1)]); return rt; }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static T[] Create<T>(int n, Func<T> f) { var rt = new T[n]; for (var i = 0; i < rt.Length; ++i) rt[i] = f(); return rt; }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static T[] Create<T>(int n, Func<int, T> f) { var rt = new T[n]; for (var i = 0; i < rt.Length; ++i) rt[i] = f(i); return rt; }
    public static void Fail<T>(T s) { Console.WriteLine(s); Console.Out.Close(); Environment.Exit(0); }
}
public class Scanner
{
    public string Str => Console.ReadLine().Trim();
    public int Int => int.Parse(Str);
    public long Long => long.Parse(Str);
    public double Double => double.Parse(Str);
    public int[] ArrInt => Str.Split(' ').Select(int.Parse).ToArray();
    public long[] ArrLong => Str.Split(' ').Select(long.Parse).ToArray();
    public char[][] Grid(int n) => Create(n, () => Str.ToCharArray());
    public int[] ArrInt1D(int n) => Create(n, () => Int);
    public long[] ArrLong1D(int n) => Create(n, () => Long);
    public int[][] ArrInt2D(int n) => Create(n, () => ArrInt);
    public long[][] ArrLong2D(int n) => Create(n, () => ArrLong);
    private Queue<string> q = new Queue<string>();
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public T Next<T>() { if (q.Count == 0) foreach (var item in Str.Split(' ')) q.Enqueue(item); return (T)Convert.ChangeType(q.Dequeue(), typeof(T)); }
    public void Make<T1>(out T1 v1) => v1 = Next<T1>();
    public void Make<T1, T2>(out T1 v1, out T2 v2) { v1 = Next<T1>(); v2 = Next<T2>(); }
    public void Make<T1, T2, T3>(out T1 v1, out T2 v2, out T3 v3) { Make(out v1, out v2); v3 = Next<T3>(); }
    public void Make<T1, T2, T3, T4>(out T1 v1, out T2 v2, out T3 v3, out T4 v4) { Make(out v1, out v2, out v3); v4 = Next<T4>(); }
    public void Make<T1, T2, T3, T4, T5>(out T1 v1, out T2 v2, out T3 v3, out T4 v4, out T5 v5) { Make(out v1, out v2, out v3, out v4); v5 = Next<T5>(); }
    public void Make<T1, T2, T3, T4, T5, T6>(out T1 v1, out T2 v2, out T3 v3, out T4 v4, out T5 v5, out T6 v6) { Make(out v1, out v2, out v3, out v4, out v5); v6 = Next<T6>(); }
    //public (T1, T2) Make<T1, T2>() { Make(out T1 v1, out T2 v2); return (v1, v2); }
    //public (T1, T2, T3) Make<T1, T2, T3>() { Make(out T1 v1, out T2 v2, out T3 v3); return (v1, v2, v3); }
    //public (T1, T2, T3, T4) Make<T1, T2, T3, T4>() { Make(out T1 v1, out T2 v2, out T3 v3, out T4 v4); return (v1, v2, v3, v4); }
}
public class Pair<T1, T2> : IComparable<Pair<T1, T2>>
{
    public T1 v1;
    public T2 v2;
    public Pair() { }
    public Pair(T1 v1, T2 v2)
    { this.v1 = v1; this.v2 = v2; }

    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public int CompareTo(Pair<T1, T2> p)
    {
        var c = Comparer<T1>.Default.Compare(v1, p.v1);
        if (c == 0)
            c = Comparer<T2>.Default.Compare(v2, p.v2);
        return c;
    }
    public override string ToString() => $"{v1.ToString()} {v2.ToString()}";
    public void Deconstruct(out T1 a, out T2 b) { a = v1; b = v2; }
}

public class Pair<T1, T2, T3> : Pair<T1, T2>, IComparable<Pair<T1, T2, T3>>
{
    public T3 v3;
    public Pair() : base() { }
    public Pair(T1 v1, T2 v2, T3 v3) : base(v1, v2)
    { this.v3 = v3; }

    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public int CompareTo(Pair<T1, T2, T3> p)
    {
        var c = base.CompareTo(p);
        if (c == 0)
            c = Comparer<T3>.Default.Compare(v3, p.v3);
        return c;
    }
    public override string ToString() => $"{base.ToString()} {v3.ToString()}";
    public void Deconstruct(out T1 a, out T2 b, out T3 c) { Deconstruct(out a, out b); c = v3; }
}
#endregion