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++14 (GCC 5.4.1) (judge verdict: Wrong Answer)

#include <bits/stdc++.h>
using namespace std;

using i64 = int64_t;

#define rep(i, x, y) for (i64 i = i64(x), i##_max_for_repmacro = i64(y); i < i##_max_for_repmacro; ++i)
#define debug(x) #x << "=" << (x)

#ifdef DEBUG
#define _GLIBCXX_DEBUG
#define print(x) std::cerr << debug(x) << " (L:" << __LINE__ << ")" << std::endl
#else
#define print(x)
#endif

template <i64 p>
class fp {
public:
    i64 x;
    fp() : x(0) {}
    fp(i64 x_) : x((x_ % p + p) % p) {}
    fp operator+() const { return fp(x); }
    fp operator-() const { return fp(-x); }
    fp& operator+=(const fp& y) {
        x += y.x;
        if (x >= p) x -= p;
        return *this;
    }
    fp& operator-=(const fp& y) { return *this += -y; }
    fp& operator*=(const fp& y) {
        x = x * y.x % p;
        return *this;
    }
    fp& operator/=(const fp& y) { return *this *= fp(inverse(y.x)); }
    fp operator+(const fp& y) const { return fp(x) += y; }
    fp operator-(const fp& y) const { return fp(x) -= y; }
    fp operator*(const fp& y) const { return fp(x) *= y; }
    fp operator/(const fp& y) const { return fp(x) /= y; }
    bool operator==(const fp& y) const { return x == y.x; }
    bool operator!=(const fp& y) const { return !(*this == y); }
    i64 extgcd(i64 a, i64 b, i64& x, i64& y) {
        i64 d = a;
        if (b != 0) {
            d = extgcd(b, a % b, y, x);
            y -= (a / b) * x;
        } else {
            x = 1;
            y = 0;
        }
        return d;
    }
    i64 inverse(i64 a) {
        i64 x, y;
        extgcd(a, p, x, y);
        return (x % p + p) % p;
    }
};

template <i64 p>
i64 abs(const fp<p>& x) { return x.x; }

template <i64 p>
ostream& operator<<(ostream& os, const fp<p>& x) {
    os << x.x;
    return os;
}

template <typename T, typename U>
ostream& operator<<(ostream& os, const pair<T, U>& p) {
    os << "(" << p.first << ", " << p.second << ")";
    return os;
}

template <typename T>
ostream& operator<<(ostream& os, const vector<T>& vec) {
    os << "[";
    for (const auto& v : vec) {
        os << v << ",";
    }
    os << "]";
    return os;
}

template <typename T>
bool chmin(T& a, const T& b) {
    if (a > b) {
        a = b;
        return true;
    }
    return false;
}

template <typename T>
bool chmax(T& a, const T& b) {
    if (a < b) {
        a = b;
        return true;
    }
    return false;
}

template <typename A, typename T, size_t size>
void fill(A (&ary)[size], const T& val) {
    fill((T*)ary, (T*)(ary + size), val);
}

// x未満の要素の個数を返す
template <typename T>
i64 count_less(const vector<T>& v, T x) {
    return lower_bound(begin(v), end(v), x) - begin(v);
}

// x以下の要素の個数を返す
template <typename T>
i64 count_less_or_equal(const vector<T>& v, T x) {
    return upper_bound(begin(v), end(v), x) - begin(v);
}

// xより大きい要素の個数を返す
template <typename T>
i64 count_greater(const vector<T>& v, T x) {
    return end(v) - upper_bound(begin(v), end(v), x);
}

// x以上の要素の個数を返す
template <typename T>
i64 count_greater_or_equal(const vector<T>& v, T x) {
    return end(v) - lower_bound(begin(v), end(v), x);
}

// x未満の要素のうち最大のものを返す
template <typename T>
T max_less(const vector<T>& v, T x) {
    return *(lower_bound(begin(v), end(v), x) - 1);
}

// x以下の要素のうち最大のものを返す
template <typename T>
T max_less_or_equal(const vector<T>& v, T x) {
    return *(upper_bound(begin(v), end(v), x) - 1);
}

// xより大きい要素のうち最小のものを返す
template <typename T>
T min_greater(const vector<T>& v, T x) {
    return *upper_bound(begin(v), end(v), x);
}

// x以上の要素のうち最小のものを返す
template <typename T>
T min_greater_or_equal(const vector<T>& v, T x) {
    return *lower_bound(begin(v), end(v), x);
}

constexpr int inf = 1.01e9;
constexpr i64 inf64 = 4.01e18;
constexpr long double eps = 1e-9;

// 実数値のn分探索のループ回数の上限(2分探索なら50でも十分かもしれない)
constexpr i64 max_loop = 100;

long double tolerance(const long double& a, const long double& b, const long double& epsilon = eps) {
    return epsilon * max({1.l, abs(a), abs(b)});
}

// a<0なら-1, a==0なら0, a>0なら1
int sgn(const long double a, const long double& epsilon = eps) {
    return (a > epsilon) - (a < -epsilon);
}

// a<bなら-1, a==bなら0, a>bなら1
int compare(const long double& a, const long double& b, const long double& epsilon = eps, bool is_relative = true) {
    return sgn(a - b, is_relative ? tolerance(a, b, epsilon) : epsilon);
}

// a==bか?
bool is_equal(const long double& a, const long double& b, const long double& epsilon = eps, bool is_relative = true) {
    return compare(a, b, epsilon, is_relative) == 0;
}

// a<bか?
bool is_less(const long double& a, const long double& b, const long double& epsilon = eps, bool is_relative = true) {
    return compare(a, b, epsilon, is_relative) == -1;
}

// a<=bか?
bool is_less_or_equal(const long double& a, const long double& b, const long double& epsilon = eps, bool is_relative = true) {
    return compare(a, b, epsilon, is_relative) <= 0;
}

// a>bか?
bool is_greater(const long double& a, const long double& b, const long double& epsilon = eps, bool is_relative = true) {
    return compare(a, b, epsilon, is_relative) == 1;
}

// a>=bか?
bool is_greater_or_equal(const long double& a, const long double& b, const long double& epsilon = eps, bool is_relative = true) {
    return compare(a, b, epsilon, is_relative) >= 0;
}

pair<int, int> farthest_vertex(const vector<vector<pair<int, int>>> &graph, int v, int p = -1) {
    pair<int, int> res(v, 0);  //(最も遠い頂点,距離)
    for (auto &e : graph[v]) {
        if (e.first == p) continue;
        auto tmp = farthest_vertex(graph, e.first, v);
        tmp.second += e.second;
        if (res.second < tmp.second) res = tmp;
    }
    return res;
}

void solve() {
    //constexpr i64 mod = 1'000'000'007;
    i64 N;
    cin >> N;

    vector<vector<i64>> tree(N);
    rep(i,0,N-1){
        i64 a,b;
        cin >> a >> b;
        --a;
        --b;
        tree[a].emplace_back(b);
        tree[b].emplace_back(a);
    }

    {
        vector<vector<pair<int,int>>> graph(N);
        rep(i,0,N){
            for(auto j:tree[i]) graph[i].emplace_back(make_pair(j,1));
        }
        auto p1=farthest_vertex(graph,0);
        auto p2=farthest_vertex(graph,p1.first);
        if(p2.second<3){
            rep(i,0,N){
                cout << i+1;
                if(i==N-1) cout << endl;
                else cout << " ";
            }
            return;
        }
    }

    vector<i64> red,green;
    function<void(i64,i64,i64)> dfs=[&](i64 u,i64 p,i64 c){
        if(c==0) red.emplace_back(u);
        else green.emplace_back(u);
        for(auto v:tree[u]){
            if(v==p) continue;
            dfs(v,u,(c+1)%2);
        }
    };

    dfs(0,-1,0);

    i64 one=0,two=0,zero=0;
    rep(i,1,N+1){
        if(i%3==1) ++one;
        else if(i%3==2) ++two;
        else ++zero;
    }

    i64 r=red.size(),g=green.size();
    /*
    if(max(i64(0),min(r,g)-min(one,two))+max(i64(0),max(r,g)-max(one,two))-zero>0){
        cout << -1 << endl;
        return;
    }
     */

    if(r<g){
        swap(red,green);
        swap(r,g);
    }

    vector<i64> color(N);
    for(auto u:red) color[u]=0;
    for(auto u:green) color[u]=1;

    vector<i64> ans(N);
    vector<i64> ones,twos,zeros;
    rep(i,1,N+1){
        if(i%3==1) ones.emplace_back(i);
        else if(i%3==2) twos.emplace_back(i);
        else zeros.emplace_back(i);
    }
    rep(i,0,N){
        if(color[i]==0){
            if(ones.size()){
                ans[i]=ones.back();
                ones.pop_back();
            }else{
                if(zeros.empty()){
                    ans[i]=twos.back();
                    twos.pop_back();
                    continue;
                }
                ans[i]=zeros.back();
                zeros.pop_back();
            }
        }
    }
    rep(i,0,N){
        if(color[i]==1){
            if(twos.size()){
                ans[i]=twos.back();
                twos.pop_back();
            }else{
                if(zeros.empty()){
                    cout << -1 << endl;
                    return;
                }
                ans[i]=zeros.back();
                zeros.pop_back();
            }
        }
    }

    rep(i,0,N){
        cout << ans[i];
        if(i==N-1) cout << endl;
        else cout << " ";
    }
}

int main() {
    std::cin.tie(0);
    std::ios::sync_with_stdio(false);
    cout.setf(ios::fixed);
    cout.precision(16);
    solve();
    return 0;
}