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 Python (3.4.3) (judge verdict: Wrong Answer)

#C
#入力
N=int(input())
X=[list(map(int,input().split())) for i in range(N-1)]

#前提関数
def c_1(X,n,j):#nとの距離がiのノードを返す
    t=X[n-1]
    s=[]
    for i in range(len(t)):
        if t[i]==j:
            s.append(i+1)
    return s

#処理
Y=[ [ -1 for i in range(N)] for j in range(N)]
Z=[ [ 0 for i in range(N)] for j in range(N)]
for i in range(N):
    Y[i][i]=0
for i in X:
    Y[i[0]-1][i[1]-1]=1
    Y[i[1]-1][i[0]-1]=1

for i in range(2):#三段探索する
    
    for j in range(N):#一列づつ見る
        t=c_1(Y,j+1,i+1)
        for l in t:
            s=c_1(Y,l,1)
            for m in s:
                if Y[j][m-1]==-1:
                    Y[j][m-1]=i+2