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 (3.8.2) (judge verdict: Wrong Answer)

import numpy

N = int(input())
color = numpy.ones(N) * -1
color[0] = 0

edges = []

for i in range(N - 1):
    root, tip = map(int, input().split())
    edges.append([root, tip])

edges = numpy.sort(numpy.array(edges))

for e in edges:
    root, tip = e[0], e[1]
    color[tip - 1] = 1 - color[root - 1]

l1 = numpy.arange(1, N + 1 ,3)
l2 = numpy.arange(2, N + 1 ,3)
l3 = numpy.arange(3, N + 1 ,3)

d1 = numpy.where(color == 0)[0]
d2 = numpy.where(color == 1)[0]

if len(d1) <= N / 3:
    l = numpy.hstack([l3, l1, l2])

elif len(d2) <= N / 3:
    l = numpy.hstack([l1, l2, l3])

else:
    l = numpy.hstack([l2, l3, l1])

d = numpy.hstack([d1, d2])
print(' '.join(map(str, l[numpy.argsort(d)])))