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 sys
sys.setrecursionlimit(10**9)
N=int(input())

tree=[[] for _ in range(N+1)]
occur_list=[0]*(N+1)
for _ in range(N-1):
  a,b=map(int,input().split())
  tree[a].append(b)
  tree[b].append(a)
  occur_list[a]+=1
  occur_list[b]+=1
#print(tree)
#print(occur_list)

leaf=0
for i in range(1,N+1):
  if occur_list[i]==1:
    leaf=i
    break
#print(leaf)

def dfs(u,p1,p2,p3):
  global min_3m0
  global min_3m1
  global min_3m2
  
  if p3==0:
    if min_3m0<=N:
      mark_list[u]=min_3m0
      min_3m0+=3
    elif min_3m1<=N:
      mark_list[u]=min_3m1
      min_3m1+=3
    elif min_3m2<=N:
      mark_list[u]=min_3m2
      min_3m2+=3
    else:
      print(-1)
      sys.exit(0)
  elif mark_list[p3]%3==0:
    if min_3m1<=N:
      mark_list[u]=min_3m1
      min_3m1+=3
    elif min_3m2<=N:
      mark_list[u]=min_3m2
      min_3m2+=3
    elif min_3m0<=N:
      mark_list[u]=min_3m0
      min_3m0+=3
    else:
      print(-1)
      sys.exit(0)
  elif mark_list[p3]%3==1:
    if min_3m2<=N:
      mark_list[u]=min_3m2
      min_3m2+=3
    elif min_3m0<=N:
      mark_list[u]=min_3m0
      min_3m0+=3
    else:
      print(-1)
      sys.exit(0)
  elif mark_list[p3]%3==2:
    if min_3m1<=N:
      mark_list[u]=min_3m1
      min_3m1+=3
    elif min_3m0<=N:
      mark_list[u]=min_3m0
      min_3m0+=3
    else:
      print(-1)
      sys.exit(0)
    
  for v in tree[u]:
    if mark_list[v]==0:
      dfs(v,u,p1,p2)

mark_list=[0]*(N+1)
min_3m0=3
min_3m1=1
min_3m2=2
dfs(leaf,0,0,0)

print(*mark_list[1:])