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)

from collections import deque
n=int(input())
E=[[] for i in range(n)]
for i in range(n-1):
  a,b=map(int,input().split())
  a=a-1
  b=b-1
  E[a].append(b)
  E[b].append(a)
D=deque([0])
V=[0]*n
V[0]=1
A=[0]*n
k=1
while len(D)>0:
  x=D[0]
  D.popleft()
  for i in range(len(E[x])):
    if V[E[x][i]]==0:
      V[E[x][i]]=1
      D.append(E[x][i])
      A[E[x][i]]=A[x]^1
f=0
if n-sum(A)>sum(A):
  f=1
g=max(n-sum(A),sum(A))
a=(n+2)//3
b=n//3
if g>a+b:
  print(-1)
else:
  Ans=[0]*n
  s=1
  t=2
  u=3
  if f==1:
    for i in range(n):
      if A[i]==0 and s<=n:
        Ans[i]=s
        s=s+3
      elif A[i]==0 and s>n:
        Ans[i]=u
        u=u+3
      elif A[i]==1 and t<=n:
        Ans[i]=t
        t=t+3
      else:
        Ans[i]=u
        u=u+3
  else:
    for i in range(n):
      if A[i]==1 and s<=n:
        Ans[i]=s
        s=s+3
      elif A[i]==1 and s>n:
        Ans[i]=u
        u=u+3
      elif A[i]==0 and t<=n:
        Ans[i]=t
        t=t+3
      else:
        Ans[i]=u
        u=u+3
  print(*Ans)