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 Rust (1.15.1) (judge verdict: Wrong Answer)

// This code is generated by [cargo-atcoder](https://github.com/tanakh/cargo-atcoder)
// Original source code:
/*
use proconio::input;
#[allow(unused_imports)]
use proconio::marker::*;
#[allow(unused_imports)]
use std::cmp::*;
#[allow(unused_imports)]
use std::collections::*;
#[allow(unused_imports)]
use std::f64::consts::*;

#[allow(unused)]
const INF: usize = std::usize::MAX / 4;
#[allow(unused)]
const M: usize = 1000000007;

fn first(set: &HashSet<usize>) -> Option<usize> {
    set.iter().nth(0).map(|&x| x)
}

fn main() {
    input! {
        n: usize,
        ab: [(Usize1, Usize1); n - 1],
    }
    let mut graph = vec![HashSet::new(); n];
    for &(ai, bi) in &ab {
        graph[ai].insert(bi);
        graph[bi].insert(ai);
    }
    let mut two_neighbors = vec![HashSet::new(); n];
    for u in 0..n {
        for &v in &graph[u] {
            for &w in &graph[v] {
                if u != w {
                    two_neighbors[u].insert(w);
                }
            }
        }
    }

    let mut three_neighbors = vec![HashSet::new(); n];
    for u in 0..n {
        for &v in &graph[u] {
            for &w in &two_neighbors[v] {
                if !graph[u].contains(&w) {
                    three_neighbors[u].insert(w);
                }
            }
        }
    }
    println!("-1");
    // for i in 0..n {
    //     eprintln!("{:?}", three_neighbors[i]);
    // }
    // let tmp = three_neighbors.clone();

    // let mut indices = (0..n).collect::<Vec<usize>>();
    // indices.sort_by_key(|&i| three_neighbors[i].len());
    // indices.reverse();

    // let mut p = vec![0; n];

    // let mut zero = (3..=n).step_by(3).collect::<HashSet<usize>>();
    // let mut one = (1..=n).step_by(3).collect::<HashSet<usize>>();
    // let mut two = (2..=n).step_by(3).collect::<HashSet<usize>>();

    // for &i in &indices {
    //     if three_neighbors[i].len() == 0 {
    //         continue;
    //     }

    //     let js = three_neighbors[i]
    //         .iter()
    //         .map(|&j| j)
    //         .collect::<Vec<usize>>();

    //     if p[i] != 0 {
    //         if p[i] % 3 == 1 {
    //             if two.len() < three_neighbors[i].len() {
    //                 println!("-1");
    //                 return;
    //             }
    //             if let Some(x) = first(&one) {
    //                 p[i] = x;
    //                 for &j in &js {
    //                     if let Some(y) = first(&two) {
    //                         p[j] = y;
    //                         three_neighbors[j].remove(&i);
    //                         two.remove(&y);
    //                     }
    //                 }
    //             }
    //         } else {
    //             if one.len() < three_neighbors[i].len() {
    //                 println!("-1");
    //                 return;
    //             }
    //             if let Some(x) = first(&two) {
    //                 p[i] = x;
    //                 for &j in &js {
    //                     if let Some(y) = first(&one) {
    //                         p[j] = y;
    //                         three_neighbors[j].remove(&i);
    //                         one.remove(&y);
    //                     }
    //                 }
    //             }
    //         }
    //         continue;
    //     }

    //     if let Some(x) = first(&zero) {
    //         p[i] = x;
    //         let js = three_neighbors[i]
    //             .iter()
    //             .map(|&j| j)
    //             .collect::<Vec<usize>>();
    //         for &j in &js {
    //             three_neighbors[j].remove(&i);
    //         }
    //         zero.remove(&x);
    //     } else {
    //         if one.is_empty() || two.is_empty() {
    //             println!("-1");
    //             return;
    //         }
    //         if one.len() > two.len() {
    //             if one.len() < three_neighbors[i].len() {
    //                 println!("-1");
    //                 return;
    //             }
    //             if let Some(x) = first(&two) {
    //                 p[i] = x;
    //                 for &j in &js {
    //                     if let Some(y) = first(&one) {
    //                         p[j] = y;
    //                         three_neighbors[j].remove(&i);
    //                         one.remove(&y);
    //                     }
    //                 }
    //                 two.remove(&x);
    //             }
    //         } else {
    //             if two.len() < three_neighbors[i].len() {
    //                 println!("-1");
    //                 return;
    //             }
    //             if let Some(x) = first(&one) {
    //                 p[i] = x;
    //                 for &j in &js {
    //                     if let Some(y) = first(&two) {
    //                         p[j] = y;
    //                         three_neighbors[j].remove(&i);
    //                         two.remove(&y);
    //                     }
    //                 }
    //                 one.remove(&x);
    //             }
    //         }
    //     }
    // }

    // for i in 0..n {
    //     if p[i] == 0 {
    //         if let Some(x) = first(&zero) {
    //             p[i] = x;
    //             zero.remove(&x);
    //         } else if let Some(x) = first(&one) {
    //             p[i] = x;
    //             one.remove(&x);
    //         } else if let Some(x) = first(&two) {
    //             p[i] = x;
    //             two.remove(&x);
    //         }
    //     }
    // }

    // for i in 0..n {
    //     for &j in &tmp[i] {
    //         if (p[i] + p[j]) % 3 != 0 && (p[i] * p[j]) % 3 != 0 {
    //             eprintln!("error");
    //         }
    //     }
    // }

    // for &pi in &p {
    //     println!("{}", pi);
    // }
}
*/

fn main() {
    let exe = "/tmp/binCBE30A93";
    std::io::Write::write_all(&mut std::fs::File::create(exe).unwrap(), &decode(BIN)).unwrap();
    std::fs::set_permissions(exe, std::os::unix::fs::PermissionsExt::from_mode(0o755)).unwrap();
    std::process::exit(std::process::Command::new(exe).status().unwrap().code().unwrap())
}

fn decode(v: &str) -> Vec<u8> {
    let mut ret = vec![];
    let mut buf = 0;
    let mut tbl = vec![64; 256];
    for i in 0..64 { tbl[TBL[i] as usize] = i as u8; }
    for (i, c) in v.bytes().filter_map(|c| { let c = tbl[c as usize]; if c < 64 { Some(c) } else { None } }).enumerate() {
        match i % 4 {
            0 => buf = c << 2,
            1 => { ret.push(buf | c >> 4); buf = c << 4; }
            2 => { ret.push(buf | c >> 2); buf = c << 6; }
            3 => ret.push(buf | c),
            _ => unreachable!(),
        }
    }
    ret
}

const TBL: &'static [u8] = b"ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz0123456789+/";
const BIN: &'static str = "<base64-encoded ELF executable removed>";