Problem:
Score: 600 points

Problem Statement

AtCoder's head office consists of N rooms numbered 1 to N. For any two rooms, there is a direct passage connecting these rooms.

For security reasons, Takahashi the president asked you to set a level for every passage, which is a positive integer and must satisfy the following condition:

For each room i\ (1 \leq i \leq N), if we leave Room i, pass through some passages whose levels are all equal and get back to Room i, the number of times we pass through a passage is always even.

Your task is to set levels to the passages so that the highest level of a passage is minimized.

Constraints

N is an integer between 2 and 500 (inclusive).

Input

Input is given from Standard Input in the following format:

N

Output

Print one way to set levels to the passages so that the objective is achieved, as follows:

a_{1,2} a_{1,3} ... a_{1,N}
a_{2,3} ... a_{2,N}
.
.
.
a_{N-1,N}

Here a_{i,j} is the level of the passage connecting Room i and Room j.

If there are multiple solutions, any of them will be accepted.

Sample Input 1

3

Sample Output 1

1 2
1

The following image describes this output:

For example, if we leave Room 2, traverse the path 2 \to 3 \to 2 \to 3 \to 2 \to 1 \to 2 while only passing passages of level 1 and get back to Room 2, we pass through a passage six times.

Solution in Rust (1.15.1) (judge verdict: Wrong Answer)

/*------------------------------------------------------------------*/
/*------------------------- begin template -------------------------*/
/*------------------------------------------------------------------*/

// ref: tanakh <https://qiita.com/tanakh/items/0ba42c7ca36cd29d0ac8>
// diff: using Parser
macro_rules! input {
    (source = $s:expr, $($r:tt)*) => {
        let mut parser = Parser::from_str($s);
        input_inner!{parser, $($r)*}
    };
    (parser = $parser:ident, $($r:tt)*) => {
        input_inner!{$parser, $($r)*}
    };
    (new_stdin_parser = $parser:ident, $($r:tt)*) => {
        let stdin = std::io::stdin();
        let reader = std::io::BufReader::new(stdin.lock());
        let mut $parser = Parser::new(reader);
        input_inner!{$parser, $($r)*}
    };
    ($($r:tt)*) => {
        input!{new_stdin_parser = parser, $($r)*}
    };
}

macro_rules! input_inner {
    ($parser:ident) => {};
    ($parser:ident, ) => {};
    ($parser:ident, $var:ident : $t:tt $($r:tt)*) => {
        let $var = read_value!($parser, $t);
        input_inner!{$parser $($r)*}
    };
}

macro_rules! read_value {
    ($parser:ident, ( $($t:tt),* )) => {
        ( $(read_value!($parser, $t)),* )
    };
    ($parser:ident, [ $t:tt ; $len:expr ]) => {
        (0..$len).map(|_| read_value!($parser, $t)).collect::<Vec<_>>()
    };
    ($parser:ident, chars) => {
        read_value!($parser, String).chars().collect::<Vec<char>>()
    };
    ($parser:ident, char_) => {
        read_value!($parser, String).chars().collect::<Vec<char>>()[0]
    };
    ($parser:ident, usize1) => {
        read_value!($parser, usize) - 1
    };
    ($parser:ident, i64_) => {
        $parser.fast_i64()
    };
    ($parser:ident, usize_) => {
        $parser.fast_i64() as usize
    };
    ($parser:ident, usize1_) => {
        ($parser.fast_i64() - 1) as usize
    };
    ($parser:ident, $t:ty) => {
        $parser.next::<$t>().expect("Parse error")
    };
}

use std::io;
use std::io::BufRead;
use std::str;

// ref: tatsuya6502 <https://qiita.com/tatsuya6502/items/cd448486f7ef7b5b8c7e>
// ref: wariuni <https://qiita.com/tatsuya6502/items/cd448486f7ef7b5b8c7e#comment-7040a5ae96305e884eb9>
// diff: using std::io::BufRead::fill_buf()
pub struct Parser<R> {
    pub reader: R,
    buf: Vec<u8>,
    pos: usize,
}

impl Parser<io::Empty> {
    pub fn from_str(s: &str) -> Parser<io::Empty> {
        Parser {
            reader: io::empty(),
            buf: s.as_bytes().to_vec(),
            pos: 0,
        }
    }
}

impl<R: BufRead> Parser<R> {
    pub fn new(reader: R) -> Parser<R> {
        Parser {
            reader: reader,
            buf: vec![],
            pos: 0,
        }
    }
    pub fn update_buf(&mut self) {
        self.buf.clear();
        self.pos = 0;
        loop {
            let (len, complete) = {
                let buf2 = self.reader.fill_buf().unwrap();
                self.buf.extend_from_slice(buf2);
                let len = buf2.len();
                (len, buf2[len - 1] <= 0x20)
            };
            self.reader.consume(len);
            if complete {
                break;
            }
        }
    }
    pub fn next<T: str::FromStr>(&mut self) -> Result<T, T::Err> {
        loop {
            let mut begin = self.pos;
            while begin < self.buf.len() && (self.buf[begin] <= 0x20) {
                begin += 1;
            }
            let mut end = begin;
            while end < self.buf.len() && (self.buf[end] > 0x20) {
                end += 1;
            }
            if begin != self.buf.len() {
                self.pos = end;
                return unsafe { str::from_utf8_unchecked(&self.buf[begin..end]) }.parse::<T>();
            } else {
                self.update_buf();
            }
        }
    }
    pub fn fast_i64(&mut self) -> i64 {
        loop {
            let mut begin = self.pos;
            while begin < self.buf.len() && (self.buf[begin] <= 0x20) {
                begin += 1;
            }
            if begin == self.buf.len() {
                self.update_buf();
                continue;
            }
            let mut res = 0;
            let (is_positive, mut end) = match self.buf[begin] {
                b'+' => (true, begin + 1),
                b'-' => (false, begin + 1),
                _ => (true, begin),
            };
            while end < self.buf.len() && (self.buf[end] > 0x20) {
                res = res * 10 + (self.buf[end] as i64 - '0' as i64);
                end += 1;
            }
            if begin != self.buf.len() {
                self.pos = end;
                return if is_positive { res } else { -res };
            } else {
                self.update_buf();
            }
        }
    }
}

#[allow(unused_imports)]
use std::cmp::*;
use std::fmt::Display;
use std::fmt::Write;

pub fn writeln_vec<T: Display>(s: &mut String, xs: &Vec<T>) {
    if xs.len() == 0 {
        writeln!(s, "");
    } else {
        write!(s, "{}", xs[0]);
        for x in xs.iter().skip(1) {
            write!(s, " {}", x);
        }
        writeln!(s, "");
    }
}

struct UnionFindNode {
    parent: usize,
    num: usize,
}

impl UnionFindNode {
    fn new(parent: usize) -> UnionFindNode {
        UnionFindNode {
            parent: parent,
            num: 1,
        }
    }
}

pub struct UnionFind(Vec<UnionFindNode>);

impl UnionFind {
    pub fn new(n: usize) -> UnionFind {
        let mut xs = vec![];
        for i in 0..n {
            xs.push(UnionFindNode::new(i));
        }
        UnionFind(xs)
    }
    pub fn root(&mut self, i: usize) -> usize {
        let parent = self.0[i].parent;
        if parent == i {
            i
        } else {
            let res = self.root(parent);
            self.0[i].parent = res;
            res
        }
    }
    pub fn unite(&mut self, i: usize, k: usize) {
        let i = self.root(i);
        let k = self.root(k);
        if i == k {
            return;
        }
        let (i, k) = if self.0[i].num < self.0[k].num {
            (i, k)
        } else {
            (k, i)
        };
        self.0[i].parent = k;
        self.0[k].num += self.0[i].num;
    }
    pub fn is_same_set(&mut self, i: usize, k: usize) -> bool {
        self.root(i) == self.root(k)
    }
    pub fn num(&mut self, i: usize) -> usize {
        let i = self.root(i);
        self.0[i].num
    }
}

fn main() {
    input! {
        new_stdin_parser = parser,
    }
    let res = solve(&mut parser);
    print!("{}", res);
}

/*------------------------------------------------------------------*/
/*-------------------------- end template --------------------------*/
/*------------------------------------------------------------------*/

fn solve<R: BufRead>(parser: &mut Parser<R>) -> String {
    let mut buf = String::new();
    input! {
        parser = parser,
        n: usize,
    }
    let mut res = vec![vec![0; n]; n];
    let mut level_table = vec![0; n];
    let mut next_level = 1;
    {
        for i in 1..n / 2 + 1 {
            let mut k = 3;
            while k < i {
                if i % k == 0 {
                    level_table[i] = level_table[i / k];
                    break;
                }
                k += 2;
            }
            if level_table[i] == 0 {
                level_table[i] = next_level;
                next_level += 1;
            }
        }
    }
    for d in 1..n / 2 + 1 {
        let mut visited = vec![false; n];
        for i in 0..n {
            let mut p = i;
            let mut count = 0;
            while !visited[p] {
                count += 1;
                let q = (p + d) % n;
                if count % 2 == 1 && visited[q] {
                    visited[p] = true;
                    p = q;
                    break;
                }
                res[min(p, q)][max(p, q)] = level_table[d];
                visited[p] = true;
                p = q;
            }
        }
    }
    for i in 0..n {
        for k in i + 1..n {
            if res[i][k] == 0 {
                res[i][k] = next_level;
            }
        }
    }
    for i in 0..n - 1 {
        writeln_vec(&mut buf, &res[i][i + 1..].to_vec());
    }
    buf
}