Problem:
Score : 500 points

Problem Statement

Snuke has two boards, each divided into a grid with N rows and N columns.
For both of these boards, the square at the i-th row from the top and the j-th column from the left is called Square (i,j).

There is a lowercase English letter written in each square on the first board. The letter written in Square (i,j) is S_{i,j}. On the second board, nothing is written yet.

Snuke will write letters on the second board, as follows:

First, choose two integers A and B ( 0 \leq A, B < N ).

Write one letter in each square on the second board.
Specifically, write the letter written in Square ( i+A, j+B ) on the first board into Square (i,j) on the second board.
Here, the k-th row is also represented as the (N+k)-th row, and the k-th column is also represented as the (N+k)-th column.

After this operation, the second board is called a good board when, for every i and j ( 1 \leq i, j \leq N ), the letter in Square (i,j) and the letter in Square (j,i) are equal.

Find the number of the ways to choose integers A and B ( 0 \leq A, B < N ) such that the second board is a good board.

Constraints

1 \leq N \leq 300

S_{i,j} ( 1 \leq i, j \leq N ) is a lowercase English letter.

Input

Input is given from Standard Input in the following format:

N
S_{1,1}S_{1,2}..S_{1,N}
S_{2,1}S_{2,2}..S_{2,N}
:
S_{N,1}S_{N,2}..S_{N,N}

Output

Print the number of the ways to choose integers A and B ( 0 \leq A, B < N ) such that the second board is a good board.

Sample Input 1

2
ab
ca

Sample Output 1

2

For each pair of A and B, the second board will look as shown below:

The second board is a good board when (A,B) = (0,1) or (A,B) = (1,0), thus the answer is 2.

Sample Input 2

4
aaaa
aaaa
aaaa
aaaa

Sample Output 2

16

Every possible choice of A and B makes the second board good.

Sample Input 3

5
abcde
fghij
klmno
pqrst
uvwxy

Sample Output 3

0

No possible choice of A and B makes the second board good.

Solution in Rust (1.15.1) (judge verdict: Time Limit Exceeded)

/**
*  _           _                 __                            _   _ _   _                                 _                    _                  _
* | |         | |               / /                           | | (_) | (_)                               | |                  (_)                | |
* | |__   __ _| |_ ___   ___   / /__ ___  _ __ ___  _ __   ___| |_ _| |_ ___   _____ ______ _ __ _   _ ___| |_ ______ ___ _ __  _ _ __  _ __   ___| |_ ___
* | '_ \ / _` | __/ _ \ / _ \ / / __/ _ \| '_ ` _ \| '_ \ / _ \ __| | __| \ \ / / _ \______| '__| | | / __| __|______/ __| '_ \| | '_ \| '_ \ / _ \ __/ __|
* | | | | (_| | || (_) | (_) / / (_| (_) | | | | | | |_) |  __/ |_| | |_| |\ V /  __/      | |  | |_| \__ \ |_       \__ \ | | | | |_) | |_) |  __/ |_\__ \
* |_| |_|\__,_|\__\___/ \___/_/ \___\___/|_| |_| |_| .__/ \___|\__|_|\__|_| \_/ \___|      |_|   \__,_|___/\__|      |___/_| |_|_| .__/| .__/ \___|\__|___/
*                                                  | |                                                                           | |   | |
*                                                  |_|                                                                           |_|   |_|
*
* https://github.com/hatoo/competitive-rust-snippets
*/
#[allow(unused_imports)]
use std::cmp::{max, min, Ordering};
#[allow(unused_imports)]
use std::collections::{BTreeMap, BTreeSet, BinaryHeap, HashMap, HashSet, VecDeque};
#[allow(unused_imports)]
use std::io::{stdin, stdout, BufWriter, Write};
#[allow(unused_imports)]
use std::iter::FromIterator;
mod util {
    use std::fmt::Debug;
    use std::io::{stdin, stdout, BufWriter, StdoutLock};
    use std::str::FromStr;
    #[allow(dead_code)]
    pub fn line() -> String {
        let mut line: String = String::new();
        stdin().read_line(&mut line).unwrap();
        line.trim().to_string()
    }
    #[allow(dead_code)]
    pub fn chars() -> Vec<char> {
        line().chars().collect()
    }
    #[allow(dead_code)]
    pub fn gets<T: FromStr>() -> Vec<T>
    where
        <T as FromStr>::Err: Debug,
    {
        let mut line: String = String::new();
        stdin().read_line(&mut line).unwrap();
        line.split_whitespace()
            .map(|t| t.parse().unwrap())
            .collect()
    }
    #[allow(dead_code)]
    pub fn with_bufwriter<F: FnOnce(BufWriter<StdoutLock>) -> ()>(f: F) {
        let out = stdout();
        let writer = BufWriter::new(out.lock());
        f(writer)
    }
}
#[allow(unused_macros)]
macro_rules ! get { ( $ t : ty ) => { { let mut line : String = String :: new ( ) ; stdin ( ) . read_line ( & mut line ) . unwrap ( ) ; line . trim ( ) . parse ::<$ t > ( ) . unwrap ( ) } } ; ( $ ( $ t : ty ) ,* ) => { { let mut line : String = String :: new ( ) ; stdin ( ) . read_line ( & mut line ) . unwrap ( ) ; let mut iter = line . split_whitespace ( ) ; ( $ ( iter . next ( ) . unwrap ( ) . parse ::<$ t > ( ) . unwrap ( ) , ) * ) } } ; ( $ t : ty ; $ n : expr ) => { ( 0 ..$ n ) . map ( | _ | get ! ( $ t ) ) . collect ::< Vec < _ >> ( ) } ; ( $ ( $ t : ty ) ,*; $ n : expr ) => { ( 0 ..$ n ) . map ( | _ | get ! ( $ ( $ t ) ,* ) ) . collect ::< Vec < _ >> ( ) } ; ( $ t : ty ;; ) => { { let mut line : String = String :: new ( ) ; stdin ( ) . read_line ( & mut line ) . unwrap ( ) ; line . split_whitespace ( ) . map ( | t | t . parse ::<$ t > ( ) . unwrap ( ) ) . collect ::< Vec < _ >> ( ) } } ; ( $ t : ty ;; $ n : expr ) => { ( 0 ..$ n ) . map ( | _ | get ! ( $ t ;; ) ) . collect ::< Vec < _ >> ( ) } ; }
#[allow(unused_macros)]
macro_rules ! debug { ( $ ( $ a : expr ) ,* ) => { eprintln ! ( concat ! ( $ ( stringify ! ( $ a ) , " = {:?}, " ) ,* ) , $ ( $ a ) ,* ) ; } }
const BIG_STACK_SIZE: bool = true;
#[allow(dead_code)]
fn main() {
    use std::thread;
    if BIG_STACK_SIZE {
        thread::Builder::new()
            .stack_size(32 * 1024 * 1024)
            .name("solve".into())
            .spawn(solve)
            .unwrap()
            .join()
            .unwrap();
    } else {
        solve();
    }
}

const TRUE: &'static bool = &true;
const FALSE: &'static bool = &false;
#[derive(Clone, Debug)]
/// Efficient bool collection
pub struct BitSet {
    buf: Vec<u64>,
    size: usize,
}
impl BitSet {
    #[allow(dead_code)]
    pub fn new(size: usize) -> BitSet {
        BitSet {
            buf: vec![0; (size + 63) / 64],
            size: size,
        }
    }
    #[allow(dead_code)]
    pub fn set(&mut self, i: usize, b: bool) {
        assert!(i < self.size);
        if b {
            self.buf[i >> 6] |= 1 << (i & 63);
        } else {
            self.buf[i >> 6] &= !(1 << (i & 63));
        }
    }
    #[allow(dead_code)]
    pub fn count_ones(&self) -> u32 {
        self.buf.iter().map(|x| x.count_ones()).sum()
    }
    #[allow(dead_code)]
    fn chomp(&mut self) {
        let r = self.size & 63;
        if r != 0 {
            if let Some(x) = self.buf.last_mut() {
                let d = 64 - r;
                *x = (*x << d) >> d;
            }
        }
    }
}
impl std::ops::Index<usize> for BitSet {
    type Output = bool;
    fn index(&self, index: usize) -> &bool {
        [FALSE, TRUE][(self.buf[index >> 6] >> (index & 63)) as usize & 1]
    }
}
impl std::ops::ShlAssign<usize> for BitSet {
    fn shl_assign(&mut self, x: usize) {
        let q = x >> 6;
        let r = x & 63;
        if q >= self.buf.len() {
            for x in &mut self.buf {
                *x = 0;
            }
            return;
        }
        if r == 0 {
            for i in (q..self.buf.len()).rev() {
                self.buf[i] = self.buf[i - q];
            }
        } else {
            for i in (q + 1..self.buf.len()).rev() {
                self.buf[i] = (self.buf[i - q] << r) | (self.buf[i - q - 1] >> (64 - r));
            }
            self.buf[q] = self.buf[0] << r;
        }
        for i in 0..q {
            self.buf[i] = 0;
        }
        self.chomp();
    }
}
impl std::ops::Shl<usize> for BitSet {
    type Output = Self;
    fn shl(mut self, x: usize) -> Self {
        self <<= x;
        self
    }
}
impl std::ops::ShrAssign<usize> for BitSet {
    fn shr_assign(&mut self, x: usize) {
        let q = x >> 6;
        let r = x & 63;
        if q >= self.buf.len() {
            for x in &mut self.buf {
                *x = 0;
            }
            return;
        }
        if r == 0 {
            for i in 0..self.buf.len() - q {
                self.buf[i] = self.buf[i + q];
            }
        } else {
            for i in 0..self.buf.len() - q - 1 {
                self.buf[i] = (self.buf[i + q] >> r) | (self.buf[i + q + 1] << (64 - r));
            }
            let len = self.buf.len();
            self.buf[len - q - 1] = self.buf[len - 1] >> r;
        }
        for i in self.buf.len() - q..self.buf.len() {
            self.buf[i] = 0;
        }
    }
}
impl std::ops::Shr<usize> for BitSet {
    type Output = Self;
    fn shr(mut self, x: usize) -> Self {
        self >>= x;
        self
    }
}
impl<'a> std::ops::BitAndAssign<&'a BitSet> for BitSet {
    fn bitand_assign(&mut self, rhs: &'a Self) {
        for (a, b) in self.buf.iter_mut().zip(rhs.buf.iter()) {
            *a &= *b;
        }
    }
}
impl<'a> std::ops::BitAnd<&'a BitSet> for BitSet {
    type Output = Self;
    fn bitand(mut self, rhs: &'a Self) -> Self {
        self &= rhs;
        self
    }
}
impl<'a> std::ops::BitOrAssign<&'a BitSet> for BitSet {
    fn bitor_assign(&mut self, rhs: &'a Self) {
        for (a, b) in self.buf.iter_mut().zip(rhs.buf.iter()) {
            *a |= *b;
        }
        self.chomp();
    }
}
impl<'a> std::ops::BitOr<&'a BitSet> for BitSet {
    type Output = Self;
    fn bitor(mut self, rhs: &'a Self) -> Self {
        self |= rhs;
        self
    }
}
impl<'a> std::ops::BitXorAssign<&'a BitSet> for BitSet {
    fn bitxor_assign(&mut self, rhs: &'a Self) {
        for (a, b) in self.buf.iter_mut().zip(rhs.buf.iter()) {
            *a ^= *b;
        }
        self.chomp();
    }
}
impl<'a> std::ops::BitXor<&'a BitSet> for BitSet {
    type Output = Self;
    fn bitxor(mut self, rhs: &'a Self) -> Self {
        self ^= rhs;
        self
    }
}

fn solve() {
    let n = get!(usize);
    let s: Vec<Vec<char>> = (0..n).map(|_| util::chars()).collect();

    let mut h = HashMap::new();

    for i in 0..n {
        for j in 0..n {
            h.entry(s[i][j]).or_insert(HashSet::new()).insert((i, j));
        }
    }

    if h.len() == 1 {
        println!("{}", n * n);
        return;
    }
    let mut ans = BitSet::new(n * n);
    for i in 0..n * n {
        ans.set(i, true);
    }
    let mut vss: Vec<HashSet<(usize, usize)>> = h.into_iter().map(|t| t.1).collect();
    vss.sort_by_key(|h| h.len());
    for vs in &vss[..vss.len() - 1] {
        let mut set = BitSet::new(n * n);
        for a in 0..n {
            'outer: for b in 0..n {
                if !ans[a * n + b] {
                    continue;
                }
                for &(x, y) in vs {
                    let v = ((n + y - b + a) % n, (n + x - a + b) % n);
                    if !vs.contains(&v) {
                        continue 'outer;
                    }
                }
                set.set(a * n + b, true);
            }
        }

        ans &= &set;
    }

    println!("{}", ans.count_ones());
}