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 Haskell (GHC 7.10.3) (judge verdict: Accepted)

import qualified Data.Vector.Unboxed as VU
import qualified Data.Vector.Unboxed.Mutable as VUM
import Control.Monad

type Cand = (Int, Int) -- (start, skip)                                         
-- a candidate set is either empty (skip == 0)                                  
-- or singleton (skip == n)                                                     
-- or { start + i * skip | i = 0, 1, ..., i0 } where skip|n                     

type VChar = VU.Vector Char
type Ring = [Int]

icmp :: Int -> Int -> Int -> Int
icmp n x y = ((y + n) `mod` n) * n + ((x + n) `mod` n)

idiv :: Int -> Int -> (Int, Int)
idiv n i = (i `mod` n, i `div` n)

solve :: Int -> [String] -> Int
solve n css =
  if skip == 0 then 0 else n * (n `div` skip)
  where
    vc = buildV n css
    (_, skip) = foldr (nextCand n vc) (0,1) allRings
    allRings = map (mkRing n) [0..n-1]

mkRing :: Int -> Int -> Ring
mkRing n i = map (\k -> icmp n (0+k) (i-k)) [0..n-1]

nextCand :: Int -> VChar -> Ring -> Cand -> Cand
nextCand n vc ring cand@(start,skip)
  | skip == 0 = (0,0)
  | otherwise = nextCand1 n vc ring cand

nextCand1 :: Int -> VChar -> Ring -> Cand -> Cand
nextCand1 n vc ring (start,skip) =
  case allGoods of []         -> (0,0)
                   [k]        -> (k,n)
                   (k0:k1:_)  -> (k0,k1-k0)
  where allGoods = [fk p | p <- [0 .. n`div`skip - 1], good n vc ring (fk p)]
          where fk p = start + skip * p

good :: Int -> VChar -> Ring -> Int -> Bool
good n vc ring k = flip all ring $ \i ->
  let (x, y) = idiv n i
      j      = icmp n (y+k) (x-k)
  in vc VU.! i == vc VU.! j

buildV :: Int -> [String] -> VChar
buildV n css =
  VU.create $ do
  v <- VUM.new (n*n)
  forM (zip [0..] (concat css)) $ \(i, c) -> do
    VUM.write v i c
  return v

main = do
  conts <- getContents
  let sn : css = lines conts
      n = read sn
  putStrLn $ show $ solve n css