Problem:
Score : 700 points

Problem Statement

Tonight, in your favourite cinema they are giving the movie Joker and all seats are occupied. In the cinema there are N rows with N seats each, forming an N\times N square. We denote with 1, 2,\dots, N the viewers in the first row (from left to right); with N+1, \dots, 2N the viewers in the second row (from left to right); and so on until the last row, whose viewers are denoted by N^2-N+1,\dots, N^2.

At the end of the movie, the viewers go out of the cinema in a certain order: the i-th viewer leaving her seat is the one denoted by the number P_i. The viewer P_{i+1} waits until viewer P_i has left the cinema before leaving her seat.
To exit from the cinema, a viewer must move from seat to seat until she exits the square of seats (any side of the square is a valid exit). A viewer can move from a seat to one of its 4 adjacent seats (same row or same column).
While leaving the cinema, it might be that a certain viewer x goes through a seat currently occupied by viewer y; in that case viewer y will hate viewer x forever. Each viewer chooses the way that minimizes the number of viewers that will hate her forever.

Compute the number of pairs of viewers (x, y) such that y will hate x forever.

Constraints

2 \le N \le 500

The sequence P_1, P_2, \dots, P_{N^2} is a permutation of \{1, 2, \dots, N^2\}.

Input

The input is given from Standard Input in the format

N
P_1 P_2 \cdots P_{N^2}

Output

If ans is the number of pairs of viewers described in the statement, you should print on Standard Output

ans

Sample Input 1

3
1 3 7 9 5 4 8 6 2

Sample Output 1

1

Before the end of the movie, the viewers are arranged in the cinema as follows:

1 2 3
4 5 6
7 8 9

The first four viewers leaving the cinema (1, 3, 7, 9) can leave the cinema without going through any seat, so they will not be hated by anybody.

Then, viewer 5 must go through one of the seats where viewers 2, 4, 6, 8 are currently seated while leaving the cinema; hence he will be hated by at least one of those viewers.

Finally the remaining viewers can leave the cinema (in the order 4, 8, 6, 2) without going through any occupied seat (actually, they can leave the cinema without going through any seat at all).

Sample Input 2

4
6 7 1 4 13 16 10 9 5 11 12 14 15 2 3 8

Sample Output 2

3

Sample Input 3

6
11 21 35 22 7 36 27 34 8 20 15 13 16 1 24 3 2 17 26 9 18 32 31 23 19 14 4 25 10 29 28 33 12 6 5 30

Sample Output 3

11

Solution in C# (.NET Core 3.1.201) (judge verdict: Wrong Answer)

using System;
using System.IO;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;

namespace AtCoder.Contest.B
{
	static class Program
	{
		public static void Main(string[] args)
		{
			var sw = new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false };
			Console.SetOut(sw);

			var cin = new Scanner();

			int n = cin.NextInt();
			var p = new List<Tuple<int, int>>();
            for (int i = 0; i < n * n; i++)
            {
				int x = cin.NextInt();
				p.Add(new Tuple<int, int>(x, i));
            }
            p.Sort();
			var ans = new List<long>();
            for (int i = 0; i < n * n; i++)
            {
                ans.Add(int.MaxValue);
            }
            /*
            var r = new List<int>();
            var l = new List<int>();
            var d = new List<int>();
            var u = new List<int>();
            for (int i = 0; i < n * n; i++)
            {
                r.Add(int.MaxValue);
                l.Add(int.MaxValue);
                d.Add(int.MaxValue);
                u.Add(int.MaxValue);
            }*/
            var s = new Set<int>();

            for (int i = 0; i < n; i++)
            {
                ans[n * i] = Math.Min(ans[n * i], 0);
                //r[n * i] = 0;
                s.Insert(p[n * i].Item2);
                for (int j = 1; j < n; j++)
                {
                    int t = s.Count() - s.LowerBound(p[n * i + j].Item2);
                    ans[n * i + j] = Math.Min(ans[n * i + j], t);
                    //r[n * i + j] = t;
                    s.Insert(p[n * i + j].Item2);
                }
                s.Clear();
            }
            
            for (int i = 0; i < n; i++)
            {
                ans[n * i + n - 1] = Math.Min(ans[n * i + n - 1], 0);
                //l[n * i + n - 1] = 0;
                s.Insert(p[n * i + n - 1].Item2);
                for (int j = n - 2; j >= 0; j--)
                {
                    int t = s.Count() - s.LowerBound(p[n * i + j].Item2);
                    ans[n * i + j] = Math.Min(ans[n * i + j], t);
                    //l[n * i + j] = t;
                    s.Insert(p[n * i + j].Item2);
                }
                s.Clear();
            }

            for (int i = 0; i < n; i++)
            {
                ans[i] = Math.Min(ans[i], 0);
               // d[i] = 0;
                s.Insert(p[i].Item2);
                for (int j = 1; j < n; j++)
                {
                    int t = s.Count() - s.LowerBound(p[i + n * j].Item2);
                    ans[i + n * j] = Math.Min(ans[i + n * j], t);
                    //d[i + n * j] = t;
                    s.Insert(p[i + n * j].Item2);
                }
                s.Clear();
            }

            for (int i = 0; i < n; i++)
            {
                ans[i + n * (n - 1)] = Math.Min(ans[i + n * (n - 1)], 0);
                //u[i + n * (n - 1)] = 0;
                s.Insert(p[i + n * (n - 1)].Item2);
                for (int j = n - 2; j >= 0; j--)
                {
                    int t = s.Count() - s.LowerBound(p[i + n * j].Item2);
                    ans[i + n * j] = Math.Min(ans[i + n * j], t);
                    //u[i + n * j] = t;
                    s.Insert(p[i + n * j].Item2);
                }
                s.Clear();
            }
            /*
            for (int i = 0; i < n * n; i++)
            {
                if (i / 10 == 0) Console.Write(" ");
                Console.WriteLine(i + " " + ans[i] + ": " + r[i] + " " + l[i] + " " + d[i] + " " + u[i]);
            }

            Console.WriteLine();

            for (int i = 0; i < n; i++)
            {
                for (int j = 0; j < n; j++)
                {
                    int x = i + n * j;
                    if (x / 10 == 0) Console.Write(" ");
                    Console.WriteLine(x + " " + ans[x] + ": " + r[x] + " " + l[x] + " " + d[x] + " " + u[x]);
                }
            }
            */

            long answer = 0;
            foreach (var e in ans)
            {
                answer += e;
            }

            Console.WriteLine(answer);

            Console.Out.Flush();
		}
	}

    /// <summary>
    /// Self-Balancing Binary Search Tree
    /// (using Randamized BST)
    /// </summary>
    public class SB_BinarySearchTree<T> where T : IComparable
    {
        public class Node
        {
            public T Value;
            public Node LChild;
            public Node RChild;
            public int Count;     //size of the sub tree

            public Node(T v)
            {
                Value = v;
                Count = 1;
            }
        }

        static Random _rnd = new Random();

        public static int Count(Node t)
        {
            return t == null ? 0 : t.Count;
        }

        static Node Update(Node t)
        {
            t.Count = Count(t.LChild) + Count(t.RChild) + 1;
            return t;
        }

        public static Node Merge(Node l, Node r)
        {
            if (l == null || r == null) return l == null ? r : l;

            if ((double)Count(l) / (double)(Count(l) + Count(r)) > _rnd.NextDouble())
            {
                l.RChild = Merge(l.RChild, r);
                return Update(l);
            }
            else
            {
                r.LChild = Merge(l, r.LChild);
                return Update(r);
            }
        }

        /// <summary>
        /// split as [0, k), [k, n)
        /// </summary>
        public static Tuple<Node, Node> Split(Node t, int k)
        {
            if (t == null) return new Tuple<Node, Node>(null, null);
            if (k <= Count(t.LChild))
            {
                var s = Split(t.LChild, k);
                t.LChild = s.Item2;
                return new Tuple<Node, Node>(s.Item1, Update(t));
            }
            else
            {
                var s = Split(t.RChild, k - Count(t.LChild) - 1);
                t.RChild = s.Item1;
                return new Tuple<Node, Node>(Update(t), s.Item2);
            }
        }

        public static Node Remove(Node t, T v)
        {
            if (Find(t, v) == null) return t;
            return RemoveAt(t, LowerBound(t, v));
        }

        public static Node RemoveAt(Node t, int k)
        {
            var s = Split(t, k);
            var s2 = Split(s.Item2, 1);
            return Merge(s.Item1, s2.Item2);
        }

        public static bool Contains(Node t, T v)
        {
            return Find(t, v) != null;
        }

        public static Node Find(Node t, T v)
        {
            while (t != null)
            {
                var cmp = t.Value.CompareTo(v);
                if (cmp > 0) t = t.LChild;
                else if (cmp < 0) t = t.RChild;
                else break;
            }
            return t;
        }

        public static Node FindByIndex(Node t, int idx)
        {
            if (t == null) return null;

            var currentIdx = Count(t) - Count(t.RChild) - 1;
            while (t != null)
            {
                if (currentIdx == idx) return t;
                if (currentIdx > idx)
                {
                    t = t.LChild;
                    currentIdx -= (Count(t == null ? null : t.RChild) + 1);
                }
                else
                {
                    t = t.RChild;
                    currentIdx += (Count(t == null ? null : t.LChild) + 1);
                }
            }

            return null;
        }

        public static int UpperBound(Node t, T v)
        {
            var torg = t;
            if (t == null) return -1;

            var ret = Int32.MaxValue;
            var idx = Count(t) - Count(t.RChild) - 1;
            while (t != null)
            {
                var cmp = t.Value.CompareTo(v);

                if (cmp > 0)
                {
                    ret = Math.Min(ret, idx);
                    t = t.LChild;
                    idx -= (Count(t == null ? null : t.RChild) + 1);
                }
                else if (cmp <= 0)
                {
                    t = t.RChild;
                    idx += (Count(t == null ? null : t.LChild) + 1);
                }
            }
            return ret == Int32.MaxValue ? Count(torg) : ret;
        }

        public static int LowerBound(Node t, T v)
        {
            var torg = t;
            if (t == null) return -1;

            var idx = Count(t) - Count(t.RChild) - 1;
            var ret = Int32.MaxValue;
            while (t != null)
            {
                var cmp = t.Value.CompareTo(v);
                if (cmp >= 0)
                {
                    if (cmp == 0) ret = Math.Min(ret, idx);
                    t = t.LChild;
                    if (t == null) ret = Math.Min(ret, idx);
                    idx -= t == null ? 0 : (Count(t.RChild) + 1);
                }
                else if (cmp < 0)
                {
                    t = t.RChild;
                    idx += (Count(t == null ? null : t.LChild) + 1);
                    if (t == null) return idx;
                }
            }
            return ret == Int32.MaxValue ? Count(torg) : ret;
        }

        public static Node Insert(Node t, T v)
        {
            var ub = LowerBound(t, v);
            return InsertByIdx(t, ub, v);
        }

        static Node InsertByIdx(Node t, int k, T v)
        {
            var s = Split(t, k);
            return Merge(Merge(s.Item1, new Node(v)), s.Item2);
        }

        public static IEnumerable<T> Enumerate(Node t)
        {
            var ret = new List<T>();
            Enumerate(t, ret);
            return ret;
        }

        static void Enumerate(Node t, List<T> ret)
        {
            if (t == null) return;
            Enumerate(t.LChild, ret);
            ret.Add(t.Value);
            Enumerate(t.RChild, ret);
        }
    }

    /// <summary>
    /// C-like set
    /// </summary>
    public class Set<T> where T : IComparable
    {
        protected SB_BinarySearchTree<T>.Node _root;

        public T this[int idx] { get { return ElementAt(idx); } }

        public int Count()
        {
            return SB_BinarySearchTree<T>.Count(_root);
        }

        public virtual void Insert(T v)
        {
            if (_root == null) _root = new SB_BinarySearchTree<T>.Node(v);
            else
            {
                if (SB_BinarySearchTree<T>.Find(_root, v) != null) return;
                _root = SB_BinarySearchTree<T>.Insert(_root, v);
            }
        }

        public void Clear()
        {
            _root = null;
        }

        public void Remove(T v)
        {
            _root = SB_BinarySearchTree<T>.Remove(_root, v);
        }

        public bool Contains(T v)
        {
            return SB_BinarySearchTree<T>.Contains(_root, v);
        }

        public T ElementAt(int k)
        {
            var node = SB_BinarySearchTree<T>.FindByIndex(_root, k);
            if (node == null) throw new IndexOutOfRangeException();
            return node.Value;
        }

        public int Count(T v)
        {
            return SB_BinarySearchTree<T>.UpperBound(_root, v) - SB_BinarySearchTree<T>.LowerBound(_root, v);
        }

        public int LowerBound(T v)
        {
            return SB_BinarySearchTree<T>.LowerBound(_root, v);
        }

        public int UpperBound(T v)
        {
            return SB_BinarySearchTree<T>.UpperBound(_root, v);
        }

        public Tuple<int, int> EqualRange(T v)
        {
            if (!Contains(v)) return new Tuple<int, int>(-1, -1);
            return new Tuple<int, int>(SB_BinarySearchTree<T>.LowerBound(_root, v), SB_BinarySearchTree<T>.UpperBound(_root, v) - 1);
        }

        public List<T> ToList()
        {
            return new List<T>(SB_BinarySearchTree<T>.Enumerate(_root));
        }
    }

    public class Pair<T, U> : IComparable, IEquatable<Pair<T, U>> where T : IComparable<T>, IEquatable<T> where U : IComparable<U>, IEquatable<U>
	{
		public T First { get; set; }
		public U Second { get; set; }

		public Pair(T first, U second)
		{
			First = first;
			Second = second;
		}

		public int CompareTo(object obj)
		{
			Pair<T, U> castedObj = (Pair<T, U>)obj;
			int x = First.CompareTo(castedObj.First);
			if (x != 0) return x;
			else return Second.CompareTo(castedObj.Second);
		}

		public static bool operator ==(Pair<T, U> x, Pair<T, U> y) => x.CompareTo(y) == 0;

		public static bool operator !=(Pair<T, U> x, Pair<T, U> y) => x.CompareTo(y) != 0;

		public static bool operator <(Pair<T, U> x, Pair<T, U> y) => x.CompareTo(y) == 1;

		public static bool operator >(Pair<T, U> x, Pair<T, U> y) => x.CompareTo(y) == -1;

		public static bool operator <=(Pair<T, U> x, Pair<T, U> y) => x.CompareTo(y) > 0;

		public static bool operator >=(Pair<T, U> x, Pair<T, U> y) => x.CompareTo(y) < 0;

		public bool Equals(Pair<T, U> x) => First.Equals(x.First) && Second.Equals(x.Second);

		public override bool Equals(object obj)
		{
			return obj is Pair<T, U> pair &&
				   EqualityComparer<T>.Default.Equals(First, pair.First) &&
				   EqualityComparer<U>.Default.Equals(Second, pair.Second);
		}

		public override int GetHashCode()
		{
			unchecked
			{
				var hashCode = 405212230;
				hashCode = hashCode * -1521134295 + EqualityComparer<T>.Default.GetHashCode(First);
				hashCode = hashCode * -1521134295 + EqualityComparer<U>.Default.GetHashCode(Second);
				return hashCode;
			}
		}
	}

	class Scanner
	{
		string[] s;
		int i;

		char[] cs = new char[] { ' ' };

		public Scanner()
		{
			s = new string[0];
			i = 0;
		}

		public string Next()
		{
			if (i < s.Length) return s[i++];
			string st = Console.ReadLine();
			while (st == "") st = Console.ReadLine();
			s = st.Split(cs, StringSplitOptions.RemoveEmptyEntries);
			if (s.Length == 0) return Next();
			i = 0;
			return s[i++];
		}

		public int NextInt()
		{
			return int.Parse(Next());
		}

		public int[] ArrayInt(int N, int add = 0)
		{
			int[] Array = new int[N];
			for (int i = 0; i < N; i++)
			{
				Array[i] = NextInt() + add;
			}
			return Array;
		}

		public long NextLong()
		{
			return long.Parse(Next());
		}

		public long[] ArrayLong(int N, long add = 0)
		{
			long[] Array = new long[N];
			for (int i = 0; i < N; i++)
			{
				Array[i] = NextLong() + add;
			}
			return Array;
		}

		public double NextDouble()
		{
			return double.Parse(Next());
		}

		public double[] ArrayDouble(int N, double add = 0)
		{
			double[] Array = new double[N];
			for (int i = 0; i < N; i++)
			{
				Array[i] = NextDouble() + add;
			}
			return Array;
		}
	}
}