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.Linq;
using System.Collections;
using System.Collections.Generic;

namespace AtCoder.Contests.AGC044
{
    /// <summary>
    /// 
    /// </summary>
    static class B
    {
        static int n;
        static bool[,] nn;


        static int[,] mm;
        static void Main(string[] args)
        {
            n = NextInt();
            var ps = ReadIntArray();
            nn = new bool[n + 2, n + 2];

            var sum = 0;

            mm = new int[n + 2, n + 2];
            Initializemm();
            for (int i = 0; i < n * n; i++)
            {
                var p = ps[i];
                var startp = new P { x = (p-1) / n + 1, y = (p-1) % n + 1, d = 0 };


                var newCost = E4(startp).Min(po => mm[po.x, po.y]);

                sum += newCost;
                mm[startp.x, startp.y] = newCost;

                nn[startp.x, startp.y] = true;
                var done = new bool[n + 2, n + 2];

                var q = new Queue<P>(n * n);
                foreach (var item in E4(startp))
                {
                    if (mm[item.x, item.y] > newCost + (nn[item.x, item.y] ? 0 : 1))
                    {
                        mm[item.x, item.y] = newCost + (nn[item.x, item.y] ? 0 : 1);
                        q.Enqueue(item);
                    }
                }
                while (q.Any())
                {
                    var next = q.Dequeue();
                    var cost = mm[next.x, next.y];
                    foreach (var item in E4(next))
                    {
                        if (mm[item.x, item.y] > cost + (nn[next.x, next.y] ? 0 : 1))
                        {
                            mm[item.x, item.y] = cost + (nn[next.x, next.y] ? 0 : 1);
                            q.Enqueue(item);
                        }
                    }

                    //var nextCost = E4(next).Min(po => mm[po.x, po.y] + (nn[po.x, po.y] ? 0 : 1));
                    //if (cost > nextCost)
                    //{
                    //    mm[next.x, next.y] = nextCost;
                    //    foreach (var item in E4(next))
                    //    {
                    //        q.Enqueue(item);
                    //    }
                    //}
                }
            }

            Console.WriteLine(sum);


            //for (int i = 0; i < n * n; i++)
            //{
            //    var p = ps[i];
            //    var startp = new P { x = (p-1) / n + 1, y = (p-1) % n + 1, d = 0 };
            //    var mm = new int[n + 2, n + 2];
            //    var done = new bool[n + 2, n + 2];
            //    for (int k = 0; k < n + 2; k++)
            //    {
            //        for (int l = 0; l < n + 2; l++)
            //        {
            //            mm[k, l] = int.MaxValue;
            //        }
            //    }
            //    var heap = new MinHeap<P>(n*n, (a,b)=>a.d.CompareTo(b.d));
            //    heap.Push(startp);

            //    while (heap.Any())
            //    {
            //        var c = heap.Pop();
            //        done[c.x, c.y] = true;
            //        SetNext(c);
            //        foreach (var next in nexts)
            //        {
            //            if (done[next.x, next.y]) continue;
            //            if (mm[next.x, next.y] > next.d)
            //            {
            //                mm[next.x, next.y] = next.d;
            //                heap.Push(next);
            //            }
            //        }
            //    }

            //    nn[startp.x, startp.y] = true;
            //    sum += mm[0, 0];
            //}

            //Console.WriteLine(sum);
        }
        static void Initializemm()
        {
            for (int i = 1; i <= n; i++)
            {
                for (int j = 1; j <= n; j++)
                {
                    mm[i, j] = Math.Min(Math.Min(i, n - i + 1), Math.Min(j, n - j + 1));
                }
            }
        }
        static IEnumerable<P> E4(P c)
        {
            yield return new P { x = c.x - 1, y = c.y, d = c.d + (nn[c.x - 1, c.y] ? 0 : 1) };
            yield return new P { x = c.x + 1, y = c.y, d = c.d + (nn[c.x + 1, c.y] ? 0 : 1) };
            yield return new P { x = c.x, y = c.y - 1, d = c.d + (nn[c.x, c.y - 1] ? 0 : 1) };
            yield return new P { x = c.x, y = c.y + 1, d = c.d + (nn[c.x, c.y + 1] ? 0 : 1) };
        }
        static List<P> nexts = new List<P>(4);

        static void SetNext(P c)
        {
            nexts.Clear();
            if (c.x == 0 || c.x == n + 1 || c.y == 0 || c.y == n + 1)
            {
                return;
            }
            if (c.x == 1 || c.x == n || c.y == 1 || c.y == n)
            {
                nexts.Add(new P
                {
                    x = 0,
                    y = 0,
                    d = c.d
                });
                return;
            }
            nexts.Add(new P { x = c.x - 1, y = c.y, d = c.d + (nn[c.x - 1, c.y] ? 0 : 1) });
            nexts.Add(new P { x = c.x + 1, y = c.y, d = c.d + (nn[c.x + 1, c.y] ? 0 : 1) });
            nexts.Add(new P { x = c.x, y = c.y - 1, d = c.d + (nn[c.x, c.y - 1] ? 0 : 1) });
            nexts.Add(new P { x = c.x, y = c.y + 1, d = c.d + (nn[c.x, c.y + 1] ? 0 : 1) });
        }

        //static IEnumerable<P> Enumerate(P current, bool[,] nn, int n)
        //{
        //    if (current.x==0 || current.x==n+1 || current.y==0 || current.y==n+1)
        //    {
        //        yield break;
        //    }

        //    if (current.x == 1 || current.x == n || current.y == 1 || current.y == n)
        //    {
        //        yield return new P
        //        {
        //            x = 0,
        //            y = 0,
        //            d = current.d
        //        };
        //        yield break;
        //    }

        //    yield return new P { x = current.x - 1, y = current.y, d = current.d + (nn[current.x - 1, current.y] ? 0 : 1) };
        //    yield return new P { x = current.x + 1, y = current.y, d = current.d + (nn[current.x + 1, current.y] ? 0 : 1) };
        //    yield return new P { x = current.x, y = current.y - 1, d = current.d + (nn[current.x, current.y - 1] ? 0 : 1) };
        //    yield return new P { x = current.x, y = current.y + 1, d = current.d + (nn[current.x, current.y + 1] ? 0 : 1) };
        //}

        struct P
        {
            public int x;
            public int y;
            public int d;
        }

        #region Console
        static int _index = 0;
        static string[] _buffer;

        static string Next()
        {
            if (_buffer == null || _index >= _buffer.Length)
            {
                _buffer = Console.ReadLine().Split();
                _index = 0;
            }

            return _buffer[_index++];
        }
        static int NextInt()
        {
            return int.Parse(Next());
        }
        static long NextLong()
        {
            return long.Parse(Next());
        }
        static int ReadInt()
        {
            return int.Parse(Console.ReadLine());
        }
        static long ReadLong()
        {
            return long.Parse(Console.ReadLine());
        }
        static int[] ReadIntArray()
        {
            return Console.ReadLine().Split(' ').Select(int.Parse).ToArray();
        }
        static int[] ReadIntArray(int n)
        {
            var a = new int[n];
            for (int i = 0; i < n; i++)
            {
                a[i] = ReadInt();
            }
            return a;
        }
        static long[] ReadLongArray()
        {
            return Console.ReadLine().Split(' ').Select(long.Parse).ToArray();
        }
        static long[] ReadLongArray(int n)
        {
            var a = new long[n];
            for (int i = 0; i < n; i++)
            {
                a[i] = ReadLong();
            }
            return a;
        }
        static List<T> ReadObjects<T>(int n, Func<int[], T> creator)
        {
            var l = new List<T>(n);

            for (int i = 0; i < n; i++)
            {
                var a = ReadIntArray();
                l.Add(creator(a));
            }

            return l;
        }

        static int[,] ReadMatrix(int n, int m)
        {
            var r = new int[n, m];
            for (int i = 0; i < n; i++)
            {
                var a = ReadIntArray();
                for (int j = 0; j < m; j++)
                {
                    r[i, j] = a[j];
                }
            }
            return r;
        }
        #endregion
    }

    public class MinHeap<T> : IEnumerable<T>
    {
        private List<T> _list;
        private Comparer<T> _comp;

        public int Count
        {
            get
            {
                return _list.Count;
            }
        }

        public MinHeap() : this(0) { }
        public MinHeap(int capacity, Comparison<T> comp = null)
        {
            _list = new List<T>(capacity);

            if (comp == null)
            {
                _comp = Comparer<T>.Default;
            }
            else
            {
                _comp = Comparer<T>.Create(comp);
            }
        }

        public void Push(T elem)
        {
            Upheap(elem);
        }

        public T Pop()
        {
            return Downheap();
        }

        public T Peek()
        {
            return _list[0];
        }


        private void Swap(int i, int j)
        {
            var a = _list[j];
            _list[j] = _list[i];
            _list[i] = a;
        }

        private void Upheap(T elem)
        {
            _list.Add(elem);
            var i = _list.Count - 1;

            while (true)
            {
                if (i == 0) return;
                var j = (i - 1) / 2;

                var e = _list[i];
                var p = _list[j];

                var c = _comp.Compare(e, p);
                if (c < 0)
                {
                    Swap(i, j);
                    i = j;
                    continue;
                }
                else
                {
                    break;
                }
            }
        }

        private T Downheap()
        {
            var result = _list[0];

            var last = _list.Count - 1;
            Swap(0, last);
            _list.RemoveAt(last);
            if (_list.Count == 0) return result;

            var i = 0;
            while (true)
            {
                if (i == _list.Count - 1) return result;
                var p = _list[i];

                var j = i * 2 + 1;
                if (j >= _list.Count) return result;

                if (j + 1 < _list.Count && _comp.Compare(_list[j], _list[j + 1]) > 0)
                {
                    j++;
                }

                if (_comp.Compare(_list[i], _list[j]) > 0)
                {
                    Swap(i, j);
                    i = j;
                }
                else
                {
                    return result;
                }
            }
        }

        public MinHeap<T> Copy(Comparison<T> comp = null)
        {
            var c = new MinHeap<T>(0, comp);
            c._list = _list.ToList();

            return c;
        }
        public MinHeap(List<T> list)
        {
            _list = list;

            _comp = Comparer<T>.Default;
        }

        public List<T> GetList()
        {
            return _list;
        }

        public IEnumerator<T> GetEnumerator()
        {
            return _list.GetEnumerator();
        }

        IEnumerator IEnumerable.GetEnumerator()
        {
            return GetEnumerator();
        }
    }
}