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: Accepted)

using System;
using System.Collections;
using System.Collections.Generic;
using System.Diagnostics;
using System.Globalization;
using System.IO;
using System.Text;
using System.Linq;

using E = System.Linq.Enumerable;
using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;

public class FastQueue<T> {
    private T[] _array;
    private int _head;
    private int _tail;
    public int Count { get; private set; }

    public FastQueue(int capacity = 16) {
        _array = new T[capacity];
        _head = 0;
        _tail = 0;
    }

    public void Clear() {
        _head = _tail = Count = 0;
    }

    private void Expand() {
        var newNodes = new T[_array.Length * 2];

        if (_head < _tail) {
            Array.Copy(_array, _head, newNodes, 0, Count);
        } else {
            Array.Copy(_array, _head, newNodes, 0, _array.Length - _head);
            Array.Copy(_array, 0, newNodes, _array.Length - _head, _tail);
        }

        _head = 0;
        _tail = Count == newNodes.Length ? 0 : Count;
        _array = newNodes;
    }

    public void Enqueue(T value) {
        if (Count + 1 == _array.Length) { // expand
            Expand();
        }
        Count++;
        _array[_tail++] = value;
        if (_tail == _array.Length) {
            _tail = 0;
        }
    }
    public T Dequeue() {
        var ret = _array[_head++];
        Count--;
        if (_head == _array.Length) {
            _head = 0;
        }
        return ret;
    }
}

internal partial class Solver {

    public int[] dx = new int[] { 0, 1, 0, -1 };
    public int[] dy = new int[] { 1, 0, -1, 0 };

    public void Run() {
        var n = ni();
        var p = ni(n * n);

        //n = 500; p = E.Range(1, n * n).ToArray();

        var a = new int[n, n];
        var cost = new int[n * n];
        var weight = new int[n * n];
        var rows = new int[n * n];
        var cols = new int[n * n];
        int ToIndex(int r, int c) => r * n + c;
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < n; j++) {
                var val = i * n + j;
                a[i, j] = val;
                var id = ToIndex(i, j);
                weight[id] = 1;
                rows[val] = i;
                cols[val] = j;
                cost[id] = Math.Min(Math.Min(i, n - 1 - i), Math.Min(j, n - 1 - j));
            }
        }
        long ans = 0;
        var queue = new FastQueue<(int, int, int)>((n + 2) * (n + 2));
        var next = new FastQueue<(int, int, int)>((n + 2) * (n + 2));
        for (int i = 0; i < p.Length; i++) {
            var row = rows[p[i] - 1];
            var col = cols[p[i] - 1];
            var ids = ToIndex(row, col);
            ans += cost[ids];
            weight[ids] = 0;

            int minAdjCost = int.MaxValue;

            for (int k = 0; k < 4; k++) {
                var nr = row + dy[k];
                var nc = col + dx[k];
                if (0 <= nr && nr < n && 0 <= nc && nc < n) {
                    minAdjCost = Math.Min(minAdjCost, cost[ToIndex(nr, nc)]);
                } else {
                    minAdjCost = -1;
                    break;
                }
            }

            if (cost[ids] > minAdjCost) {
                queue.Enqueue((row, col, minAdjCost));
                while (queue.Count > 0) {
                    while (queue.Count > 0) {
                        var (crow, ccol, step) = queue.Dequeue();
                        var id = ToIndex(crow, ccol);
                        if (cost[id] <= step) continue;
                        cost[id] = step;
                        {
                            var nr = crow + 1;
                            var nc = ccol + 0;
                            if (0 <= nr && nr < n && 0 <= nc && nc < n) {
                                var nid = ToIndex(nr, nc);
                                var w = weight[nid];
                                if (cost[nid] > step + w) {
                                    if (w == 0) {
                                        queue.Enqueue((nr, nc, step));
                                    } else {
                                        next.Enqueue((nr, nc, step + 1));
                                    }
                                }
                            }
                        }
                        {
                            var nr = crow + 0;
                            var nc = ccol + 1;
                            if (0 <= nr && nr < n && 0 <= nc && nc < n) {
                                var nid = ToIndex(nr, nc);
                                var w = weight[nid];
                                if (cost[nid] > step + w) {
                                    if (w == 0) {
                                        queue.Enqueue((nr, nc, step));
                                    } else {
                                        next.Enqueue((nr, nc, step + 1));
                                    }
                                }
                            }
                        }
                        {
                            var nr = crow - 1;
                            var nc = ccol + 0;
                            if (0 <= nr && nr < n && 0 <= nc && nc < n) {
                                var nid = ToIndex(nr, nc);
                                var w = weight[nid];
                                if (cost[nid] > step + w) {
                                    if (w == 0) {
                                        queue.Enqueue((nr, nc, step));
                                    } else {
                                        next.Enqueue((nr, nc, step + 1));
                                    }
                                }
                            }
                        }
                        {
                            var nr = crow + 0;
                            var nc = ccol - 1;
                            if (0 <= nr && nr < n && 0 <= nc && nc < n) {
                                var nid = ToIndex(nr, nc);
                                var w = weight[nid];
                                if (cost[nid] > step + w) {
                                    if (w == 0) {
                                        queue.Enqueue((nr, nc, step));
                                    } else {
                                        next.Enqueue((nr, nc, step + 1));
                                    }
                                }
                            }
                        }
                    }
                    Swap(ref queue, ref next);
                }
                //cost.Dump();
            }
        }

        cout.WriteLine(ans);
    }

    public static void Swap<T>(ref T lhs, ref T rhs) { T temp; temp = lhs; lhs = rhs; rhs = temp; }
}
public class BooleanArray {
    private readonly int[] array;
    private int trueValue = 1;

    public BooleanArray(int size) {
        array = new int[size];
    }

    public void Clear() {
        trueValue++;
    }

    public bool this[int index] {
        get {
            return array[index] == trueValue;
        }
        set {
            array[index] = value ? trueValue : trueValue - 1;
        }
    }

    public int Length {
        get {
            return array.Length;
        }
    }
}

// PREWRITEN CODE BEGINS FROM HERE
internal partial class Solver : Scanner {
    public static void Main() {
#if LOCAL
        byte[] inputBuffer = new byte[1000000];
        var inputStream = Console.OpenStandardInput(inputBuffer.Length);
        using (var reader = new StreamReader(inputStream, Console.InputEncoding, false, inputBuffer.Length)) {
            Console.SetIn(reader);
            new Solver(Console.In, Console.Out).Run();
        }
#else
        Console.SetOut(new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false });
        new Solver(Console.In, Console.Out).Run();
        Console.Out.Flush();
#endif
    }

#pragma warning disable IDE0052
    private readonly TextReader cin;
    private readonly TextWriter cout;
#pragma warning restore IDE0052

    public Solver(TextReader reader, TextWriter writer)
        : base(reader) {
        cin = reader;
        cout = writer;
    }
    public Solver(string input, TextWriter writer)
        : this(new StringReader(input), writer) {
    }

#pragma warning disable IDE1006
#pragma warning disable IDE0051
    private int ni() { return NextInt(); }
    private int[] ni(int n) { return NextIntArray(n); }
    private long nl() { return NextLong(); }
    private long[] nl(int n) { return NextLongArray(n); }
    private double nd() { return NextDouble(); }
    private double[] nd(int n) { return NextDoubleArray(n); }
    private string ns() { return Next(); }
    private string[] ns(int n) { return NextArray(n); }
#pragma warning restore IDE1006
#pragma warning restore IDE0051
}

internal static class LinqPadExtension {
    [Conditional("DEBUG")]
    public static void Dump<T>(this T obj) {
#if DEBUG
        LINQPad.Extensions.Dump(obj);
#endif
    }
}

public class Scanner {
    private readonly TextReader Reader;
    private readonly CultureInfo ci = CultureInfo.InvariantCulture;

    private readonly char[] buffer = new char[2 * 1024];
    private int cursor = 0, length = 0;
    private string Token;
    private readonly StringBuilder sb = new StringBuilder(1024);

    public Scanner()
        : this(Console.In) {
    }

    public Scanner(TextReader reader) {
        Reader = reader;
    }

    public int NextInt() { return checked((int)NextLong()); }
    public long NextLong() {
        var s = Next();
        long r = 0;
        int i = 0;
        bool negative = false;
        if (s[i] == '-') {
            negative = true;
            i++;
        }
        for (; i < s.Length; i++) {
            r = r * 10 + (s[i] - '0');
#if DEBUG
            if (!char.IsDigit(s[i])) throw new FormatException();
#endif
        }
        return negative ? -r : r;
    }
    public double NextDouble() { return double.Parse(Next(), ci); }
    public string[] NextArray(int size) {
        string[] array = new string[size];
        for (int i = 0; i < size; i++) {
            array[i] = Next();
        }

        return array;
    }
    public int[] NextIntArray(int size) {
        int[] array = new int[size];
        for (int i = 0; i < size; i++) {
            array[i] = NextInt();
        }

        return array;
    }

    public long[] NextLongArray(int size) {
        long[] array = new long[size];
        for (int i = 0; i < size; i++) {
            array[i] = NextLong();
        }

        return array;
    }

    public double[] NextDoubleArray(int size) {
        double[] array = new double[size];
        for (int i = 0; i < size; i++) {
            array[i] = NextDouble();
        }

        return array;
    }

    public string Next() {
        if (Token == null) {
            if (!StockToken()) {
                throw new Exception();
            }
        }
        var token = Token;
        Token = null;
        return token;
    }

    public bool HasNext() {
        if (Token != null) {
            return true;
        }

        return StockToken();
    }

    private bool StockToken() {
        while (true) {
            sb.Clear();
            while (true) {
                if (cursor >= length) {
                    cursor = 0;
                    if ((length = Reader.Read(buffer, 0, buffer.Length)) <= 0) {
                        break;
                    }
                }
                var c = buffer[cursor++];
                if (33 <= c && c <= 126) {
                    sb.Append(c);
                } else {
                    if (sb.Length > 0) break;
                }
            }

            if (sb.Length > 0) {
                Token = sb.ToString();
                return true;
            }

            return false;
        }
    }
}