Problem:
Score : 1100 points

Problem Statement

You are given an integer N.

Construct any one N-by-N matrix a that satisfies the conditions below. It can be proved that a solution always exists under the constraints of this problem.

1 \leq a_{i,j} \leq 10^{15}

a_{i,j} are pairwise distinct integers.

There exists a positive integer m such that the following holds: Let x and y be two elements of the matrix that are vertically or horizontally adjacent. Then, {\rm max}(x,y) {\rm mod} {\rm min}(x,y) is always m.

Constraints

2 \leq N \leq 500

Input

Input is given from Standard Input in the following format:

N

Output

Print your solution in the following format:

a_{1,1} ... a_{1,N}
:
a_{N,1} ... a_{N,N}

Sample Input 1

2

Sample Output 1

4 7
23 10

For any two elements x and y that are vertically or horizontally adjacent, {\rm max}(x,y) {\rm mod} {\rm min}(x,y) is always 3.

Solution in C++ (GCC 9.2.1) (judge verdict: Wrong Answer)

#pragma region Macros
#pragma GCC optimize("O3")
#include <bits/stdc++.h>
#define ll long long
#define ld long double
#define rep2(i, a, b) for(ll i = a; i <= b; ++i)
#define rep(i, n) for(ll i = 0; i < n; ++i)
#define rep3(i, a, b) for(ll i = a; i >= b; --i)
#define pii pair<int, int>
#define pll pair<ll, ll>
#define pb push_back
#define eb emplace_back
#define vi vector<int>
#define vll vector<ll>
#define vpi vector<pii>
#define vpll vector<pll>
#define overload2(_1, _2, name, ...) name
#define vec(type, name, ...) vector<type> name(__VA_ARGS__)
#define VEC(type, name, size)                                                                                                                                  \
    vector<type> name(size);                                                                                                                                   \
    IN(name)
#define vv(type, name, h, ...) vector<vector<type>> name(h, vector<type>(__VA_ARGS__))
#define VV(type, name, h, w)                                                                                                                                   \
    vector<vector<type>> name(h, vector<type>(w));                                                                                                             \
    IN(name)
#define vvv(type, name, h, w, ...) vector<vector<vector<type>>> name(h, vector<vector<type>>(w, vector<type>(__VA_ARGS__)))
#define fi first
#define se second
#define all(c) begin(c), end(c)
#define ios ios_base::sync_with_stdio(0), cin.tie(0), cout.tie(0);
#define lb(c, x) distance((c).begin(), lower_bound(all(c), (x)))
#define ub(c, x) distance((c).begin(), upper_bound(all(c), (x)))
using namespace std;
template <class T> using pq = priority_queue<T>;
template <class T> using pqg = priority_queue<T, vector<T>, greater<T>>;
#define Size(c) (int)(c).size()
#define INT(...)                                                                                                                                               \
    int __VA_ARGS__;                                                                                                                                           \
    IN(__VA_ARGS__)
#define LL(...)                                                                                                                                                \
    ll __VA_ARGS__;                                                                                                                                            \
    IN(__VA_ARGS__)
#define ULL(...)                                                                                                                                               \
    ull __VA_ARGS__;                                                                                                                                           \
    IN(__VA_ARGS__)
#define STR(...)                                                                                                                                               \
    string __VA_ARGS__;                                                                                                                                        \
    IN(__VA_ARGS__)
#define CHR(...)                                                                                                                                               \
    char __VA_ARGS__;                                                                                                                                          \
    IN(__VA_ARGS__)
#define DBL(...)                                                                                                                                               \
    double __VA_ARGS__;                                                                                                                                        \
    IN(__VA_ARGS__)
#define LD(...)                                                                                                                                                \
    ld __VA_ARGS__;                                                                                                                                            \
    IN(__VA_ARGS__)
int scan() { return getchar(); }
template <class T> void scan(T a) { cin >> a; }
void scan(int &a) { cin >> a; }
void scan(long long &a) { cin >> a; }
void scan(char &a) { cin >> a; }
void scan(double &a) { cin >> a; }
void scan(long double &a) { cin >> a; }
void scan(char a[]) { scanf("%s", a); }
void scan(string &a) { cin >> a; }
template <class T> void scan(vector<T> &);
template <class T, size_t size> void scan(array<T, size> &);
template <class T, class L> void scan(pair<T, L> &);
template <class T, size_t size> void scan(T (&)[size]);
template <class T> void scan(vector<T> &a) {
    for(auto &i : a) scan(i);
}
template <class T> void scan(deque<T> &a) {
    for(auto &i : a) scan(i);
}
template <class T, size_t size> void scan(array<T, size> &a) {
    for(auto &i : a) scan(i);
}
template <class T, class L> void scan(pair<T, L> &p) {
    scan(p.first);
    scan(p.second);
}
template <class T, size_t size> void scan(T (&a)[size]) {
    for(auto &i : a) scan(i);
}
template <class T> void scan(T &a) { cin >> a; }
void IN() {}
template <class Head, class... Tail> void IN(Head &head, Tail &... tail) {
    scan(head);
    IN(tail...);
}
string stin() {
    string s;
    cin >> s;
    return s;
}
template <class T> inline bool chmax(T &a, T b) {
    if(a < b) {
        a = b;
        return 1;
    }
    return 0;
}
template <class T> inline bool chmin(T &a, T b) {
    if(a > b) {
        a = b;
        return 1;
    }
    return 0;
}
vi iota(int n) {
    vi a(n);
    iota(all(a), 0);
    return a;
}
template <class T> void UNIQUE(vector<T> &x) {
    sort(all(x));
    x.erase(unique(all(x)), x.end());
}
int in() {
    int x;
    cin >> x;
    return x;
}
ll lin() {
    unsigned long long x;
    cin >> x;
    return x;
}
void print() { putchar(' '); }
void print(bool a) { cout << a; }
void print(int a) { cout << a; }
void print(long long a) { cout << a; }
void print(char a) { cout << a; }
void print(string &a) { cout << a; }
void print(double a) { cout << a; }
template <class T> void print(const vector<T> &);
template <class T, size_t size> void print(const array<T, size> &);
template <class T, class L> void print(const pair<T, L> &p);
template <class T, size_t size> void print(const T (&)[size]);
template <class T> void print(const vector<T> &a) {
    if(a.empty()) return;
    print(a[0]);
    for(auto i = a.begin(); ++i != a.end();) {
        cout << " ";
        print(*i);
    }
    cout << endl;
}
template <class T> void print(const deque<T> &a) {
    if(a.empty()) return;
    print(a[0]);
    for(auto i = a.begin(); ++i != a.end();) {
        cout << " ";
        print(*i);
    }
}
template <class T, size_t size> void print(const array<T, size> &a) {
    print(a[0]);
    for(auto i = a.begin(); ++i != a.end();) {
        cout << " ";
        print(*i);
    }
}
template <class T, class L> void print(const pair<T, L> &p) {
    cout << '(';
    print(p.first);
    cout << ",";
    print(p.second);
    cout << ')';
}
template <class T> void print(set<T> &x) {
    for(auto e : x) print(e), cout << " ";
    cout << endl;
}
template <class T> void print(multiset<T> &x) {
    for(auto e : x) print(e), cout << " ";
    cout << endl;
}
template <class T, size_t size> void print(const T (&a)[size]) {
    print(a[0]);
    for(auto i = a; ++i != end(a);) {
        cout << " ";
        print(*i);
    }
}
template <class T> void print(const T &a) { cout << a; }
int out() {
    putchar('\n');
    return 0;
}
template <class T> int out(const T &t) {
    print(t);
    putchar('\n');
    return 0;
}
template <class Head, class... Tail> int out(const Head &head, const Tail &... tail) {
    print(head);
    putchar(' ');
    out(tail...);
    return 0;
}
ll gcd(ll a, ll b) {
    while(b) {
        ll c = b;
        b = a % b;
        a = c;
    }
    return a;
}
ll lcm(ll a, ll b) {
    if(!a || !b) return 0;
    return a * b / gcd(a, b);
}
vector<pll> factor(ll x) {
    vector<pll> ans;
    for(ll i = 2; i * i <= x; i++)
        if(x % i == 0) {
            ans.push_back({i, 1});
            while((x /= i) % i == 0) ans.back().second++;
        }
    if(x != 1) ans.push_back({x, 1});
    return ans;
}
vector<int> divisor(int x) {
    vector<int> ans;
    for(int i = 1; i * i <= x; i++)
        if(x % i == 0) {
            ans.pb(i);
            if(i * i != x) ans.pb(x / i);
        }
    return ans;
}
template <typename T> void zip(vector<T> &x) {
    vector<T> y = x;
    sort(all(y));
    for(int i = 0; i < x.size(); ++i) { x[i] = lb(y, x[i]); }
}
int popcount(ll x) { return __builtin_popcountll(x); }
mt19937 rng(chrono::steady_clock::now().time_since_epoch().count());
int rnd(int n) { return uniform_int_distribution<int>(0, n - 1)(rng); }
template <typename T> void shuffle(vector<T> &v) {
    rep3(i, v.size() - 1, 1) { swap(v[i], v[rnd(i)]); }
}
#define endl '\n'
vector<string> YES{"NO", "YES"};
vector<string> Yes{"No", "Yes"};
vector<string> yes{"no", "yes"};

#ifdef _LOCAL
#undef endl
#define debug(x)                                                                                                                                               \
    cout << #x << ": ";                                                                                                                                        \
    print(x);                                                                                                                                                  \
    cout << endl;
void err() {}
template <class T> void err(const T &t) {
    print(t);
    cout << " ";
}
template <class Head, class... Tail> void err(const Head &head, const Tail &... tail) {
    print(head);
    putchar(' ');
    out(tail...);
}
#else
#define debug(x)
template <class... T> void err(const T &...) {}
#endif

template <typename T> struct edge {
    int from, to;
    T cost;
    int id;
    edge(int to, T cost) : from(-1), to(to), cost(cost) {}
    edge(int from, int to, T cost) : from(from), to(to), cost(cost) {}
    // edge(int from,int to,T cost,int id):from(from),to(to),cost(cost),id(id){}
    edge &operator=(const int &x) {
        to = x;
        return *this;
    }
    operator int() const { return to; }
};

template <typename T> using Edges = vector<edge<T>>;
template <typename T> using WeightedTree = vector<Edges<T>>;
template <typename T> using Edges = vector<edge<T>>;
template <typename T> using WeightedGraph = vector<Edges<T>>;
template <typename T> using Matrix = vector<vector<T>>;
using Graph = vector<vector<int>>;
using tree = vector<vector<int>>;

tree getTree(int n, int offset = 1) {
    tree res(n);
    for(int i = 0; i < n - 1; i++) {
        int a, b;
        cin >> a >> b;
        a -= offset, b -= offset;
        res[a].emplace_back(b);
        res[b].emplace_back(a);
    }
    return res;
}
template <typename T> WeightedTree<T> getWeightedTree(int n, int offset = 1) {
    WeightedTree<T> res(n);
    for(int i = 0; i < n - 1; i++) {
        int a, b;
        cin >> a >> b;
        a -= offset, b -= offset;
        T c;
        cin >> c;
        res[a].emplace_back(b, c);
        res[b].emplace_back(a, c);
    }
    return res;
}

Graph getGraph(int n, int m, int offset = 1) {
    Graph res(n);
    for(int i = 0; i < m; ++i) {
        int a, b;
        cin >> a >> b;
        a -= offset, b -= offset;
        res[a].emplace_back(b);
        res[b].emplace_back(a);
    }
    return res;
}

template <typename T> WeightedGraph<T> getWeightedGraph(int n, int m, int offset = 1) {
    WeightedGraph<T> res(n);
    for(int i = 0; i < m; ++i) {
        int a, b;
        cin >> a >> b;
        T c;
        cin >> c;
        res[a].emplace_back(b, c);
        res[b].emplace_back(a, c);
    }
}
struct Setup_io {
    Setup_io() {
        ios_base::sync_with_stdio(0), cin.tie(0), cout.tie(0);
        cout << fixed << setprecision(15);
    }
} setup_io;
#pragma endregion

// _________コーディングはここから！！___________
// ... 最小側の制約も確認した？
// ... オーバーフローしない？
// from https://judge.yosupo.jp/submission/5147
vector<int> prime_sieve(const int N, const int Q = 17, const int L = 1 << 15) {
    using u8 = unsigned char;
    static const int rs[] = {1, 7, 11, 13, 17, 19, 23, 29};
    struct P {
        P(int p) : p(p) {}
        int p;
        int pos[8];
    };
    auto approx_prime_count = [](const int N) -> int { return N > 60184 ? N / (log(N) - 1.1) : max(1., N / (log(N) - 1.11)) + 1; };

    const int v = sqrt(N), vv = sqrt(v);
    vector<bool> isp(v + 1, true);
    for(int i = 2; i <= vv; ++i)
        if(isp[i]) {
            for(int j = i * i; j <= v; j += i) isp[j] = false;
        }

    const int rsize = approx_prime_count(N + 30);
    vector<int> primes = {2, 3, 5};
    int psize = 3;
    primes.resize(rsize);

    vector<P> sprimes;
    size_t pbeg = 0;
    int prod = 1;
    for(int p = 7; p <= v; ++p) {
        if(!isp[p]) continue;
        if(p <= Q) prod *= p, ++pbeg, primes[psize++] = p;
        auto pp = P(p);
        for(int t = 0; t < 8; ++t) {
            int j = (p <= Q) ? p : p * p;
            while(j % 30 != rs[t]) j += p << 1;
            pp.pos[t] = j / 30;
        }
        sprimes.push_back(pp);
    }

    vector<u8> pre(prod, 0xFF);
    for(size_t pi = 0; pi < pbeg; ++pi) {
        auto pp = sprimes[pi];
        const int p = pp.p;
        for(int t = 0; t < 8; ++t) {
            const u8 m = ~(1 << t);
            for(int i = pp.pos[t]; i < prod; i += p) pre[i] &= m;
        }
    }

    const int block_size = (L + prod - 1) / prod * prod;
    vector<u8> block(block_size);
    u8 *pblock = block.data();
    const int M = (N + 29) / 30;

    for(int beg = 0; beg < M; beg += block_size, pblock -= block_size) {
        int end = min(M, beg + block_size);
        for(int i = beg; i < end; i += prod) { copy(pre.begin(), pre.end(), pblock + i); }
        if(beg == 0) pblock[0] &= 0xFE;
        for(size_t pi = pbeg; pi < sprimes.size(); ++pi) {
            auto &pp = sprimes[pi];
            const int p = pp.p;
            for(int t = 0; t < 8; ++t) {
                int i = pp.pos[t];
                const u8 m = ~(1 << t);
                for(; i < end; i += p) pblock[i] &= m;
                pp.pos[t] = i;
            }
        }
        for(int i = beg; i < end; ++i) {
            for(int m = pblock[i]; m > 0; m &= m - 1) { primes[psize++] = i * 30 + rs[__builtin_ctz(m)]; }
        }
    }
    assert(psize <= rsize);
    while(psize > 0 && primes[psize - 1] > N) --psize;
    primes.resize(psize);
    return primes;
}
main() {
    auto P = prime_sieve(10000000);
    int n = 500;
    vv(ll, a, n, n);
    int t = 0;
    vll p;
    while(P[t] < 1000) t++;
    rep(i, n) p.eb(P[t++]);
    shuffle(p);
    auto inc = [&](int x, int y) { return 0 <= x and x < n and 0 <= y and y < n; };
    rep(i, n) a[i][i] = p[i];
    vll even, odd;
    rep(i, 125) {
        even.eb(i * 2 + 2);
        even.eb(500 - i * 2);
        odd.eb(i * 2 + 3);
        odd.eb(501 - i * 2);
    }
    rep2(i, 1, n - 1) {
        rep(j, n) {
            int x = j, y = i + j;
            if(!inc(x, y)) break;
            if(i & 1) {
                a[x][y] = a[x][y - 1] * a[x + 1][y] * even[i / 2] + 1;
            } else
                a[x][y] = p[(x + y) / 2] * even[i / 2 - 1];
        }
    }
    rep2(i, 1, n - 1) {
        rep(j, n) {
            int x = i + j, y = j;
            if(!inc(x, y)) break;
            if(i & 1) {
                a[x][y] = a[x - 1][y] * a[x][y + 1] * odd[i / 2] + 1;
            } else
                a[x][y] = p[(x + y) / 2] * odd[i / 2 - 1];
        }
    }
    INT(N);
    map<ll, pii> s;
    rep(i, N) {
        // rep(j, N) { cout << a[i][j] << " "; }
        rep(j, N) { s[a[i][j]] = {i, j}; }
        // cout << endl;
    }
    cout << (--s.end())->first << endl;
    cout << s.size() << endl;
}