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++14 (GCC 5.4.1) (judge verdict: Wrong Answer)

#include<cstdio>
typedef long long ll;
const int MAXN = 500;
const int MAXM = 10000;
bool vis[MAXM + 5];
int prm[MAXM + 5], pcnt = 0;
ll GCD(ll x, ll y) {
	return y == 0 ? x : GCD(y, x%y);
}
ll LCM(ll x, ll y) {
	return x / GCD(x, y) * y;
}
void sieve() {
	for(int i=2;i<=MAXM;i++) {
		if( !vis[i] ) prm[++pcnt] = i;
		for(int j=1;i*prm[j]<=MAXM;j++) {
			vis[i*prm[j]] = true;
			if( i % prm[j] == 0 ) break;
		}
	}
}
ll Mat[MAXN + 5][MAXN + 5];
int main() {
	sieve(); int N;
	scanf("%d", &N);
	for(int i=1;i<=N;i++)
		for(int j=1;j<=N;j++)
			Mat[i][j] = 1;
	for(int i=1;i<=N;i++)
		for(int j=1;j<=N;j++)
			if( (i & 1) == (j & 1) ) {
				Mat[i][j] = prm[(i + j) / 2] * prm[(i+N+1-j)/2 + N];
				Mat[i+1][j] = LCM(Mat[i+1][j], Mat[i][j]);
				Mat[i-1][j] = LCM(Mat[i-1][j], Mat[i][j]);
				Mat[i][j+1] = LCM(Mat[i][j+1], Mat[i][j]);
				Mat[i][j-1] = LCM(Mat[i][j-1], Mat[i][j]);
			}
	for(int i=1;i<=N;i++)
		for(int j=1;j<=N;j++)
			if( (i & 1) != (j & 1) )
				Mat[i][j]++;
	for(int i=1;i<=N;i++) {
		for(int j=1;j<N;j++)
			printf("%lld ", Mat[i][j]);
		printf("%lld\n", Mat[i][N]);
	}
}