Students went into a class to write a test and sat in some way. The teacher thought: "Probably they sat in this order to copy works of each other. I need to rearrange them in such a way that students that were neighbors are not neighbors in a new seating."

The class can be represented as a matrix with n rows and m columns with a student in each cell. Two students are neighbors if cells in which they sit have a common side.

Let's enumerate students from 1 to n·m in order of rows. So a student who initially sits in the cell in row i and column j has a number (i - 1)·m + j. You have to find a matrix with n rows and m columns in which all numbers from 1 to n·m appear exactly once and adjacent numbers in the original matrix are not adjacent in it, or determine that there is no such matrix.


-----Input-----

The only line contains two integers n and m (1 ≤ n, m ≤ 10^5; n·m ≤ 10^5) — the number of rows and the number of columns in the required matrix.


-----Output-----

If there is no such matrix, output "NO" (without quotes). 

Otherwise in the first line output "YES" (without quotes), and in the next n lines output m integers which form the required matrix.


-----Examples-----
Input
2 4

Output
YES
5 4 7 2 
3 6 1 8 

Input
2 1

Output
NO



-----Note-----

In the first test case the matrix initially looks like this:

1 2 3 4

5 6 7 8



It's easy to see that there are no two students that are adjacent in both matrices.

In the second test case there are only two possible seatings and in both of them students with numbers 1 and 2 are neighbors.

Source: codeforces | Difficulty: HARD
Tags: Complete search

Mathematics

Constructive algorithms


Solutions:
def get_answer(m, n):
	if (m, n) in [(1, 2), (2, 1), (1, 3), (3, 1), (2, 2), (2, 3), (3, 2)]:
		return ('NO', [])
	elif m == 1:
		mat = [[i for i in range(2, n + 1, 2)] + [i for i in range(1, n + 1, 2)]]
		return ('YES', mat)
	elif n == 1:
		mat = [[i] for i in range(2, m + 1, 2)] + [[i] for i in range(1, m + 1, 2)]
		return ('YES', mat)
	elif n == 2:
		bs = [[2, 3], [7, 6], [4, 1], [8, 5]]
		mat = []
		for i in range(m // 4):
			for u in bs:
				if i % 2 == 0:
					mat.append([x + 8 * i for x in u])
				else:
					mat.append([x + 8 * i for x in reversed(u)])
		if m % 4 == 1:
			mat.insert(4, [0, 0])
			for i in range(4, 0, -1):
				mat[i][0] = mat[i - 1][0]
			mat[0][0] = m * n
			mat[4][1] = m * n - 1
		elif m % 4 == 2:
			if m // 4 % 2 == 1:
				mat = [[m * n - 3, m * n]] + mat + [[m * n - 1, m * n - 2]]
			else:
				mat = [[m * n - 3, m * n]] + mat + [[m * n - 2, m * n - 1]]
		elif m % 4 == 3:
			mat.insert(4, [0, 0])
			for i in range(4, 0, -1):
				mat[i][0] = mat[i - 1][0]
			mat[0][0] = m * n - 4
			mat[4][1] = m * n - 5
			mat = [[m * n - 1, m * n - 2]] + mat + [[m * n - 3, m * n]]
		return ('YES', mat)
	elif n == 3:
		bs = [[6, 1, 8], [7, 5, 3], [2, 9, 4]]
		mat = []
		for i in range(m // 3):
			for u in bs:
				mat.append([x + 9 * i for x in u])
		if m % 3 == 1:
			mat = [[m * n - 1, m * n - 2, m * n]] + mat
			(mat[0][1], mat[1][1]) = (mat[1][1], mat[0][1])
		elif m % 3 == 2:
			mat = [[m * n - 4, m * n - 5, m * n - 3]] + mat + [[m * n - 1, m * n - 2, m * n]]
			(mat[0][1], mat[1][1]) = (mat[1][1], mat[0][1])
			(mat[m - 2][1], mat[m - 1][1]) = (mat[m - 1][1], mat[m - 2][1])
		return ('YES', mat)
	mat = []
	for i in range(m):
		if i % 2 == 0:
			mat.append([i * n + j for j in range(2, n + 1, 2)] + [i * n + j for j in range(1, n + 1, 2)])
		elif n != 4:
			mat.append([i * n + j for j in range(1, n + 1, 2)] + [i * n + j for j in range(2, n + 1, 2)])
		else:
			mat.append([i * n + j for j in range(n - (n % 2 == 0), 0, -2)] + [i * n + j for j in range(n - (n % 2 == 1), 0, -2)])
	return ('YES', mat)
(m, n) = input().split()
m = int(m)
n = int(n)
res = get_answer(m, n)
print(res[0])
if res[0] == 'YES':
	for i in range(m):
		for j in range(n):
			print(res[1][i][j], end=' ')
		print()


def getNumber(x, y):
	x %= xsize
	y %= ysize
	return x + 1 + y * xsize
(ysize, xsize) = (int(e) for e in input().split(' '))
if xsize > 3 and ysize > 1:
	print('YES')
	res = ''
	for y in range(ysize):
		for x in range(xsize):
			(tx, ty) = (x, y)
			ty += x % 2
			tx += 2 * (ty % ysize % 2)
			res += str(getNumber(tx, ty))
			res += ' '
		res += '\n'
	print(res)
elif xsize > 1 and ysize > 3:
	print('YES')
	res = ''
	for y in range(ysize):
		for x in range(xsize):
			(tx, ty) = (x, y)
			tx += y % 2
			ty += 2 * (tx % xsize % 2)
			res += str(getNumber(tx, ty))
			res += ' '
		res += '\n'
	print(res)
elif xsize == 1 and ysize == 4:
	print('YES')
	print('3\n1\n4\n2')
elif xsize == 4 and ysize == 1:
	print('YES')
	print('3 1 4 2')
elif xsize == 1 and ysize > 4:
	print('YES')
	res = ''
	tsize = ysize - (not ysize % 2)
	for y in range(tsize):
		res += str(y * 2 % tsize + 1)
		res += '\n'
	if not ysize % 2:
		res += str(ysize)
	print(res)
elif xsize > 4 and ysize == 1:
	print('YES')
	res = ''
	tsize = xsize - (not xsize % 2)
	for x in range(tsize):
		res += str(x * 2 % tsize + 1)
		res += ' '
	if not xsize % 2:
		res += str(xsize)
	print(res)
elif xsize == 3 and ysize == 3:
	print('YES')
	print('7 2 6\n3 4 8\n5 9 1')
elif xsize == 1 and ysize == 1:
	print('YES')
	print('1')
else:
	print('NO')


import random
(n, m) = map(int, input().split())
if n < 4 and m < 4 and (not (n == 1 and m == 1 or (m == 3 and n == 3))):
	print('NO')
	quit()
pedy = [list() for i in range(n * m + 1)]
for i in range(n * m):
	if i % m != 0:
		pedy[i + 1].append(i)
	if i % m != m - 1:
		pedy[i + 1].append(i + 2)
	if i >= m:
		pedy[i + 1].append(i - m + 1)
	if i < (n - 1) * m:
		pedy[i + 1].append(i + m + 1)
Arr = [x for x in range(1, n * m + 1)]
Arr = Arr[::2] + Arr[1::2]
pp = 0
s = ''
while not pp:
	pp = 1
	for i in range(n):
		for j in range(m):
			if i + 1 != n and Arr[i * m + m + j] in pedy[Arr[i * m + j]] or (j + 1 != m and Arr[i * m + 1 + j] in pedy[Arr[i * m + j]]):
				pp = 0
				break
		if not pp:
			break
	if not pp:
		random.shuffle(Arr)
print('YES')
for i in range(n):
	for j in range(m):
		s += str(Arr[i * m + j]) + ' '
	print(s)
	s = ''


import sys, itertools
f = sys.stdin

def near(i, n, m):
	x = i // m
	y = i % m
	d = [[0, -1], [0, 1], [-1, 0], [1, 0]]
	ns = []
	for (dx, dy) in d:
		nx = x + dx
		ny = y + dy
		if nx >= 0 and nx < n and (ny >= 0) and (ny < m):
			ns.append(nx * m + ny)
	return ns

def check(p, n, m):
	d = [[0, -1], [0, 1], [-1, 0], [1, 0]]
	for x in range(n):
		for y in range(m):
			ns = near(p[x * m + y], n, m)
			for (dx, dy) in d:
				nx = x + dx
				ny = y + dy
				if nx >= 0 and nx < n and (ny >= 0) and (ny < m) and (p[nx * m + ny] in ns):
					return True
	return False
(n, m) = map(int, f.readline().split())
reverse = False
if n > m:
	t1 = list(range(n * m))
	t2 = []
	for i in range(m):
		for j in range(n):
			t2.append(t1[j * m + i])
	t = n
	n = m
	m = t
	reverse = True

def ans(n, m):
	if m >= 5:
		p = []
		for i in range(n):
			t3 = []
			for j in range(m):
				if j * 2 + i % 2 >= m:
					break
				t3.append(j * 2 + i % 2)
			for j in range(m - len(t3)):
				if j * 2 + 1 - i % 2 >= m:
					break
				t3.append(j * 2 + 1 - i % 2)
			p += [x + i * m for x in t3]
		return p
	if n == 1 and m == 1:
		return [0]
	if n == 1 and m <= 3:
		return False
	if n == 2 and m <= 3:
		return False
	if n == 3 and m == 4:
		return [4, 3, 6, 1, 2, 5, 0, 7, 9, 11, 8, 10]
	if n == 4:
		return [4, 3, 6, 1, 2, 5, 0, 7, 12, 11, 14, 9, 15, 13, 11, 14]
	for p in list(itertools.permutations(list(range(n * m)), n * m)):
		failed = check(p, n, m)
		if not failed:
			return p
	return True
p = ans(n, m)
if p:
	print('YES')
	if reverse:
		for j in range(m):
			print(' '.join((str(t2[p[i * m + j]] + 1) for i in range(n))))
	else:
		for i in range(n):
			print(' '.join((str(p[i * m + j] + 1) for j in range(m))))
else:
	print('NO')


import bisect

def list_output(s):
	print(' '.join(map(str, s)))

def list_input(s='int'):
	if s == 'int':
		return list(map(int, input().split()))
	elif s == 'float':
		return list(map(float, input().split()))
	return list(map(str, input().split()))
(n, m) = map(int, input().split())
swapped = False
if n > m:
	(n, m) = (m, n)
	swapped = True

def check(M):
	for i in range(n):
		for j in range(m):
			if i - 1 >= 0 and M[i - 1][j] + m == M[i][j]:
				return False
			if i + 1 < n and M[i + 1][j] == M[i][j] + m:
				return False
			if j - 1 >= 0 and M[i][j - 1] + 1 == M[i][j]:
				return False
			if j + 1 < m and M[i][j + 1] == M[i][j] + 1:
				return False
	return True

def transpose(M):
	n = len(M)
	m = len(M[0])
	R = [[0 for i in range(n)] for j in range(m)]
	for i in range(n):
		for j in range(m):
			R[j][i] = M[i][j]
	return R
if n == 1 and m == 1:
	print('YES')
	print('1')
	exit(0)
if n <= 2 and m <= 3:
	print('NO')
	exit(0)
R = list()
if n == 3 and m == 3:
	R.append([4, 3, 8])
	R.append([9, 1, 6])
	R.append([5, 7, 2])
elif m == 4:
	if n == 1:
		R.append([3, 1, 4, 2])
	elif n == 2:
		R.append([5, 4, 7, 2])
		R.append([3, 6, 1, 8])
	elif n == 3:
		R.append([5, 4, 7, 2])
		R.append([3, 6, 1, 8])
		R.append([11, 9, 12, 10])
	elif n == 4:
		R.append([5, 4, 7, 2])
		R.append([3, 6, 1, 8])
		R.append([11, 9, 12, 10])
		R.append([14, 16, 15, 13])
else:
	M = [[(i - 1) * m + j for j in range(1, m + 1)] for i in range(1, n + 1)]
	for i in range(n):
		row = list()
		if i % 2 == 0:
			for j in range(0, m, 2):
				row.append(M[i][j])
			for j in range(1, m, 2):
				row.append(M[i][j])
		else:
			for j in range(1, m, 2):
				row.append(M[i][j])
			for j in range(0, m, 2):
				row.append(M[i][j])
		R.append(row)
if swapped:
	M = [[(i - 1) * n + j for j in range(1, n + 1)] for i in range(1, m + 1)]
	M = transpose(M)
	S = [[0 for j in range(m)] for i in range(n)]
	for i in range(n):
		for j in range(m):
			r = (R[i][j] - 1) // m
			c = (R[i][j] - 1) % m
			S[i][j] = M[r][c]
	R = transpose(S)
	(n, m) = (m, n)
print('YES')
for i in range(n):
	print(' '.join(map(str, R[i])))


import os, sys
from io import BytesIO, IOBase

def main():
	(n, m) = map(int, input().split())
	arr = [[i * m + j for j in range(1, m + 1)] for i in range(n)]
	if m >= 4:
		print('YES')
		for i in range(n):
			(k, k1) = ([], [])
			for j in range(0, m, 2):
				k.append(arr[i][j])
			for j in range(1, m, 2):
				k1.append(arr[i][j])
			if not i & 1:
				if m == 4:
					arr[i] = k[::-1] + k1[::-1]
				else:
					arr[i] = k + k1
			else:
				arr[i] = k1 + k
		for i in arr:
			print(*i)
	elif n >= 4:
		print('YES')
		for i in range(m):
			(k, k1) = ([], [])
			for j in range(0, n, 2):
				k.append(arr[j][i])
			for j in range(1, n, 2):
				k1.append(arr[j][i])
			if not i & 1:
				if n == 4:
					for (ind, x) in enumerate(k[::-1] + k1[::-1]):
						arr[ind][i] = x
				else:
					for (ind, x) in enumerate(k + k1):
						arr[ind][i] = x
			else:
				for (ind, x) in enumerate(k1 + k):
					arr[ind][i] = x
		for i in arr:
			print(*i)
	elif n == 1 and m == 1:
		print('YES')
		print(1)
	elif n == 3 and m == 3:
		print('YES')
		print(9, 3, 5)
		print(2, 7, 1)
		print(4, 6, 8)
	else:
		print('NO')
BUFSIZE = 8192

class FastIO(IOBase):
	newlines = 0

	def __init__(self, file):
		self._fd = file.fileno()
		self.buffer = BytesIO()
		self.writable = 'x' in file.mode or 'r' not in file.mode
		self.write = self.buffer.write if self.writable else None

	def read(self):
		while True:
			b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFSIZE))
			if not b:
				break
			ptr = self.buffer.tell()
			(self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr))
		self.newlines = 0
		return self.buffer.read()

	def readline(self):
		while self.newlines == 0:
			b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFSIZE))
			self.newlines = b.count(b'\n') + (not b)
			ptr = self.buffer.tell()
			(self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr))
		self.newlines -= 1
		return self.buffer.readline()

	def flush(self):
		if self.writable:
			os.write(self._fd, self.buffer.getvalue())
			(self.buffer.truncate(0), self.buffer.seek(0))

class IOWrapper(IOBase):

	def __init__(self, file):
		self.buffer = FastIO(file)
		self.flush = self.buffer.flush
		self.writable = self.buffer.writable
		self.write = lambda s: self.buffer.write(s.encode('ascii'))
		self.read = lambda : self.buffer.read().decode('ascii')
		self.readline = lambda : self.buffer.readline().decode('ascii')
(sys.stdin, sys.stdout) = (IOWrapper(sys.stdin), IOWrapper(sys.stdout))
input = lambda : sys.stdin.readline().rstrip('\r\n')
main()


(n, m) = map(int, input().strip().split(' '))
arr = []
cnt = 1
if n * m == 1:
	print('YES')
	print(1)
elif n == 1 and (m == 2 or m == 3) or (n == 2 and (m == 1 or m == 2 or m == 3)) or (n == 3 and (m == 1 or m == 2)):
	print('NO')
elif n == 3 and m == 3:
	print('YES')
	print(6, 1, 8)
	print(7, 5, 3)
	print(2, 9, 4)
else:
	print('YES')
	if m >= n:
		for i in range(n):
			l = []
			for j in range(m):
				l.append(cnt)
				cnt += 1
			arr.append(l)
		ans = []
		for i in range(n):
			l = arr[i]
			odd = []
			even = []
			for j in range(m):
				if (j + 1) % 2 == 0:
					even.append(l[j])
				else:
					odd.append(l[j])
			if (i + 1) % 2 == 1:
				if n % 2 == 1 and m % 2 == 1:
					odd.extend(even)
					ans.append(odd)
				elif m % 2 == 1 and n % 2 == 0:
					odd.extend(even)
					ans.append(odd)
				else:
					even.extend(odd)
					ans.append(even)
			elif n % 2 == 1 and m % 2 == 1:
				even.extend(odd)
				ans.append(even)
			elif m % 2 == 1 and n % 2 == 0:
				even.extend(odd)
				ans.append(even)
			else:
				even.reverse()
				odd.reverse()
				odd.extend(even)
				ans.append(odd)
		for i in range(n):
			for j in range(m):
				print(ans[i][j], end=' ')
			print()
	else:
		temp = n
		n = m
		m = temp
		cnt = 1
		for i in range(n):
			l = []
			p = cnt
			for j in range(m):
				l.append(p)
				p += n
			cnt += 1
			arr.append(l)
		ans = []
		for i in range(n):
			l = arr[i]
			odd = []
			even = []
			for j in range(m):
				if (j + 1) % 2 == 0:
					even.append(l[j])
				else:
					odd.append(l[j])
			if (i + 1) % 2 == 1:
				if n % 2 == 1 and m % 2 == 1:
					odd.extend(even)
					ans.append(odd)
				elif m % 2 == 1 and n % 2 == 0:
					odd.extend(even)
					ans.append(odd)
				else:
					even.extend(odd)
					ans.append(even)
			elif n % 2 == 1 and m % 2 == 1:
				even.extend(odd)
				ans.append(even)
			elif m % 2 == 1 and n % 2 == 0:
				even.extend(odd)
				ans.append(even)
			else:
				even.reverse()
				odd.reverse()
				odd.extend(even)
				ans.append(odd)
		for i in range(m):
			for j in range(n):
				print(ans[j][i], end=' ')
			print()


def transpos(a, n, m):
	b = []
	for i in range(m):
		b.append([a[j][i] for j in range(n)])
	return b

def printarr(a):
	for i in range(len(a)):
		for j in range(len(a[i])):
			print(a[i][j], end=' ')
		print('')
(n, m) = [int(i) for i in input().split()]
a = []
for i in range(n):
	a.append([i * m + j for j in range(1, m + 1)])
transp_flag = False
if n > m:
	tmp = m
	m = n
	n = tmp
	a = transpos(a, m, n)
	transp_flag = True
if m < 3 and m != 1:
	print('NO')
elif m == 1:
	print('YES')
	print(1)
else:
	if m == 3:
		if n < 3:
			print('NO')
		else:
			print('YES')
			printarr([[5, 9, 3], [7, 2, 4], [1, 6, 8]])
	elif m == 4:
		for i in range(n):
			tmp = a[i][:]
			if i % 2 == 0:
				a[i] = [tmp[1], tmp[3], tmp[0], tmp[2]]
			else:
				a[i] = [tmp[2], tmp[0], tmp[3], tmp[1]]
	else:
		for i in range(n):
			if i % 2 == 0:
				tmp1 = [a[i][j] for j in range(m) if j % 2 == 0]
				tmp2 = [a[i][j] for j in range(m) if j % 2 == 1]
			if i % 2 == 1:
				tmp1 = [a[i][j] for j in range(m) if j % 2 == 1]
				tmp2 = [a[i][j] for j in range(m) if j % 2 == 0]
			a[i] = tmp1 + tmp2
	if m > 3:
		if transp_flag:
			a = transpos(a, n, m)
		print('YES')
		printarr(a)


(n, m) = map(int, input().split())
if n == 1 and m == 1:
	print('YES\n1')
elif n == 3 and m == 3:
	print('YES')
	print(6, 1, 8)
	print(7, 5, 3)
	print(2, 9, 4)
elif n < 4 and m < 4:
	print('NO')
elif n == 1 or m == 1:
	t = max(n, m)
	a = [i for i in range(2, t + 1, 2)]
	a += [i for i in range(1, t + 1, 2)]
	print('YES')
	for i in a:
		print(i, end='')
		print([' ', '\n'][m == 1], end='')
else:
	a = []
	for j in range(n):
		a.append([int(i) + int(m * j) for i in range(1, m + 1)])
	if n <= m:
		for j in range(1, m, 2):
			t = a[0][j]
			for i in range(1, n):
				a[i - 1][j] = a[i][j]
			a[n - 1][j] = t
		for i in range(1, n, 2):
			(r, s) = (a[i][0], a[i][1])
			for j in range(2, m):
				a[i][j - 2] = a[i][j]
			(a[i][m - 2], a[i][m - 1]) = (r, s)
	else:
		for j in range(1, m, 2):
			(r, s) = (a[0][j], a[1][j])
			for i in range(2, n):
				a[i - 2][j] = a[i][j]
			(a[n - 2][j], a[n - 1][j]) = (r, s)
		for i in range(1, n, 2):
			t = a[i][0]
			for j in range(1, m):
				a[i][j - 1] = a[i][j]
			a[i][m - 1] = t
	print('YES')
	for i in range(n):
		print(*a[i])