Bob is a dance teacher and he started dance classes recently. He observes a strange attendance pattern among his students. Initially, there are no students. On day i, a new student starts attending the class. The student stops attending the class, if and only if he has attended the class for i consecutive days. 
Also, the student resumes attending the class, if and only if he has not attended the class for i consecutive days. 

We denote the student who starts coming on day i as student i. 

To mark attendance, o denotes present and x denotes absent.  

For example, the schedule for student 1 from day 1 is as follows: 

oxoxoxoxoxoxoxoxoxox...  

The schedule for the student 3 from day 1 is as follows:  

xxoooxxxoooxxxoooxxx... 

(Student 3 starts attending the class from day 3, and stops attending from day 6, and then starts attending from day 9, and so on. )  

The schedule for the student 5 from day 1 is as follows.
xxxxoooooxxxxxoooooxxxxx...

Bob wants his students to dance in pairs. However, if the number of students coming on day i is odd, then there will be someone who can't find a partner. So Bob wants to know if the number of students coming on day i is even or odd. We denote the number of students coming on day i as N(i).  Please find out whether N(i) is even or odd.

Input format  

The first line contains an integer, T, which denotes the number of test cases. 

For each test case, there is an integer i

Output Format 

For each test case, if N(i) is even, then print even. 

If N(i) is odd, then print one line odd.

Constraints 

1 ≤ T ≤ 100 

1 ≤ i ≤ 10^{18}  

Sample Input

4
1
2
3
4

Sample Output  

odd
odd
odd
even

Explanation 

The number of students coming on day 1 is 1: only student #1 attends the class. So N(1)=1 and it is odd. 

The number of students coming on day 2 is 1:  student #2, so n(2)=1 and it is odd. 

The number of students coming on day 3 is 3: student #1, student #2, and student #3. So N(3)=3 and it is odd. 

The number of students coming on day 4 is 2: student #3 and student #4. So N(4)=2 and it is even.

Difficulty: MEDIUM_HARD
Tags: Number theory

Starter code:

Solutions:
from math import ceil
T = int(input())
for _ in range(T):
	x = int(input())
	n = int(ceil((1 + x) ** 0.5 - 1))
	if n % 2 == 1:
		print('odd')
	else:
		print('even')


from math import sqrt
T = int(input())
for t in range(T):
	i = int(input())
	ans = int(sqrt(i))
	while ans * ans > i:
		ans -= 1
	while (ans + 1) * (ans + 1) <= i:
		ans += 1
	print('odd' if ans % 2 else 'even')


import math
test = int(input())
for _ in range(test):
	value = int(input())
	n = math.ceil(math.sqrt(value + 1)) - 1
	if n % 2 == 0:
		print('even')
	else:
		print('odd')


def isqrt(n):
	x = n
	y = x + n // x >> 1
	while y < x:
		x = y
		y = x + n // x >> 1
	return x

def nbStudentParity(d):
	r = isqrt(d)
	p = 0
	for i in range(r | 1, d // r + 1):
		p += d // i
	if p % 2 == 0:
		return 'even'
	else:
		return 'odd'
T = int(input())
for _ in range(T):
	i = int(input())
	print(nbStudentParity(i))


import math
t = int(input())

def binarysearch(i, a, b):
	mid = (a + b) // 2
	if i == a ** 2:
		return a
	elif i == b ** 2:
		return b
	elif i > mid ** 2 and i < (mid + 1) ** 2:
		return mid
	elif i <= mid ** 2 and i > a ** 2:
		return binarysearch(i, a, mid)
	elif i >= (mid + 1) ** 2 and i < b ** 2:
		return binarysearch(i, mid + 1, b)
for i in range(t):
	j = int(input())
	k = binarysearch(j, 1, 10 ** 9)
	if k % 2 == 0:
		print('even')
	else:
		print('odd')


from sys import stdin as cin, stdout as cout
from math import sqrt
from fractions import gcd

def sqrt1(i, a, b):
	if i == a ** 2:
		return a
	if i == b ** 2:
		return b
	mid = (a + b) // 2
	if i > mid ** 2 and i < (mid + 1) ** 2:
		return mid
	if i < mid ** 2:
		return sqrt1(i, a, mid)
	return sqrt1(i, mid, b)
for t in range(int(cin.readline())):
	i = int(cin.readline())
	m = sqrt1(i, 1, 10 ** 9)
	if m % 2 == 0:
		print('even')
	else:
		print('odd')


def binarysearch(i, a, b):
	mid = (a + b) // 2
	if i == a ** 2:
		return a
	elif i == b ** 2:
		return b
	elif i > mid ** 2 and i < (mid + 1) ** 2:
		return mid
	elif i <= mid ** 2 and i > a ** 2:
		return binarysearch(i, a, mid)
	elif i >= (mid + 1) ** 2 and i < b ** 2:
		return binarysearch(i, mid + 1, b)
t = int(input())
for k in range(t):
	n = int(input())
	a = 1
	b = 10 ** 9
	count = binarysearch(n, a, b)
	if count % 2 == 0:
		print('even')
	else:
		print('odd')


import sys, math

def Summ(N):
	return (1 + N) * N // 2

def RemSumm(N, m):
	int(N // m * Summ(m - 1)) + int(Summ(N % m))

def CheckDay(i):
	y = (0, 577215664901532860)

def CheckDayBrute(i):
	n = 0
	for s in range(1, i + 1):
		n += i // s % 2
	return n

def SummN(a1, d, n):
	return (2 * a1 + d * (n - 1)) * n / 2

def RoundUp(k):
	if k % 1 == 0:
		return int(k)
	else:
		return int(k + 1)

def CheckDaySemiBrute(i):
	t = 3
	per = 1
	while True:
		if i > t:
			t = 3 * (per + 1) + 2 * per
			per += 1
		else:
			return per % 2 == 0
inp = sys.stdin.read()
inp = inp.split(sep='\n')
if inp[-1] == '':
	inp.pop(-1)
days = inp.pop(0)
for i in inp:
	if RoundUp(math.sqrt(int(i) + 1) - 1) % 2 == 1:
		print('odd')
	else:
		print('even')


import decimal, sys
T = int(sys.stdin.readline())
for _ in range(T):
	i = int(sys.stdin.readline())
	print('even' if int(decimal.Decimal(i).sqrt()) % 2 == 0 else 'odd')


import sys

def isqrt(n):
	x = n
	y = (n + 1) // 2
	while y < x:
		x = y
		y = (x + n // x) // 2
	return x
T = int(sys.stdin.readline())
for _ in range(T):
	N = int(sys.stdin.readline())
	print('even' if isqrt(N) % 2 == 0 else 'odd')


import sys

def isqrt(n):
	x = n
	y = (n + 1) // 2
	while y < x:
		x = y
		y = (x + n // x) // 2
	return x
T = int(sys.stdin.readline())
for _ in range(T):
	N = int(sys.stdin.readline())
	print('even' if isqrt(N) % 2 == 0 else 'odd')


import sys

def isqrt(n):
	x = n
	y = (n + 1) // 2
	while y < x:
		x = y
		y = (x + n // x) // 2
	return x
T = int(sys.stdin.readline())
for _ in range(T):
	N = int(sys.stdin.readline())
	print('even' if isqrt(N) % 2 == 0 else 'odd')


def intsqrt(n):
	(a, b) = (1, n // 2)
	while abs(b - a) > 1:
		(a, b) = (b, b // 2 + N // (2 * b))
	while a * a <= n:
		a += 1
	while a * a > n:
		a -= 1
	return a
T = int(input().strip())
for _ in range(T):
	N = int(input().strip())
	s = intsqrt(N)
	print(s % 2 == 0 and 'even' or 'odd')


from decimal import Decimal as d
a = []
for cas in range(int(input())):
	a.append(['even', 'odd'][int(d(input()).sqrt()) & 1])
for i in range(len(a)):
	print(a[i])


import math
n = int(input())
for i0 in range(n):
	num = int(input())
	count = 0
	m = math.sqrt(num + 1)
	if m % 2 > 1 or m % 2 == 0:
		print('odd')
	else:
		print('even')


import math
for i in range(int(input())):
	x = int(input())
	root = int(math.sqrt(x))
	if root * root > x:
		root -= 1
	print('odd' if root % 2 == 1 else 'even')


import math
t = int(input())
for i in range(t):
	j = int(input())
	d = 4 * (j + 1)
	n = math.ceil((math.sqrt(d) - 2) / 2)
	if int(n) % 2 != 0:
		print('odd')
	else:
		print('even')


from math import sqrt

def sqrt_s(v):
	t = int(sqrt(v))
	return t if t ** 2 <= v else t - 1
for _ in range(int(input())):
	print('even' if int(sqrt_s(int(input()))) % 2 == 0 else 'odd')


from decimal import *
getcontext().prec = 100
for _ in range(int(input())):
	n = int(input())
	n = Decimal(n).sqrt()
	print(('even', 'odd')[int(n) & 1])


from math import sqrt
for _ in range(int(input())):
	x = int(input())
	y = int(sqrt(x))
	if y * y > x:
		y -= 1
	print('even' if y % 2 == 0 else 'odd')


from decimal import Decimal
for _ in range(int(input())):
	n = int(input())
	print('odd' if int(Decimal(n).sqrt()) & 1 else 'even')


import sys
from math import floor, sqrt
T = int(input().strip())
for _ in range(T):
	i = int(input().strip())
	k = floor(sqrt(i))
	if k * k > i:
		k = k - 1
	print('odd' if k % 2 else 'even')


T = int(input())
for run in range(T):
	n = int(input())
	(a, b) = (1, 10 ** 9)
	while a < b:
		m = (a + b) // 2 + 1
		if m * m <= n:
			a = m
		else:
			b = m - 1
	if a % 2 == 0:
		print('even')
	else:
		print('odd')


def sqrt(n):
	low = 0.0
	high = float(n + 1)
	while high - low > 1e-06:
		mid = (low + high) / 2
		if mid * mid < n:
			low = mid
		else:
			high = mid
	return high
x = int(input().strip())
for i in range(x):
	y = int(input().strip())
	accu = int(sqrt(y))
	x = abs(accu * accu - y - 1)
	accu -= 1
	y = abs(accu * accu - y - 1)
	if min(x, y) == x:
		accu += 1
	if y / accu < 2:
		accu -= 1
	if accu % 2 == 0:
		print('even')
	else:
		print('odd')


from decimal import *
getcontext().prec = 100
for _ in range(int(input())):
	n = int(input())
	s = int(Decimal(n) ** Decimal(0.5))
	print('even' if s % 2 == 0 else 'odd')


import math
for _ in range(int(input())):
	days = int(input())
	v = math.floor(pow(days, 0.5))
	if v ** 2 > days:
		v -= 1
	print('even' if v % 2 == 0 else 'odd')


import sys
import decimal

def square_root(n, epsilon):
	c = 1
	while abs(c * c - n) > epsilon:
		c = (n + c * c) / (2 * c)
	return c

def solve(n):
	r = int(decimal.Decimal(n).sqrt())
	return r % 2

def test(n):
	mm = list()
	mm.append([0] * n)
	for i in range(1, n):
		row = list()
		for j in range(n):
			d = j // i
			row.append(d % 2)
		mm.append(row)
	sm = list()
	for j in range(n):
		u = sum((mm[i][j] for i in range(n))) % 2
		sm.append(u)
	print('sm=%s' % ''.join(map(str, sm)), file=sys.stderr)
	return 0
t = int(input())
for _ in range(t):
	n = int(input())
	if solve(n):
		print('odd')
	else:
		print('even')


from math import floor, sqrt
from decimal import *
getcontext().prec = 100
for _ in range(int(input())):
	print('odd' if int(Decimal(int(input())).sqrt()) % 2 == 1 else 'even')


from math import sqrt, floor
t = int(input())
for _ in range(t):
	n = int(input())
	num = floor(sqrt(n))
	if num * num > n:
		num -= 1
	else:
		num
	if num % 2 == 1:
		print('odd')
	else:
		print('even')


from decimal import Decimal

def main():
	T = int(input())
	for _ in range(T):
		N = int(input())
		print('odd' if N == 1 or int(Decimal(N).sqrt()) & 1 else 'even')
main()


import sys
import math

def pattern(i, n):
	s = ('0' * i + '1' * i) * (n // (2 * i) + 1)
	return s[:n + 1] + ' %d' % i

def showPattern(n):
	for i in range(1, 1 + (n + 1) // 2):
		print(pattern(i, n))

def isqrt(n):
	x = n
	y = (x + 1) // 2
	while y < x:
		x = y
		y = (x + n // x) // 2
	return x

def DirichletSumParity(x):
	return int(math.floor(math.sqrt(x))) % 2

def DirichletSumParity(x):
	return isqrt(x) % 2
T = int(input())
for i in range(T):
	i = int(input())
	p = DirichletSumParity(i)
	if p == 1:
		print('odd')
	else:
		print('even')


import math
for _ in range(int(input())):
	n = int(input())
	p = math.ceil(math.sqrt(n + 1) - 1)
	print(['even', 'odd'][p % 2])


from math import sqrt
from decimal import *
getcontext().prec = 100
num = int(input())
days = list()
for i in range(0, num):
	days.append(int(input()))
for day in days:
	tmp_0 = Decimal(day).sqrt()
	tmp_1 = int(tmp_0)
	if tmp_1 % 2 != 0:
		print('odd')
	else:
		print('even')


import os
import sys
from decimal import Decimal
for _ in range(int(input())):
	n = Decimal(input())
	p = n.sqrt()
	print(['even', 'odd'][int(p) % 2])


import sys
from math import sqrt

def dance(line):
	for _ in range(int(next(line))):
		q = int(next(line))
		n = int(sqrt(q))
		n -= n * n > q
		print('odd' if n & 1 else 'even')
dance(sys.stdin)


T = 2 ** 32
for _ in range(int(input())):
	n = int(input())
	r = 0
	t = T
	while t:
		rt = r + t
		if rt * rt <= n:
			r = rt
		t //= 2
	print('odd' if r & 1 else 'even')


import os
import sys
import math
t = int(input())
for i in range(0, t):
	n = int(input())
	s = math.sqrt(n)
	s = int(s)
	if s * s > n:
		s = s - 1
	if s % 2 == 1:
		print('odd')
	else:
		print('even')


from decimal import Decimal
for _ in range(int(input())):
	n = Decimal(input())
	p = n.sqrt()
	print(['even', 'odd'][int(p) % 2])


from decimal import Decimal
for _ in range(int(input())):
	n = Decimal(input())
	p = n.sqrt()
	print(['even', 'odd'][int(p) % 2])


from decimal import Decimal
for _ in range(int(input())):
	n = Decimal(input())
	p = n.sqrt()
	print(['even', 'odd'][int(p) % 2])


import decimal, sys
T = int(sys.stdin.readline())
for _ in range(T):
	i = int(sys.stdin.readline())
	print('even' if int(decimal.Decimal(i).sqrt()) % 2 == 0 else 'odd')


from decimal import Decimal
for _ in range(int(input())):
	n = Decimal(input())
	p = n.sqrt()
	print(['even', 'odd'][int(p) % 2])


from math import ceil
T = int(input())
for _ in range(T):
	x = int(input())
	n = int(ceil((1 + x) ** 0.5 - 1))
	if n % 2 == 1:
		print('odd')
	else:
		print('even')


from decimal import Decimal
for t in range(int(input())):
	N = Decimal(input())
	print(['even', 'odd'][int(Decimal.sqrt(N)) % 2])


t = int(input())
import math
for i in range(t):
	n = int(input())
	if (math.ceil(math.sqrt(n + 1)) - 1) % 2 == 0:
		print('even')
	else:
		print('odd')


import os
import sys
import math
t = input()
i = []
for x in range(int(t)):
	i.append(int(input()))
for x in i:
	d = 4 * (x + 1)
	n = math.ceil((math.sqrt(d) - 2) / 2)
	if int(n) % 2 == 0:
		print('even')
	else:
		print('odd')


from math import ceil, sqrt
for _ in range(int(input())):
	i = int(input())
	i += 1
	print('odd' if i == 2 or ceil(sqrt(i) - 1) % 2 == 1 else 'even')


import math
t = int(input())
i = []
for x in range(t):
	i.append(input())
for x in i:
	d = 4 * (int(x) + 1)
	n = math.ceil((math.sqrt(d) - 2) / 2)
	if int(n) % 2 == 0:
		print('even')
	else:
		print('odd')