You are given a regular N-gon with vertices at (cos(2πi / N), sin(2πi / N)), ∀ i ∈ [0,N-1]. Some of these vertices are blocked and all others are unblocked. We consider triangles with vertices at the vertices of N-gon and with at least one vertex at unblocked point. Can you find how many pairs of such triangles have equal area?

Input Format 

The first line of input contains single integer T - number of testcases. 2T lines follow. 

Each testcase has two lines. 

The first line of testcase contains a single integer N - the number of vertices in N-gon. The second line contains string S with length N. If S[j] equals '1' it means that the vertex (cos(2πj / N), sin(2πj / N)) is unblocked, and if S[j] equals '0' it means that the vertex (cos(2πj / N), sin(2πj / N)) is blocked.

Output Format 

For each testcase output single line with an answer.

Constraints 

1 <= T <= 100 

3 <= N <= 10^{4}    

There will be no more than 50 blocked vertices in each of the testcase.  

Sample Input  

1
4
1111

Sample Output  

6

Explanation

The testcase given is a square and there are 4 triangles that have the same area. So, the number of pairs are 4C2 = 6.

Difficulty: HARD
Tags: Geometry

Starter code:

Solutions:
import math
from collections import defaultdict
err = pow(10, -12)
t = int(input())
for _ in range(t):
	n = int(input())
	minang = math.pi / n
	s = input()
	blocked = []
	for i in range(n):
		if s[i] == '0':
			blocked.append(i)
	global numdiff
	numdiff = 0
	numars = 0
	for a in range(1, n // 3 + 1):
		numars += (n - a) // 2 + 1 - a
	diffar = set()
	triangarea = defaultdict(lambda : 6 * n)
	triangrep = dict()
	for a in range((n - 1) % 2 + 1, n, 2):
		b = (n - a) // 2
		c = b
		[alta, altb, altc] = sorted([a, b, c])
		area = round(math.sin(a * minang) * math.sin(b * minang) * math.sin(c * minang), 12)
		triangarea[alta, altb, altc] = 3 * n
		diffar.add((alta, altb, altc))
	if n % 4 == 0:
		for a in range(1, (n + 4) // 8):
			b = n // 2 - 2 * a
			c = n // 2 + a
			[diffa, diffb, diffc] = sorted([a, b, c])
			area = round(math.sin(a * minang) * math.sin(b * minang) * math.sin(c * minang), 12)
			triangarea[diffa, diffb, diffc] = 12 * n
			numars -= 1
			[alta, altb, altc] = sorted([n // 4 - a, 3 * (n // 4) - a, 2 * a])
			triangrep[alta, altb, altc] = (diffa, diffb, diffc)
			diffar.add((diffa, diffb, diffc))
	if n % 30 == 0:
		area = round(math.sin(4 * math.pi / 30) * math.sin(12 * math.pi / 30) * math.sin(14 * math.pi / 30), 12)
		triangarea[4 * n // 30, 12 * n // 30, 14 * n // 30] = 12 * n
		numars -= 1
		triangrep[6 * n // 30, 7 * n // 30, 17 * n // 30] = (4 * n // 30, 12 * n // 30, 14 * n // 30)
		diffar.add((4 * n // 30, 12 * n // 30, 14 * n // 30))
		area = round(math.sin(1 * math.pi / 30) * math.sin(11 * math.pi / 30) * math.sin(18 * math.pi / 30), 12)
		triangarea[1 * n // 30, 11 * n // 30, 18 * n // 30] = 12 * n
		numars -= 1
		triangrep[2 * n // 30, 6 * n // 30, 22 * n // 30] = (1 * n // 30, 11 * n // 30, 18 * n // 30)
		diffar.add((1 * n // 30, 11 * n // 30, 18 * n // 30))
	if n % 24 == 0:
		area = 0.25
		triangarea[3 * n // 24, 6 * n // 24, 15 * n // 24] = 12 * n
		numars -= 1
		triangrep[2 * n // 24, 10 * n // 24, 12 * n // 24] = (3 * n // 24, 6 * n // 24, 15 * n // 24)
		diffar.add((3 * n // 24, 6 * n // 24, 15 * n // 24))
	if n % 12 == 0:
		area = round(math.sqrt(3) / 8, 12)
		triangarea[1 * n // 12, 4 * n // 12, 7 * n // 12] = 9 * n
		triangarea.pop((2 * n // 12, 2 * n // 12, 8 * n // 12))
		diffar.remove((2 * n // 12, 2 * n // 12, 8 * n // 12))
		diffar.add((1 * n // 12, 4 * n // 12, 7 * n // 12))
	if n % 3 == 0:
		area = round(math.sqrt(27) / 8, 12)
		triangarea[n // 3, n // 3, n // 3] = n
		diffar.add((n // 3, n // 3, n // 3))
	b = len(blocked)
	for i in range(b):
		x = blocked[i]
		for j in range(i):
			y = blocked[j]
			for k in range(j):
				z = blocked[k]
				a = (x - y) % n
				b = (y - z) % n
				c = (z - x) % n
				[a, b, c] = sorted([a, b, c])
				area = round(math.sin(a * minang) * math.sin(b * minang) * math.sin(c * minang), 12)
				try:
					coords = triangrep[a, b, c]
				except:
					coords = (a, b, c)
				triangarea[coords] -= 3
				diffar.add(coords)
	toprint = sum(map(lambda x: x // 3 * (x // 3 - 1) // 2, triangarea.values()))
	toprint += (numars - len(diffar)) * (n * (2 * n - 1))
	print(toprint)


import math
from collections import defaultdict
err = pow(10, -12)
t = int(input())
for _ in range(t):
	n = int(input())
	minang = math.pi / n
	s = input()
	blocked = []
	for i in range(n):
		if s[i] == '0':
			blocked.append(i)
	global numdiff
	numdiff = 0
	numars = 0
	for a in range(1, n // 3 + 1):
		numars += (n - a) // 2 + 1 - a
	diffar = set()
	triangarea = defaultdict(lambda : 6 * n)
	triangrep = dict()
	for a in range((n - 1) % 2 + 1, n, 2):
		b = (n - a) // 2
		c = b
		[alta, altb, altc] = sorted([a, b, c])
		area = round(math.sin(a * minang) * math.sin(b * minang) * math.sin(c * minang), 12)
		triangarea[alta, altb, altc] = 3 * n
		diffar.add((alta, altb, altc))
	if n % 4 == 0:
		for a in range(1, (n + 4) // 8):
			b = n // 2 - 2 * a
			c = n // 2 + a
			[diffa, diffb, diffc] = sorted([a, b, c])
			area = round(math.sin(a * minang) * math.sin(b * minang) * math.sin(c * minang), 12)
			triangarea[diffa, diffb, diffc] = 12 * n
			numars -= 1
			[alta, altb, altc] = sorted([n // 4 - a, 3 * (n // 4) - a, 2 * a])
			triangrep[alta, altb, altc] = (diffa, diffb, diffc)
			diffar.add((diffa, diffb, diffc))
	if n % 30 == 0:
		area = round(math.sin(4 * math.pi / 30) * math.sin(12 * math.pi / 30) * math.sin(14 * math.pi / 30), 12)
		triangarea[4 * n // 30, 12 * n // 30, 14 * n // 30] = 12 * n
		numars -= 1
		triangrep[6 * n // 30, 7 * n // 30, 17 * n // 30] = (4 * n // 30, 12 * n // 30, 14 * n // 30)
		diffar.add((4 * n // 30, 12 * n // 30, 14 * n // 30))
		area = round(math.sin(1 * math.pi / 30) * math.sin(11 * math.pi / 30) * math.sin(18 * math.pi / 30), 12)
		triangarea[1 * n // 30, 11 * n // 30, 18 * n // 30] = 12 * n
		numars -= 1
		triangrep[2 * n // 30, 6 * n // 30, 22 * n // 30] = (1 * n // 30, 11 * n // 30, 18 * n // 30)
		diffar.add((1 * n // 30, 11 * n // 30, 18 * n // 30))
	if n % 24 == 0:
		area = 0.25
		triangarea[3 * n // 24, 6 * n // 24, 15 * n // 24] = 12 * n
		numars -= 1
		triangrep[2 * n // 24, 10 * n // 24, 12 * n // 24] = (3 * n // 24, 6 * n // 24, 15 * n // 24)
		diffar.add((3 * n // 24, 6 * n // 24, 15 * n // 24))
	if n % 12 == 0:
		area = round(math.sqrt(3) / 8, 12)
		triangarea[1 * n // 12, 4 * n // 12, 7 * n // 12] = 9 * n
		triangarea.pop((2 * n // 12, 2 * n // 12, 8 * n // 12))
		diffar.remove((2 * n // 12, 2 * n // 12, 8 * n // 12))
		diffar.add((1 * n // 12, 4 * n // 12, 7 * n // 12))
	if n % 3 == 0:
		area = round(math.sqrt(27) / 8, 12)
		triangarea[n // 3, n // 3, n // 3] = n
		diffar.add((n // 3, n // 3, n // 3))
	b = len(blocked)
	for i in range(b):
		x = blocked[i]
		for j in range(i):
			y = blocked[j]
			for k in range(j):
				z = blocked[k]
				a = (x - y) % n
				b = (y - z) % n
				c = (z - x) % n
				[a, b, c] = sorted([a, b, c])
				area = round(math.sin(a * minang) * math.sin(b * minang) * math.sin(c * minang), 12)
				try:
					coords = triangrep[a, b, c]
				except:
					coords = (a, b, c)
				triangarea[coords] -= 3
				diffar.add(coords)
	toprint = sum(map(lambda x: x // 3 * (x // 3 - 1) // 2, triangarea.values()))
	toprint += (numars - len(diffar)) * (n * (2 * n - 1))
	print(toprint)


import math
from collections import defaultdict
err = pow(10, -12)
t = int(input())
for _ in range(t):
	n = int(input())
	minang = math.pi / n
	s = input()
	blocked = []
	for i in range(n):
		if s[i] == '0':
			blocked.append(i)
	global numdiff
	numdiff = 0
	numars = 0
	for a in range(1, n // 3 + 1):
		numars += (n - a) // 2 + 1 - a
	diffar = set()
	triangarea = defaultdict(lambda : 6 * n)
	triangrep = dict()
	for a in range((n - 1) % 2 + 1, n, 2):
		b = (n - a) // 2
		c = b
		[alta, altb, altc] = sorted([a, b, c])
		area = round(math.sin(a * minang) * math.sin(b * minang) * math.sin(c * minang), 12)
		triangarea[alta, altb, altc] = 3 * n
		diffar.add((alta, altb, altc))
	if n % 4 == 0:
		for a in range(1, (n + 4) // 8):
			b = n // 2 - 2 * a
			c = n // 2 + a
			[diffa, diffb, diffc] = sorted([a, b, c])
			area = round(math.sin(a * minang) * math.sin(b * minang) * math.sin(c * minang), 12)
			triangarea[diffa, diffb, diffc] = 12 * n
			numars -= 1
			[alta, altb, altc] = sorted([n // 4 - a, 3 * (n // 4) - a, 2 * a])
			triangrep[alta, altb, altc] = (diffa, diffb, diffc)
			diffar.add((diffa, diffb, diffc))
	if n % 30 == 0:
		area = round(math.sin(4 * math.pi / 30) * math.sin(12 * math.pi / 30) * math.sin(14 * math.pi / 30), 12)
		triangarea[4 * n // 30, 12 * n // 30, 14 * n // 30] = 12 * n
		numars -= 1
		triangrep[6 * n // 30, 7 * n // 30, 17 * n // 30] = (4 * n // 30, 12 * n // 30, 14 * n // 30)
		diffar.add((4 * n // 30, 12 * n // 30, 14 * n // 30))
		area = round(math.sin(1 * math.pi / 30) * math.sin(11 * math.pi / 30) * math.sin(18 * math.pi / 30), 12)
		triangarea[1 * n // 30, 11 * n // 30, 18 * n // 30] = 12 * n
		numars -= 1
		triangrep[2 * n // 30, 6 * n // 30, 22 * n // 30] = (1 * n // 30, 11 * n // 30, 18 * n // 30)
		diffar.add((1 * n // 30, 11 * n // 30, 18 * n // 30))
	if n % 24 == 0:
		area = 0.25
		triangarea[3 * n // 24, 6 * n // 24, 15 * n // 24] = 12 * n
		numars -= 1
		triangrep[2 * n // 24, 10 * n // 24, 12 * n // 24] = (3 * n // 24, 6 * n // 24, 15 * n // 24)
		diffar.add((3 * n // 24, 6 * n // 24, 15 * n // 24))
	if n % 12 == 0:
		area = round(math.sqrt(3) / 8, 12)
		triangarea[1 * n // 12, 4 * n // 12, 7 * n // 12] = 9 * n
		triangarea.pop((2 * n // 12, 2 * n // 12, 8 * n // 12))
		diffar.remove((2 * n // 12, 2 * n // 12, 8 * n // 12))
		diffar.add((1 * n // 12, 4 * n // 12, 7 * n // 12))
	if n % 3 == 0:
		area = round(math.sqrt(27) / 8, 12)
		triangarea[n // 3, n // 3, n // 3] = n
		diffar.add((n // 3, n // 3, n // 3))
	b = len(blocked)
	for i in range(b):
		x = blocked[i]
		for j in range(i):
			y = blocked[j]
			for k in range(j):
				z = blocked[k]
				a = (x - y) % n
				b = (y - z) % n
				c = (z - x) % n
				[a, b, c] = sorted([a, b, c])
				area = round(math.sin(a * minang) * math.sin(b * minang) * math.sin(c * minang), 12)
				try:
					coords = triangrep[a, b, c]
				except:
					coords = (a, b, c)
				triangarea[coords] -= 3
				diffar.add(coords)
	toprint = sum(map(lambda x: x // 3 * (x // 3 - 1) // 2, triangarea.values()))
	toprint += (numars - len(diffar)) * (n * (2 * n - 1))
	print(toprint)


import math
from collections import defaultdict
err = pow(10, -12)
t = int(input())
for _ in range(t):
	n = int(input())
	minang = math.pi / n
	s = input()
	blocked = []
	for i in range(n):
		if s[i] == '0':
			blocked.append(i)
	global numdiff
	numdiff = 0
	numars = 0
	for a in range(1, n // 3 + 1):
		numars += (n - a) // 2 + 1 - a
	diffar = set()
	triangarea = defaultdict(lambda : 6 * n)
	triangrep = dict()
	for a in range((n - 1) % 2 + 1, n, 2):
		b = (n - a) // 2
		c = b
		[alta, altb, altc] = sorted([a, b, c])
		area = round(math.sin(a * minang) * math.sin(b * minang) * math.sin(c * minang), 12)
		triangarea[alta, altb, altc] = 3 * n
		diffar.add((alta, altb, altc))
	if n % 4 == 0:
		for a in range(1, (n + 4) // 8):
			b = n // 2 - 2 * a
			c = n // 2 + a
			[diffa, diffb, diffc] = sorted([a, b, c])
			area = round(math.sin(a * minang) * math.sin(b * minang) * math.sin(c * minang), 12)
			triangarea[diffa, diffb, diffc] = 12 * n
			numars -= 1
			[alta, altb, altc] = sorted([n // 4 - a, 3 * (n // 4) - a, 2 * a])
			triangrep[alta, altb, altc] = (diffa, diffb, diffc)
			diffar.add((diffa, diffb, diffc))
	if n % 30 == 0:
		area = round(math.sin(4 * math.pi / 30) * math.sin(12 * math.pi / 30) * math.sin(14 * math.pi / 30), 12)
		triangarea[4 * n // 30, 12 * n // 30, 14 * n // 30] = 12 * n
		numars -= 1
		triangrep[6 * n // 30, 7 * n // 30, 17 * n // 30] = (4 * n // 30, 12 * n // 30, 14 * n // 30)
		diffar.add((4 * n // 30, 12 * n // 30, 14 * n // 30))
		area = round(math.sin(1 * math.pi / 30) * math.sin(11 * math.pi / 30) * math.sin(18 * math.pi / 30), 12)
		triangarea[1 * n // 30, 11 * n // 30, 18 * n // 30] = 12 * n
		numars -= 1
		triangrep[2 * n // 30, 6 * n // 30, 22 * n // 30] = (1 * n // 30, 11 * n // 30, 18 * n // 30)
		diffar.add((1 * n // 30, 11 * n // 30, 18 * n // 30))
	if n % 24 == 0:
		area = 0.25
		triangarea[3 * n // 24, 6 * n // 24, 15 * n // 24] = 12 * n
		numars -= 1
		triangrep[2 * n // 24, 10 * n // 24, 12 * n // 24] = (3 * n // 24, 6 * n // 24, 15 * n // 24)
		diffar.add((3 * n // 24, 6 * n // 24, 15 * n // 24))
	if n % 12 == 0:
		area = round(math.sqrt(3) / 8, 12)
		triangarea[1 * n // 12, 4 * n // 12, 7 * n // 12] = 9 * n
		triangarea.pop((2 * n // 12, 2 * n // 12, 8 * n // 12))
		diffar.remove((2 * n // 12, 2 * n // 12, 8 * n // 12))
		diffar.add((1 * n // 12, 4 * n // 12, 7 * n // 12))
	if n % 3 == 0:
		area = round(math.sqrt(27) / 8, 12)
		triangarea[n // 3, n // 3, n // 3] = n
		diffar.add((n // 3, n // 3, n // 3))
	b = len(blocked)
	for i in range(b):
		x = blocked[i]
		for j in range(i):
			y = blocked[j]
			for k in range(j):
				z = blocked[k]
				a = (x - y) % n
				b = (y - z) % n
				c = (z - x) % n
				[a, b, c] = sorted([a, b, c])
				area = round(math.sin(a * minang) * math.sin(b * minang) * math.sin(c * minang), 12)
				try:
					coords = triangrep[a, b, c]
				except:
					coords = (a, b, c)
				triangarea[coords] -= 3
				diffar.add(coords)
	toprint = sum(map(lambda x: x // 3 * (x // 3 - 1) // 2, triangarea.values()))
	toprint += (numars - len(diffar)) * (n * (2 * n - 1))
	print(toprint)


import math
from collections import defaultdict
err = pow(10, -12)
t = int(input())
for _ in range(t):
	n = int(input())
	minang = math.pi / n
	s = input()
	blocked = []
	for i in range(n):
		if s[i] == '0':
			blocked.append(i)
	global numdiff
	numdiff = 0
	numars = 0
	for a in range(1, n // 3 + 1):
		numars += (n - a) // 2 + 1 - a
	diffar = set()
	triangarea = defaultdict(lambda : 6 * n)
	triangrep = dict()
	for a in range((n - 1) % 2 + 1, n, 2):
		b = (n - a) // 2
		c = b
		[alta, altb, altc] = sorted([a, b, c])
		area = round(math.sin(a * minang) * math.sin(b * minang) * math.sin(c * minang), 12)
		triangarea[alta, altb, altc] = 3 * n
		diffar.add((alta, altb, altc))
	if n % 4 == 0:
		for a in range(1, (n + 4) // 8):
			b = n // 2 - 2 * a
			c = n // 2 + a
			[diffa, diffb, diffc] = sorted([a, b, c])
			area = round(math.sin(a * minang) * math.sin(b * minang) * math.sin(c * minang), 12)
			triangarea[diffa, diffb, diffc] = 12 * n
			numars -= 1
			[alta, altb, altc] = sorted([n // 4 - a, 3 * (n // 4) - a, 2 * a])
			triangrep[alta, altb, altc] = (diffa, diffb, diffc)
			diffar.add((diffa, diffb, diffc))
	if n % 30 == 0:
		area = round(math.sin(4 * math.pi / 30) * math.sin(12 * math.pi / 30) * math.sin(14 * math.pi / 30), 12)
		triangarea[4 * n // 30, 12 * n // 30, 14 * n // 30] = 12 * n
		numars -= 1
		triangrep[6 * n // 30, 7 * n // 30, 17 * n // 30] = (4 * n // 30, 12 * n // 30, 14 * n // 30)
		diffar.add((4 * n // 30, 12 * n // 30, 14 * n // 30))
		area = round(math.sin(1 * math.pi / 30) * math.sin(11 * math.pi / 30) * math.sin(18 * math.pi / 30), 12)
		triangarea[1 * n // 30, 11 * n // 30, 18 * n // 30] = 12 * n
		numars -= 1
		triangrep[2 * n // 30, 6 * n // 30, 22 * n // 30] = (1 * n // 30, 11 * n // 30, 18 * n // 30)
		diffar.add((1 * n // 30, 11 * n // 30, 18 * n // 30))
	if n % 24 == 0:
		area = 0.25
		triangarea[3 * n // 24, 6 * n // 24, 15 * n // 24] = 12 * n
		numars -= 1
		triangrep[2 * n // 24, 10 * n // 24, 12 * n // 24] = (3 * n // 24, 6 * n // 24, 15 * n // 24)
		diffar.add((3 * n // 24, 6 * n // 24, 15 * n // 24))
	if n % 12 == 0:
		area = round(math.sqrt(3) / 8, 12)
		triangarea[1 * n // 12, 4 * n // 12, 7 * n // 12] = 9 * n
		triangarea.pop((2 * n // 12, 2 * n // 12, 8 * n // 12))
		diffar.remove((2 * n // 12, 2 * n // 12, 8 * n // 12))
		diffar.add((1 * n // 12, 4 * n // 12, 7 * n // 12))
	if n % 3 == 0:
		area = round(math.sqrt(27) / 8, 12)
		triangarea[n // 3, n // 3, n // 3] = n
		diffar.add((n // 3, n // 3, n // 3))
	b = len(blocked)
	for i in range(b):
		x = blocked[i]
		for j in range(i):
			y = blocked[j]
			for k in range(j):
				z = blocked[k]
				a = (x - y) % n
				b = (y - z) % n
				c = (z - x) % n
				[a, b, c] = sorted([a, b, c])
				area = round(math.sin(a * minang) * math.sin(b * minang) * math.sin(c * minang), 12)
				try:
					coords = triangrep[a, b, c]
				except:
					coords = (a, b, c)
				triangarea[coords] -= 3
				diffar.add(coords)
	toprint = sum(map(lambda x: x // 3 * (x // 3 - 1) // 2, triangarea.values()))
	toprint += (numars - len(diffar)) * (n * (2 * n - 1))
	print(toprint)


import math
from collections import defaultdict
err = pow(10, -12)
t = int(input())
for _ in range(t):
	n = int(input())
	minang = math.pi / n
	s = input()
	blocked = []
	for i in range(n):
		if s[i] == '0':
			blocked.append(i)
	global numdiff
	numdiff = 0
	numars = 0
	for a in range(1, n // 3 + 1):
		numars += (n - a) // 2 + 1 - a
	diffar = set()
	triangarea = defaultdict(lambda : 6 * n)
	triangrep = dict()
	for a in range((n - 1) % 2 + 1, n, 2):
		b = (n - a) // 2
		c = b
		[alta, altb, altc] = sorted([a, b, c])
		area = round(math.sin(a * minang) * math.sin(b * minang) * math.sin(c * minang), 12)
		triangarea[alta, altb, altc] = 3 * n
		diffar.add((alta, altb, altc))
	if n % 4 == 0:
		for a in range(1, (n + 4) // 8):
			b = n // 2 - 2 * a
			c = n // 2 + a
			[diffa, diffb, diffc] = sorted([a, b, c])
			area = round(math.sin(a * minang) * math.sin(b * minang) * math.sin(c * minang), 12)
			triangarea[diffa, diffb, diffc] = 12 * n
			numars -= 1
			[alta, altb, altc] = sorted([n // 4 - a, 3 * (n // 4) - a, 2 * a])
			triangrep[alta, altb, altc] = (diffa, diffb, diffc)
			diffar.add((diffa, diffb, diffc))
	if n % 30 == 0:
		area = round(math.sin(4 * math.pi / 30) * math.sin(12 * math.pi / 30) * math.sin(14 * math.pi / 30), 12)
		triangarea[4 * n // 30, 12 * n // 30, 14 * n // 30] = 12 * n
		numars -= 1
		triangrep[6 * n // 30, 7 * n // 30, 17 * n // 30] = (4 * n // 30, 12 * n // 30, 14 * n // 30)
		diffar.add((4 * n // 30, 12 * n // 30, 14 * n // 30))
		area = round(math.sin(1 * math.pi / 30) * math.sin(11 * math.pi / 30) * math.sin(18 * math.pi / 30), 12)
		triangarea[1 * n // 30, 11 * n // 30, 18 * n // 30] = 12 * n
		numars -= 1
		triangrep[2 * n // 30, 6 * n // 30, 22 * n // 30] = (1 * n // 30, 11 * n // 30, 18 * n // 30)
		diffar.add((1 * n // 30, 11 * n // 30, 18 * n // 30))
	if n % 24 == 0:
		area = 0.25
		triangarea[3 * n // 24, 6 * n // 24, 15 * n // 24] = 12 * n
		numars -= 1
		triangrep[2 * n // 24, 10 * n // 24, 12 * n // 24] = (3 * n // 24, 6 * n // 24, 15 * n // 24)
		diffar.add((3 * n // 24, 6 * n // 24, 15 * n // 24))
	if n % 12 == 0:
		area = round(math.sqrt(3) / 8, 12)
		triangarea[1 * n // 12, 4 * n // 12, 7 * n // 12] = 9 * n
		triangarea.pop((2 * n // 12, 2 * n // 12, 8 * n // 12))
		diffar.remove((2 * n // 12, 2 * n // 12, 8 * n // 12))
		diffar.add((1 * n // 12, 4 * n // 12, 7 * n // 12))
	if n % 3 == 0:
		area = round(math.sqrt(27) / 8, 12)
		triangarea[n // 3, n // 3, n // 3] = n
		diffar.add((n // 3, n // 3, n // 3))
	b = len(blocked)
	for i in range(b):
		x = blocked[i]
		for j in range(i):
			y = blocked[j]
			for k in range(j):
				z = blocked[k]
				a = (x - y) % n
				b = (y - z) % n
				c = (z - x) % n
				[a, b, c] = sorted([a, b, c])
				area = round(math.sin(a * minang) * math.sin(b * minang) * math.sin(c * minang), 12)
				try:
					coords = triangrep[a, b, c]
				except:
					coords = (a, b, c)
				triangarea[coords] -= 3
				diffar.add(coords)
	toprint = sum(map(lambda x: x // 3 * (x // 3 - 1) // 2, triangarea.values()))
	toprint += (numars - len(diffar)) * (n * (2 * n - 1))
	print(toprint)