You have a rectangular chocolate bar consisting of n × m single squares. You want to eat exactly k squares, so you may need to break the chocolate bar. 

In one move you can break any single rectangular piece of chocolate in two rectangular pieces. You can break only by lines between squares: horizontally or vertically. The cost of breaking is equal to square of the break length.

For example, if you have a chocolate bar consisting of 2 × 3 unit squares then you can break it horizontally and get two 1 × 3 pieces (the cost of such breaking is 3^2 = 9), or you can break it vertically in two ways and get two pieces: 2 × 1 and 2 × 2 (the cost of such breaking is 2^2 = 4).

For several given values n, m and k find the minimum total cost of breaking. You can eat exactly k squares of chocolate if after all operations of breaking there is a set of rectangular pieces of chocolate with the total size equal to k squares. The remaining n·m - k squares are not necessarily form a single rectangular piece.


-----Input-----

The first line of the input contains a single integer t (1 ≤ t ≤ 40910) — the number of values n, m and k to process.

Each of the next t lines contains three integers n, m and k (1 ≤ n, m ≤ 30, 1 ≤ k ≤ min(n·m, 50)) — the dimensions of the chocolate bar and the number of squares you want to eat respectively.


-----Output-----

For each n, m and k print the minimum total cost needed to break the chocolate bar, in order to make it possible to eat exactly k squares.


-----Examples-----
Input
4
2 2 1
2 2 3
2 2 2
2 2 4

Output
5
5
4
0



-----Note-----

In the first query of the sample one needs to perform two breaks:  to split 2 × 2 bar into two pieces of 2 × 1 (cost is 2^2 = 4),  to split the resulting 2 × 1 into two 1 × 1 pieces (cost is 1^2 = 1). 

In the second query of the sample one wants to eat 3 unit squares. One can use exactly the same strategy as in the first query of the sample.

Difficulty: HARD
Tags: Dynamic programming

Complete search
Skill types: Dynamic programming

Complete search

Starter code:

Solutions:
d = [[[0 for i in range(51)] for j in range(31)] for k in range(31)]

def rec(n, m, k):
	nonlocal d
	if n * m == k or k == 0:
		return 0
	if d[n][m][k] > 0:
		return d[n][m][k]
	if n * m < k:
		return 10 ** 10
	cost = 10 ** 10
	for i in range(1, n // 2 + 1):
		for j in range(k + 1):
			cost = min(cost, m * m + rec(n - i, m, k - j) + rec(i, m, j))
	for i in range(1, m // 2 + 1):
		for j in range(k + 1):
			cost = min(cost, n * n + rec(n, m - i, k - j) + rec(n, i, j))
	d[n][m][k] = cost
	return cost
p = []
t = int(input())
for i in range(t):
	(n, m, k) = list(map(int, input().split()))
	p.append(rec(n, m, k))
print('\n'.join((str(x) for x in p)))


import sys
input = sys.stdin.readline
d = {}
testnumber = int(input())

def calc(n, m, k):
	if k <= 0 or k == m * n:
		return 0
	if k > n * m:
		return 1000000000
	nonlocal d
	if n < m:
		(n, m) = (m, n)
	if k > m * n - m:
		return m * m + 1
	if k < m:
		return m * m + 1
	if k % m == 0:
		return m * m
	if (n, m, k) in d:
		return d[n, m, k]
	d[n, m, k] = min(calc2(n, m, k), calc2(m, n, k))
	return d[n, m, k]

def calc2(n, m, k):
	m2 = m * m
	ans = m2 * 2 + 1
	for i in range(1, n):
		if i * m >= k:
			ans = min(ans, m2 + calc(m, i, k))
		else:
			ans = min(ans, m2 + calc(m, n - i, k - i * m))
	return ans
for ntest in range(testnumber):
	(n, m, k) = map(int, input().split())
	if k == n * m:
		print(0)
		continue
	print(calc(n, m, k))


t = int(input())
dp = [[[0 for i in range(51)] for j in range(31)] for k in range(31)]

def cost(n, m, k):
	if dp[n][m][k] or k == 0 or n * m == k:
		return dp[n][m][k]
	c = 10 ** 9
	for i in range(1, n // 2 + 1):
		for j in range(k + 1):
			c = min(c, cost(n - i, m, k - j) + cost(i, m, j) + m * m)
	for i in range(1, m // 2 + 1):
		for j in range(k + 1):
			c = min(c, cost(n, m - i, k - j) + cost(n, i, j) + n * n)
	dp[n][m][k] = c
	return c
for _ in range(t):
	(n, m, k) = map(int, input().split())
	print(cost(n, m, k))


import sys
from collections import deque
from queue import PriorityQueue
from math import gcd
from math import log
from math import ceil
from math import pi
input_ = lambda : sys.stdin.readline().strip('\r\n')
ii = lambda : int(input_())
il = lambda : list(map(int, input_().split()))
ilf = lambda : list(map(float, input_().split()))
ip = lambda : input_()
fi = lambda : float(input_())
ap = lambda ab, bc, cd: ab[bc].append(cd)
li = lambda : list(input_())
pr = lambda x: print(x)
prinT = lambda x: print(x)
f = lambda : sys.stdout.flush()
mod = 10 ** 9 + 7
dp = [[[0 for k in range(51)] for j in range(31)] for i in range(31)]

def calc(n, m, k):
	if dp[n][m][k] != 0 or n * m == k or k == 0:
		return dp[n][m][k]
	ans = 10 ** 9
	for i in range(1, n // 2 + 1):
		for j in range(k + 1):
			if i * m >= k - j and (n - i) * m >= j:
				ans = min(ans, m * m + calc(i, m, k - j) + calc(n - i, m, j))
	for i in range(1, m // 2 + 1):
		for j in range(k + 1):
			if i * n >= k - j and (m - i) * n >= j:
				ans = min(ans, n * n + calc(n, i, k - j) + calc(n, m - i, j))
	dp[n][m][k] = ans
	return ans
for _ in range(ii()):
	(n, m, k) = il()
	print(calc(n, m, k))


t = int(input())
d = []
for i in range(31):
	dd = []
	for j in range(31):
		dd.append([0] * 51)
	d.append(dd)
d[1][1][1] = 0
for i in range(1, 31):
	for j in range(1, 31):
		for k in range(1, min(i * j, 50) + 1):
			if k > i * j // 2:
				d[i][j][k] = d[i][j][i * j - k]
			elif i > j:
				d[i][j][k] = d[j][i][k]
			elif (i, j) != (1, 1):
				k = min(k, i * j - k)
				kk = i * j - k
				jj = i ** 2 * j * j ** 2 * i
				for l in range(1, i):
					if k <= l * j:
						jj = min(jj, d[l][j][k] + j ** 2)
					else:
						k1 = k - l * j
						jj = min(jj, d[i - l][j][k1] + j ** 2)
					if kk <= l * j:
						if kk <= 50:
							jj = min(jj, d[l][j][kk] + j ** 2)
					else:
						kk1 = kk - l * j
						if kk1 <= 50:
							jj = min(jj, d[i - l][j][kk1] + j ** 2)
				for l in range(1, j):
					if k <= l * i:
						jj = min(jj, d[i][l][k] + i ** 2)
					else:
						k1 = k - l * i
						jj = min(jj, d[i][j - l][k1] + i ** 2)
					if kk <= l * i:
						if kk <= 50:
							jj = min(jj, d[i][l][kk] + i ** 2)
					else:
						kk1 = kk - l * i
						if kk1 <= 50:
							jj = min(jj, d[i][j - l][kk1] + i ** 2)
				d[i][j][k] = jj
for i in range(t):
	(n, m, k) = list(map(int, input().split()))
	jj = d[n][m][k]
	print(jj)


mem = [[[0 for i in range(51)] for j in range(31)] for k in range(31)]

def f(n, m, k):
	if mem[n][m][k]:
		return mem[n][m][k]
	if n * m == k or k == 0:
		return 0
	cost = 10 ** 9
	for x in range(1, n // 2 + 1):
		for z in range(k + 1):
			cost = min(cost, m * m + f(n - x, m, k - z) + f(x, m, z))
	for y in range(1, m // 2 + 1):
		for z in range(k + 1):
			cost = min(cost, n * n + f(n, m - y, k - z) + f(n, y, z))
	mem[n][m][k] = cost
	return cost
t = int(input())
for i in range(t):
	(n, m, k) = list(map(int, input().split()))
	print(f(n, m, k))


dp = {}

def getDP(n, m, k):
	if (n, m, k) in dp:
		return dp[n, m, k]
	elif (m, n, k) in dp:
		return dp[m, n, k]
	return None

def solve(n, m, k):
	if n == 2 and m == 3 and (k == 5):
		h = 5
	if k == m * n or k == 0:
		dp[n, m, k] = 0
	elif k % min(n, m) == 0:
		dp[n, m, k] = min(n, m) ** 2
	elif k == 1:
		dp[n, m, k] = min(n, m) ** 2 + 1
	elif getDP(n, m, k) is not None:
		return getDP(n, m, k)
	else:
		bestAns = float('inf')
		for i in range(1, n):
			if k <= i * m:
				bestAns = min(bestAns, getDP(i, m, k) + m ** 2)
			else:
				bestAns = min(bestAns, getDP(n - i, m, k - i * m) + m ** 2)
		for i in range(1, m):
			if k <= i * n:
				bestAns = min(bestAns, getDP(i, n, k) + n ** 2)
			else:
				bestAns = min(bestAns, getDP(m - i, n, k - i * n) + n ** 2)
		dp[n, m, k] = bestAns
for i in range(1, 31):
	for j in range(1, 31):
		for k in range(min(i * j, 50) + 1):
			solve(i, j, k)
toPrint = []
t = int(input())
for i in range(t):
	(n, m, k) = [int(x) for x in input().split(' ')]
	toPrint.append(getDP(n, m, k))
for x in toPrint:
	print(x)


d = [[[0 for i in range(51)] for j in range(31)] for k in range(31)]
for i in range(31):
	d.append([])
	for j in range(31):
		d[i].append([])
		for k in range(50):
			d[i][j].append(0)

def rec(n, m, k):
	global d
	if n * m == k or k == 0:
		return 0
	if d[n][m][k] > 0:
		return d[n][m][k]
	if n * m < k:
		return 10 ** 10
	cost = 10 ** 10
	for i in range(1, n // 2 + 1):
		for j in range(k + 1):
			cost = min(cost, m * m + rec(n - i, m, k - j) + rec(i, m, j))
	for i in range(1, m // 2 + 1):
		for j in range(k + 1):
			cost = min(cost, n * n + rec(n, m - i, k - j) + rec(n, i, j))
	d[n][m][k] = cost
	return cost
for i in range(int(input())):
	(a, b, c) = map(int, input().split())
	print(rec(a, b, c))


d = [0] * 49011

def g(n, m, k):
	t = 1000000000.0
	for i in range(1, m // 2 + 1):
		for j in range(k + 1):
			t = min(t, f(n, m - i, k - j) + f(n, i, j))
	return n * n + t

def f(n, m, k):
	if n > m:
		(n, m) = (m, n)
	k = min(k, n * m - k)
	if k == 0:
		return 0
	if k < 0:
		return 1000000000.0
	q = n + 31 * m + 961 * k
	if d[q] == 0:
		d[q] = min(g(n, m, k), g(m, n, k))
	return d[q]
for q in range(int(input())):
	(n, m, k) = map(int, input().split())
	print(f(n, m, k))


d = [[[0 for i in range(51)] for j in range(31)] for g in range(31)]

def rec(n, m, k):
	global d
	if k == 0 or n * m == k:
		return 0
	if d[n][m][k] > 0:
		return d[n][m][k]
	if n * m < k:
		return 10 ** 10
	cost = 10 ** 10
	for i in range(1, n // 2 + 1):
		for j in range(k + 1):
			cost = min(cost, m ** 2 + rec(i, m, j) + rec(n - i, m, k - j))
	for i in range(1, m // 2 + 1):
		for j in range(0, k + 1):
			cost = min(cost, n ** 2 + rec(n, i, j) + rec(n, m - i, k - j))
	d[n][m][k] = cost
	return cost
t = int(input())
a = []
for c in range(t):
	(n, m, k) = map(int, input().split())
	a.append(rec(n, m, k))
print('\n'.join((str(x) for x in a)))


import sys
mem = []
for i in range(32):
	mem.append([[-1] * 52 for u in range(32)])

def solve(x, y, z):
	if x > y:
		mem[x][y][z] = solve(y, x, z)
		return mem[x][y][z]
	if x * y == z or z == 0:
		mem[x][y][z] = 0
		return 0
	if x * y < z:
		mem[x][y][z] = -2
		return -2
	res = -2
	for i in range(1, x // 2 + 1):
		for eaten in range(z + 1):
			t1 = mem[i][y][eaten] if mem[i][y][eaten] != -1 else solve(i, y, eaten)
			if t1 == -2:
				continue
			t2 = mem[x - i][y][z - eaten] if mem[x - i][y][z - eaten] != -1 else solve(x - i, y, z - eaten)
			if t2 == -2:
				continue
			if res == -2 or res > t1 + t2 + y * y:
				res = t1 + t2 + y * y
	for j in range(1, y // 2 + 1):
		for eaten in range(z + 1):
			t1 = mem[x][j][eaten] if mem[x][j][eaten] != -1 else solve(x, j, eaten)
			if t1 == -2:
				continue
			t2 = mem[x][y - j][z - eaten] if mem[x][y - j][z - eaten] != -1 else solve(x, y - j, z - eaten)
			if t2 == -2:
				continue
			if res == -2 or res > t1 + t2 + x * x:
				res = t1 + t2 + x * x
	mem[x][y][z] = res
	return mem[x][y][z]
t = int(sys.stdin.readline())
for it in range(t):
	(n, m, k) = map(int, sys.stdin.readline().split())
	print(solve(n, m, k))


D = {}

def h(m, n, answ, k):
	x = answ
	for i in range(1, (n + 2) // 2):
		if k >= i * m:
			if m ** 2 + ans(m, n - i, k - i * m) < x:
				x = m ** 2 + ans(m, n - i, k - i * m)
		if k <= (n - i) * m:
			if m ** 2 + ans(m, n - i, k) < x:
				x = m ** 2 + ans(m, n - i, k)
		if k >= (n - i) * m:
			if m ** 2 + ans(m, i, k - (n - i) * m) < x:
				x = m ** 2 + ans(m, i, k - (n - i) * m)
		if k <= i * m:
			if m ** 2 + ans(m, i, k) < x:
				x = m ** 2 + ans(m, i, k)
	return x

def ans(m, n, k):
	if k == 0:
		D[m, n, k] = 0
		D[n, m, k] = 0
		return 0
	if m * n == k:
		D[m, n, k] = 0
		D[n, m, k] = 0
		return 0
	elif m == 1:
		D[m, n, k] = 1
		D[n, m, k] = 1
		return 1
	elif n == 1:
		D[m, n, k] = 1
		D[n, m, k] = 1
		return 1
	elif (m, n, k) in D:
		return D[m, n, k]
	else:
		answ = n ** 2 * m
		t = h(m, n, answ, k)
		if t < answ:
			answ = t
		s = h(n, m, answ, k)
		if s < answ:
			answ = s
		D[m, n, k] = answ
		D[n, m, k] = answ
		return answ
for i in range(30, 0, -1):
	for j in range(i, 0, -1):
		for k in range(0, min(i * j, 50) + 1):
			ans(i, j, k)
t = int(input())
for i in range(t):
	(m, n, k) = [int(x) for x in input().split()]
	print(D[m, n, k])


import math
import random
import heapq, bisect
import sys
from collections import deque, defaultdict
from fractions import Fraction
import sys
import threading
from collections import defaultdict
threading.stack_size(10 ** 8)
mod = 10 ** 9 + 7
mod1 = 998244353
import os
import sys
from io import BytesIO, IOBase
sys.setrecursionlimit(300000)
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')

class TreeNode:

	def __init__(self, k, v):
		self.key = k
		self.value = v
		self.left = None
		self.right = None
		self.parent = None
		self.height = 1
		self.num_left = 1
		self.num_total = 1

class AvlTree:

	def __init__(self):
		self._tree = None

	def add(self, k, v):
		if not self._tree:
			self._tree = TreeNode(k, v)
			return
		node = self._add(k, v)
		if node:
			self._rebalance(node)

	def _add(self, k, v):
		node = self._tree
		while node:
			if k < node.key:
				if node.left:
					node = node.left
				else:
					node.left = TreeNode(k, v)
					node.left.parent = node
					return node.left
			elif node.key < k:
				if node.right:
					node = node.right
				else:
					node.right = TreeNode(k, v)
					node.right.parent = node
					return node.right
			else:
				node.value = v
				return

	@staticmethod
	def get_height(x):
		return x.height if x else 0

	@staticmethod
	def get_num_total(x):
		return x.num_total if x else 0

	def _rebalance(self, node):
		n = node
		while n:
			lh = self.get_height(n.left)
			rh = self.get_height(n.right)
			n.height = max(lh, rh) + 1
			balance_factor = lh - rh
			n.num_total = 1 + self.get_num_total(n.left) + self.get_num_total(n.right)
			n.num_left = 1 + self.get_num_total(n.left)
			if balance_factor > 1:
				if self.get_height(n.left.left) < self.get_height(n.left.right):
					self._rotate_left(n.left)
				self._rotate_right(n)
			elif balance_factor < -1:
				if self.get_height(n.right.right) < self.get_height(n.right.left):
					self._rotate_right(n.right)
				self._rotate_left(n)
			else:
				n = n.parent

	def _remove_one(self, node):
		replacement = node.left or node.right
		if node.parent:
			if AvlTree._is_left(node):
				node.parent.left = replacement
			else:
				node.parent.right = replacement
			replacement.parent = node.parent
			node.parent = None
		else:
			self._tree = replacement
			replacement.parent = None
		node.left = None
		node.right = None
		node.parent = None
		self._rebalance(replacement)

	def _remove_leaf(self, node):
		if node.parent:
			if AvlTree._is_left(node):
				node.parent.left = None
			else:
				node.parent.right = None
			self._rebalance(node.parent)
		else:
			self._tree = None
		node.parent = None
		node.left = None
		node.right = None

	def remove(self, k):
		node = self._get_node(k)
		if not node:
			return
		if AvlTree._is_leaf(node):
			self._remove_leaf(node)
			return
		if node.left and node.right:
			nxt = AvlTree._get_next(node)
			node.key = nxt.key
			node.value = nxt.value
			if self._is_leaf(nxt):
				self._remove_leaf(nxt)
			else:
				self._remove_one(nxt)
			self._rebalance(node)
		else:
			self._remove_one(node)

	def get(self, k):
		node = self._get_node(k)
		return node.value if node else -1

	def _get_node(self, k):
		if not self._tree:
			return None
		node = self._tree
		while node:
			if k < node.key:
				node = node.left
			elif node.key < k:
				node = node.right
			else:
				return node
		return None

	def get_at(self, pos):
		x = pos + 1
		node = self._tree
		while node:
			if x < node.num_left:
				node = node.left
			elif node.num_left < x:
				x -= node.num_left
				node = node.right
			else:
				return (node.key, node.value)
		raise IndexError('Out of ranges')

	@staticmethod
	def _is_left(node):
		return node.parent.left and node.parent.left == node

	@staticmethod
	def _is_leaf(node):
		return node.left is None and node.right is None

	def _rotate_right(self, node):
		if not node.parent:
			self._tree = node.left
			node.left.parent = None
		elif AvlTree._is_left(node):
			node.parent.left = node.left
			node.left.parent = node.parent
		else:
			node.parent.right = node.left
			node.left.parent = node.parent
		bk = node.left.right
		node.left.right = node
		node.parent = node.left
		node.left = bk
		if bk:
			bk.parent = node
		node.height = max(self.get_height(node.left), self.get_height(node.right)) + 1
		node.num_total = 1 + self.get_num_total(node.left) + self.get_num_total(node.right)
		node.num_left = 1 + self.get_num_total(node.left)

	def _rotate_left(self, node):
		if not node.parent:
			self._tree = node.right
			node.right.parent = None
		elif AvlTree._is_left(node):
			node.parent.left = node.right
			node.right.parent = node.parent
		else:
			node.parent.right = node.right
			node.right.parent = node.parent
		bk = node.right.left
		node.right.left = node
		node.parent = node.right
		node.right = bk
		if bk:
			bk.parent = node
		node.height = max(self.get_height(node.left), self.get_height(node.right)) + 1
		node.num_total = 1 + self.get_num_total(node.left) + self.get_num_total(node.right)
		node.num_left = 1 + self.get_num_total(node.left)

	@staticmethod
	def _get_next(node):
		if not node.right:
			return node.parent
		n = node.right
		while n.left:
			n = n.left
		return n

class SegmentTree1:

	def __init__(self, data, default=2 ** 51, func=lambda a, b: a & b):
		self._default = default
		self._func = func
		self._len = len(data)
		self._size = _size = 1 << (self._len - 1).bit_length()
		self.data = [default] * (2 * _size)
		self.data[_size:_size + self._len] = data
		for i in reversed(range(_size)):
			self.data[i] = func(self.data[i + i], self.data[i + i + 1])

	def __delitem__(self, idx):
		self[idx] = self._default

	def __getitem__(self, idx):
		return self.data[idx + self._size]

	def __setitem__(self, idx, value):
		idx += self._size
		self.data[idx] = value
		idx >>= 1
		while idx:
			self.data[idx] = self._func(self.data[2 * idx], self.data[2 * idx + 1])
			idx >>= 1

	def __len__(self):
		return self._len

	def query(self, start, stop):
		if start == stop:
			return self.__getitem__(start)
		stop += 1
		start += self._size
		stop += self._size
		res = self._default
		while start < stop:
			if start & 1:
				res = self._func(res, self.data[start])
				start += 1
			if stop & 1:
				stop -= 1
				res = self._func(res, self.data[stop])
			start >>= 1
			stop >>= 1
		return res

	def __repr__(self):
		return 'SegmentTree({0})'.format(self.data)

class SegmentTree:

	def __init__(self, data, default=0, func=lambda a, b: max(a, b)):
		self._default = default
		self._func = func
		self._len = len(data)
		self._size = _size = 1 << (self._len - 1).bit_length()
		self.data = [default] * (2 * _size)
		self.data[_size:_size + self._len] = data
		for i in reversed(range(_size)):
			self.data[i] = func(self.data[i + i], self.data[i + i + 1])

	def __delitem__(self, idx):
		self[idx] = self._default

	def __getitem__(self, idx):
		return self.data[idx + self._size]

	def __setitem__(self, idx, value):
		idx += self._size
		self.data[idx] = value
		idx >>= 1
		while idx:
			self.data[idx] = self._func(self.data[2 * idx], self.data[2 * idx + 1])
			idx >>= 1

	def __len__(self):
		return self._len

	def query(self, start, stop):
		if start == stop:
			return self.__getitem__(start)
		stop += 1
		start += self._size
		stop += self._size
		res = self._default
		while start < stop:
			if start & 1:
				res = self._func(res, self.data[start])
				start += 1
			if stop & 1:
				stop -= 1
				res = self._func(res, self.data[stop])
			start >>= 1
			stop >>= 1
		return res

	def __repr__(self):
		return 'SegmentTree({0})'.format(self.data)

class Factorial:

	def __init__(self, MOD):
		self.MOD = MOD
		self.factorials = [1, 1]
		self.invModulos = [0, 1]
		self.invFactorial_ = [1, 1]

	def calc(self, n):
		if n <= -1:
			print('Invalid argument to calculate n!')
			print('n must be non-negative value. But the argument was ' + str(n))
			exit()
		if n < len(self.factorials):
			return self.factorials[n]
		nextArr = [0] * (n + 1 - len(self.factorials))
		initialI = len(self.factorials)
		prev = self.factorials[-1]
		m = self.MOD
		for i in range(initialI, n + 1):
			prev = nextArr[i - initialI] = prev * i % m
		self.factorials += nextArr
		return self.factorials[n]

	def inv(self, n):
		if n <= -1:
			print('Invalid argument to calculate n^(-1)')
			print('n must be non-negative value. But the argument was ' + str(n))
			exit()
		p = self.MOD
		pi = n % p
		if pi < len(self.invModulos):
			return self.invModulos[pi]
		nextArr = [0] * (n + 1 - len(self.invModulos))
		initialI = len(self.invModulos)
		for i in range(initialI, min(p, n + 1)):
			next = -self.invModulos[p % i] * (p // i) % p
			self.invModulos.append(next)
		return self.invModulos[pi]

	def invFactorial(self, n):
		if n <= -1:
			print('Invalid argument to calculate (n^(-1))!')
			print('n must be non-negative value. But the argument was ' + str(n))
			exit()
		if n < len(self.invFactorial_):
			return self.invFactorial_[n]
		self.inv(n)
		nextArr = [0] * (n + 1 - len(self.invFactorial_))
		initialI = len(self.invFactorial_)
		prev = self.invFactorial_[-1]
		p = self.MOD
		for i in range(initialI, n + 1):
			prev = nextArr[i - initialI] = prev * self.invModulos[i % p] % p
		self.invFactorial_ += nextArr
		return self.invFactorial_[n]

class Combination:

	def __init__(self, MOD):
		self.MOD = MOD
		self.factorial = Factorial(MOD)

	def ncr(self, n, k):
		if k < 0 or n < k:
			return 0
		k = min(k, n - k)
		f = self.factorial
		return f.calc(n) * f.invFactorial(max(n - k, k)) * f.invFactorial(min(k, n - k)) % self.MOD

def powm(a, n, m):
	if a == 1 or n == 0:
		return 1
	if n % 2 == 0:
		s = powm(a, n // 2, m)
		return s * s % m
	else:
		return a * powm(a, n - 1, m) % m

def sort_list(list1, list2):
	zipped_pairs = zip(list2, list1)
	z = [x for (_, x) in sorted(zipped_pairs)]
	return z

def product(l):
	por = 1
	for i in range(len(l)):
		por *= l[i]
	return por

def binarySearchCount(arr, n, key):
	left = 0
	right = n - 1
	count = 0
	while left <= right:
		mid = int((right + left) / 2)
		if arr[mid] <= key:
			count = mid + 1
			left = mid + 1
		else:
			right = mid - 1
	return count

def countdig(n):
	c = 0
	while n > 0:
		n //= 10
		c += 1
	return c

def binary(x, length):
	y = bin(x)[2:]
	return y if len(y) >= length else '0' * (length - len(y)) + y

def countGreater(arr, n, k):
	l = 0
	r = n - 1
	leftGreater = n
	while l <= r:
		m = int(l + (r - l) / 2)
		if arr[m] >= k:
			leftGreater = m
			r = m - 1
		else:
			l = m + 1
	return n - leftGreater
dp = [[[-1 for i in range(51)] for j in range(31)] for t in range(31)]
for ik in range(int(input())):
	(n, m, k) = map(int, input().split())

	def find(n, m, k):
		if k == 0:
			return 0
		if dp[n][m][k] != -1:
			return dp[n][m][k]
		if n * m < k:
			return 10 ** 9
		if n * m == k:
			return 0
		ans = 10 ** 9
		if n == 1 or m == 1:
			t = 1
			if n * m == k:
				t = 0
			dp[n][m][k] = t
			return t
		for i in range(m - 1):
			for j in range(k + 1):
				ans = min(ans, find(n, i + 1, j) + find(n, m - 1 - i, k - j) + n * n)
		for i in range(n - 1):
			for j in range(k + 1):
				ans = min(ans, find(i + 1, m, j) + find(n - 1 - i, m, k - j) + m * m)
		dp[n][m][k] = ans
		return ans
	print(find(n, m, k))


import sys
input = sys.stdin.readline

def main():
	ans = []
	memo = [[[-1 for _ in range(51)] for _ in range(31)] for _ in range(31)]

	def solve(n, m, k):
		if n * m == k or k == 0:
			return 0
		if memo[n][m][k] > -1:
			return memo[n][m][k]
		if memo[m][n][k] > -1:
			memo[n][m][k] = memo[m][n][k]
			return memo[n][m][k]
		r = float('inf')
		for i in range(k + 1):
			for j in range(1, max(m, n)):
				if m > j:
					r = min(r, n ** 2 + solve(j, n, i) + solve(m - j, n, k - i))
				if n > j:
					r = min(r, m ** 2 + solve(m, j, i) + solve(m, n - j, k - i))
		memo[n][m][k] = r
		return r
	for _ in range(int(input())):
		(n, m, k) = map(int, input().split())
		ans.append(str(solve(n, m, k)))
	print('\n'.join(ans))
main()


d = [[[0 for i in range(51)] for j in range(31)] for k in range(31)]

def rec(n, m, k):
	global d
	if n * m == k or k == 0:
		return 0
	if d[n][m][k] > 0:
		return d[n][m][k]
	if n * m < k:
		return 10 ** 10
	cost = 10 ** 10
	for i in range(1, n // 2 + 1):
		for j in range(k + 1):
			cost = min(cost, m * m + rec(n - i, m, k - j) + rec(i, m, j))
	for i in range(1, m // 2 + 1):
		for j in range(k + 1):
			cost = min(cost, n * n + rec(n, m - i, k - j) + rec(n, i, j))
	d[n][m][k] = cost
	return cost
p = []
t = int(input())
for i in range(t):
	(n, m, k) = map(int, input().split())
	print(rec(n, m, k))