# Task

The function `fold_cube(number_list)` should return a boolean based on a net (given as a number list), if the net can fold into a cube. 
Your code must be effecient and complete the tests within 500ms. 

## Input

Imagine a net such as the one below.

```
@
@ @ @ @
    @
```
Then, put it on the table with each `@` being one cell.

```
--------------------------
|  1 |  2 |  3 |  4 |  5 |
--------------------------
|  6 |  7 |  8 |  9 | 10 |
--------------------------
| 11 | 12 | 13 | 14 | 15 |
--------------------------
| 16 | 17 | 18 | 19 | 20 |
--------------------------
| 21 | 22 | 23 | 24 | 25 |
--------------------------
```
The number list for a net will be the numbers the net falls on when placed on the table. Note that the numbers in the list won't always be in order. The position of the net can be anywhere on the table, it can also be rotated.

For the example above, a possible number list will be `[1, 7, 8, 6, 9, 13]`.

## Output

`fold_cube` should return `True` if the net can be folded into a cube. `False` if not.

---

# Examples

### Example 1:

`number_list` = `[24, 20, 14, 19, 18, 9]`

Shape and rotation of net:
```
  @
  @
@ @ @
  @
```
Displayed on the table:
```
--------------------------
|  1 |  2 |  3 |  4 |  5 |
--------------------------
|  6 |  7 |  8 |  @ | 10 |
--------------------------
| 11 | 12 | 13 |  @ | 15 |
--------------------------
| 16 | 17 |  @ |  @ |  @ |
--------------------------
| 21 | 22 | 23 |  @ | 25 |
--------------------------
```
So, `fold_cube([24, 20, 14, 19, 18, 9])` should return `True`.

### Example 2:

`number_list` = `[1, 7, 6, 17, 12, 16]`

Shape and rotation of net:
```
@
@ @
  @ 
@ @
```
Displayed on the table:
```
--------------------------
|  @ |  2 |  3 |  4 |  5 |
--------------------------
|  @ |  @ |  8 |  9 | 10 |
--------------------------
| 11 |  @ | 13 | 14 | 15 |
--------------------------
|  @ |  @ | 18 | 19 | 20 |
--------------------------
| 21 | 22 | 23 | 24 | 25 |
--------------------------
```
`fold_cube([1, 7, 6, 17, 12, 16])` should return `False`.

#### *Translations appreciated!!*

---

### If you liked this kata.... check out these!             
- [Folding a 4D Cube (Tesseract)](https://www.codewars.com/kata/5f3b561bc4a71f000f191ef7)  
- [Wraping a net around a cube](https://www.codewars.com/kata/5f4af9c169f1cd0001ae764d)

Difficulty: EASY
Starter code: def fold_cube(nums):
	

Solutions:
def fold_cube(nums):
	return expand(nums.pop(), set(nums), 1, 2, 3) == {1, 2, 3, -1, -2, -3}

def expand(val, nums, x, y, z):
	dirs = {z}
	for num in nums.copy():
		if abs(val - num) not in (1, 5) or {val % 5, num % 5} == {0, 1}:
			continue
		nums.discard(num)
		diff = val - num
		sign = diff // abs(diff)
		(nx, ny, nz) = (x, z * sign, -y * sign) if abs(diff) == 1 else (-z * sign, y, x * sign)
		dirs |= expand(num, nums, nx, ny, nz)
	return dirs


def fold_cube(nums):
	stack = [nums[0]]
	for _ in range(6):
		for i in range(len(stack)):
			if (stack[i] - 1) % 5 > 0 and stack[i] - 1 in nums and (stack[i] - 1 not in stack):
				stack.append(stack[i] - 1)
			if (stack[i] - 1) % 5 < 4 and stack[i] + 1 in nums and (stack[i] + 1 not in stack):
				stack.append(stack[i] + 1)
			if stack[i] - 1 > 5 and stack[i] - 5 in nums and (stack[i] - 5 not in stack):
				stack.append(stack[i] - 5)
			if stack[i] - 1 < 21 and stack[i] + 5 in nums and (stack[i] + 5 not in stack):
				stack.append(stack[i] + 5)
	print(stack)
	print(nums)
	if len(stack) != 6:
		return False
	cols = []
	for n in stack:
		if (n - 1) % 5 not in cols:
			cols.append((n - 1) % 5)
	rows = []
	for n in stack:
		if int((n - 1) / 5) not in rows:
			rows.append(int((n - 1) / 5))
	if len(rows) + len(cols) != 7:
		return False
	if len(rows) == 2:
		if (sum(rows) + 1) / 2 * 5 + sum(cols) / 5 + 1 not in stack or (sum(rows) - 1) / 2 * 5 + sum(cols) / 5 + 1 not in stack:
			return False
	if len(rows) == 3:
		if sum(rows) / 3 * 5 + (sum(cols) + 2) / 4 + 1 not in stack or sum(rows) / 3 * 5 + (sum(cols) - 2) / 4 + 1 not in stack:
			print(3)
			return False
	if len(rows) == 4:
		if (sum(rows) + 2) / 4 * 5 + sum(cols) / 3 + 1 not in stack or (sum(rows) - 2) / 4 * 5 + sum(cols) / 3 + 1 not in stack:
			print(4)
			return False
	if len(rows) == 5:
		if sum(rows) + (sum(cols) + 1) / 2 + 1 not in stack or sum(rows) + (sum(cols) - 1) / 2 + 1 not in stack:
			print(5)
			return False
	if len(rows) == 2:
		if sum(stack) % 30 != 3:
			return False
	if len(rows) == 5:
		if sum(stack) % 6 != 3:
			return False
	print(True)
	return True


nets = [[[1, 0, 0, 0], [1, 1, 1, 1], [1, 0, 0, 0]], [[1, 0, 0, 0], [1, 1, 1, 1], [0, 1, 0, 0]], [[1, 0, 0, 0], [1, 1, 1, 1], [0, 0, 1, 0]], [[1, 0, 0, 0], [1, 1, 1, 1], [0, 0, 0, 1]], [[0, 1, 0, 0], [1, 1, 1, 1], [0, 0, 1, 0]], [[0, 1, 0, 0], [1, 1, 1, 1], [0, 1, 0, 0]], [[1, 1, 0, 0], [0, 1, 1, 1], [0, 1, 0, 0]], [[1, 1, 0, 0], [0, 1, 1, 1], [0, 0, 1, 0]], [[1, 0, 0, 0], [1, 1, 1, 0], [0, 0, 1, 1]], [[1, 1, 0, 0], [0, 1, 1, 0], [0, 0, 1, 1]], [[1, 1, 1, 0, 0], [0, 0, 1, 1, 1]]]

def h_mirror(m):
	return m[::-1]

def v_mirror(m):
	return [r[::-1] for r in m]

def transpose(m):
	return list(map(list, zip(*m)))

def rotate_cw(m):
	coords = [(x, y) for (y, row) in enumerate(m) for (x, v) in enumerate(row) if v == 1]
	new_coords = normalise([(y, -x) for (x, y) in coords])
	result = [[0 for _ in range(len(m))] for _ in range(len(m[0]))]
	for (x, y) in new_coords:
		result[y][x] = 1
	return result

def rotate_ccw(m):
	coords = [(x, y) for (y, row) in enumerate(m) for (x, v) in enumerate(row) if v == 1]
	new_coords = normalise([(-y, x) for (x, y) in coords])
	result = [[0 for _ in range(len(m))] for _ in range(len(m[0]))]
	for (x, y) in new_coords:
		result[y][x] = 1
	return result

def identity(coords):
	return coords

def normalise(coords):
	min_x = min((x for (x, y) in coords))
	min_y = min((y for (x, y) in coords))
	return tuple([(x - min_x, y - min_y) for (x, y) in coords])

def compose(*fns):

	def apply(*args, **kwargs):
		result = fns[0](*args, **kwargs)
		for fn in fns[1:]:
			result = fn(result)
		return result
	return apply
compositions = [identity, h_mirror, v_mirror, transpose, rotate_cw, rotate_ccw, compose(rotate_cw, rotate_cw), compose(rotate_cw, rotate_cw, transpose)]
solutions = []
for net in nets:
	for composition in compositions:
		net_prime = composition(net)
		if net_prime not in solutions:
			solutions.append(net_prime)

def to_relative(m):
	return tuple(sorted([(x, y) for (y, row) in enumerate(m) for (x, v) in enumerate(row) if v == 1]))
relative_coordinate_solutions = set((to_relative(s) for s in solutions))

def fold_cube(number_list):
	relative_coordinates = normalise(sorted([((n - 1) // 5, (n - 1) % 5) for n in number_list]))
	return relative_coordinates in relative_coordinate_solutions


def folding(grid, face, list_on_face, remain_list):
	faces = [face]
	dirs = [1, -1, 5, -5]
	if list_on_face % 5 == 1:
		dirs.remove(-1)
	if list_on_face % 5 == 0:
		dirs.remove(1)
	goto_dirs = []
	print(remain_list)
	for direction in dirs:
		goto_cell = direction + list_on_face
		print((goto_cell, remain_list))
		if goto_cell in remain_list:
			remain_list.remove(goto_cell)
			goto_dirs.append(direction)
	print(('face', face, 'grid', grid, 'listCELL', list_on_face, 'to', goto_dirs))
	for direction in goto_dirs:
		faces.extend(folding(grid=new_grid(face, direction, grid), face=new_face(grid, direction), list_on_face=list_on_face + direction, remain_list=remain_list))
	return faces

def new_face(grid, direction):
	return grid[[1, -1, 5, -5].index(direction)]

def new_grid(face, direction, grid):
	opposite_face = {1: 6, 2: 4, 6: 1, 4: 2, 5: 3, 3: 5}
	dir_index = {1: 0, -1: 1, 5: 2, -5: 3}
	newgrid = grid.copy()
	newgrid[dir_index[-direction]] = face
	newgrid[dir_index[direction]] = opposite_face[face]
	return newgrid

def fold_cube(number_list):
	faces = folding(grid=[3, 5, 2, 4], face=1, list_on_face=number_list[0], remain_list=number_list[1:])
	return sorted(faces) == list(range(1, 7))


BASE = {'  x\nxxx\n x \n x ', ' x\n x\nxx\nx \nx ', ' x \n xx\nxx \n x ', 'xx \n xx\n x \n x ', ' x \n xx\nxx \nx  ', 'xxx\n x \n x \n x ', 'xx \n x \n xx\n x ', 'xx \n x \n xx\n  x', 'xx \n x \n x \n xx', '  xx\n xx \nxx  ', ' x \nxxx\n x \n x '}

def reflect(s):
	return '\n'.join((r[::-1] for r in s.split('\n')))

def rotate(s):
	return '\n'.join((''.join(c[::-1]) for c in zip(*s.split('\n'))))

def fold_cube(arr):
	table = [[' '] * 5 for _ in range(5)]
	(x0, y0, x1, y1) = (4, 4, 0, 0)
	for p in arr:
		(x, y) = divmod(p - 1, 5)
		table[x][y] = 'x'
		(x0, y0) = (min(x, x0), min(y, y0))
		(x1, y1) = (max(x, x1), max(y, y1))
	net = '\n'.join((''.join(r[y0:y1 + 1]) for r in table[x0:x1 + 1]))
	net2 = reflect(net)
	for i in range(4):
		if net in BASE or net2 in BASE:
			return True
		(net, net2) = (rotate(net), rotate(net2))
	return False


db = [{9, 14, 18, 19, 20, 24}, {13, 14, 15, 16, 17, 18}, {2, 3, 8, 9, 10, 13}, {3, 6, 7, 8, 9, 12}, {4, 7, 8, 9, 10, 14}, {4, 6, 7, 8, 9, 13}, {8, 9, 13, 17, 18, 23}, {5, 10, 14, 15, 19, 24}, {3, 8, 12, 13, 18, 19}, {8, 12, 13, 14, 18, 23}, {7, 11, 12, 13, 17, 22}, {8, 13, 17, 18, 23, 24}, {8, 12, 13, 14, 18, 23}, {3, 8, 12, 13, 18, 19}, {13, 16, 17, 18, 19, 23}, {4, 9, 13, 14, 15, 19}, {2, 7, 8, 11, 12, 16}, {1, 6, 7, 8, 13, 14}, {13, 17, 18, 19, 20, 22}, {8, 12, 13, 14, 15, 18}, {15, 17, 18, 19, 20, 23}, {3, 6, 7, 8, 9, 12}, {8, 13, 17, 18, 19, 24}, {3, 4, 8, 13, 17, 18}, {6, 11, 12, 13, 14, 18}, {8, 12, 13, 18, 19, 24}, {2, 7, 8, 9, 10, 14}, {15, 17, 18, 19, 20, 25}, {10, 14, 15, 18, 19, 24}, {15, 17, 18, 19, 20, 24}, {11, 16, 17, 18, 19, 23}, {4, 8, 9, 14, 15, 19}, {5, 8, 9, 10, 14, 19}, {9, 13, 14, 19, 20, 24}, {8, 13, 14, 19, 20, 24}, {11, 16, 17, 18, 19, 22}, {2, 6, 7, 8, 13, 14}, {8, 12, 13, 18, 19, 23}, {8, 9, 14, 15, 19, 24}, {9, 13, 14, 19, 20, 24}, {3, 8, 12, 13, 14, 18}, {7, 12, 13, 17, 21, 22}, {7, 12, 16, 17, 18, 23}, {1, 2, 3, 7, 12, 17}, {9, 14, 18, 19, 24, 25}, {9, 11, 12, 13, 14, 16}, {14, 17, 18, 19, 20, 24}, {4, 9, 13, 14, 19, 20}, {2, 7, 8, 9, 14, 15}, {2, 3, 8, 9, 10, 13}, {6, 11, 12, 13, 17, 22}, {12, 16, 17, 18, 23, 24}, {7, 12, 16, 17, 18, 21}, {13, 14, 17, 18, 21, 22}, {4, 6, 7, 8, 9, 12}, {12, 17, 18, 19, 21, 22}, {8, 13, 18, 19, 22, 23}, {7, 12, 13, 18, 19, 23}, {1, 2, 7, 8, 9, 12}, {9, 14, 19, 20, 23, 24}, {11, 16, 17, 18, 19, 21}, {8, 13, 14, 17, 18, 23}, {9, 14, 15, 19, 23, 24}, {7, 12, 13, 14, 15, 19}, {6, 11, 12, 13, 17, 22}, {2, 6, 7, 8, 12, 17}, {3, 6, 7, 8, 9, 13}, {4, 5, 7, 8, 9, 12}, {3, 8, 9, 12, 13, 17}, {2, 3, 8, 9, 10, 15}, {6, 11, 12, 17, 22, 23}, {13, 17, 18, 19, 20, 24}, {8, 9, 14, 15, 19, 24}, {6, 7, 12, 13, 17, 22}, {9, 13, 14, 15, 17, 18}, {7, 12, 16, 17, 18, 22}, {8, 9, 14, 15, 19, 24}, {4, 9, 10, 14, 18, 19}, {8, 12, 13, 18, 19, 24}, {3, 4, 9, 10, 14, 19}, {8, 13, 14, 19, 24, 25}, {7, 12, 16, 17, 22, 23}, {8, 11, 12, 13, 17, 22}, {7, 12, 16, 17, 18, 22}, {2, 6, 7, 8, 13, 14}, {3, 4, 6, 7, 8, 12}, {3, 8, 9, 10, 14, 19}, {8, 13, 14, 15, 19, 24}, {4, 8, 9, 10, 12, 13}, {8, 13, 17, 18, 19, 23}, {2, 6, 7, 8, 9, 13}, {13, 16, 17, 18, 19, 23}, {3, 8, 12, 13, 14, 17}, {3, 7, 8, 9, 13, 18}, {9, 14, 15, 18, 19, 24}, {3, 6, 7, 8, 9, 12}, {11, 16, 17, 18, 19, 23}, {8, 9, 11, 12, 13, 18}, {7, 12, 13, 14, 15, 19}, {10, 12, 13, 14, 15, 18}, {8, 13, 17, 18, 23, 24}, {4, 9, 13, 14, 15, 19}, {13, 16, 17, 18, 23, 24}, {1, 2, 3, 7, 12, 17}, {9, 12, 13, 14, 15, 17}, {4, 9, 10, 14, 18, 19}, {12, 13, 18, 19, 24, 25}, {2, 7, 11, 12, 17, 18}, {2, 6, 7, 12, 13, 17}, {3, 7, 8, 9, 10, 13}, {1, 6, 7, 8, 9, 14}, {2, 7, 8, 9, 10, 12}, {3, 7, 8, 9, 10, 13}, {3, 7, 8, 13, 14, 19}, {5, 9, 10, 13, 14, 19}, {8, 13, 14, 17, 18, 23}, {4, 8, 9, 14, 15, 19}, {4, 8, 9, 14, 19, 20}, {3, 7, 8, 13, 14, 19}, {1, 2, 7, 8, 9, 12}, {7, 8, 13, 14, 19, 20}, {14, 15, 17, 18, 19, 23}, {8, 11, 12, 13, 17, 22}, {4, 8, 9, 12, 13, 18}, {4, 8, 9, 14, 19, 20}, {4, 9, 13, 14, 15, 18}, {7, 12, 13, 14, 15, 17}, {2, 7, 8, 11, 12, 16}, {1, 6, 7, 8, 9, 13}, {15, 17, 18, 19, 20, 24}, {8, 9, 10, 11, 12, 13}, {12, 16, 17, 18, 19, 21}, {9, 14, 18, 19, 20, 25}, {3, 4, 8, 12, 13, 18}, {9, 12, 13, 14, 15, 17}, {6, 7, 12, 13, 17, 22}, {10, 14, 15, 18, 19, 24}, {2, 3, 6, 7, 12, 17}, {4, 9, 13, 14, 15, 19}, {2, 7, 8, 9, 13, 18}, {3, 8, 13, 14, 17, 18}, {9, 10, 13, 14, 19, 24}, {3, 8, 9, 12, 13, 18}, {8, 12, 13, 14, 15, 18}, {14, 15, 18, 19, 22, 23}, {12, 16, 17, 18, 19, 21}, {7, 11, 12, 17, 18, 22}, {3, 6, 7, 8, 9, 11}, {7, 8, 12, 16, 17, 22}, {9, 13, 14, 15, 17, 18}, {6, 11, 12, 13, 14, 17}, {13, 17, 18, 19, 21, 22}, {12, 16, 17, 18, 19, 24}, {6, 11, 12, 13, 14, 18}, {14, 16, 17, 18, 19, 23}, {3, 7, 8, 9, 10, 13}, {4, 6, 7, 8, 9, 13}, {9, 12, 13, 14, 15, 20}, {2, 3, 6, 7, 12, 17}, {12, 13, 18, 19, 20, 23}, {8, 9, 10, 11, 12, 13}, {8, 13, 14, 17, 18, 23}, {4, 6, 7, 8, 9, 14}, {9, 12, 13, 14, 15, 19}, {3, 7, 8, 9, 11, 12}, {3, 8, 9, 12, 13, 18}, {9, 14, 15, 18, 19, 23}, {8, 12, 13, 14, 15, 18}, {3, 4, 9, 14, 15, 19}, {4, 8, 9, 14, 15, 20}, {7, 11, 12, 13, 14, 18}, {4, 9, 14, 18, 19, 20}, {2, 6, 7, 12, 13, 18}, {13, 16, 17, 18, 19, 23}, {9, 14, 19, 23, 24, 25}, {6, 11, 12, 13, 14, 18}, {7, 12, 16, 17, 18, 23}, {10, 12, 13, 14, 15, 19}, {4, 8, 9, 10, 14, 19}, {13, 16, 17, 18, 19, 22}, {4, 7, 8, 9, 10, 13}, {14, 17, 18, 19, 20, 23}, {12, 17, 18, 19, 20, 23}, {16, 17, 18, 23, 24, 25}, {8, 11, 12, 13, 14, 18}, {3, 8, 12, 13, 14, 18}, {9, 12, 13, 14, 15, 18}, {9, 13, 14, 19, 20, 25}, {7, 12, 13, 14, 15, 20}, {11, 12, 13, 18, 19, 20}, {7, 8, 13, 14, 18, 23}, {3, 6, 7, 8, 9, 12}, {10, 13, 14, 15, 17, 18}, {6, 7, 12, 13, 17, 22}, {4, 7, 8, 9, 10, 15}, {10, 13, 14, 15, 19, 24}, {8, 9, 11, 12, 13, 16}, {8, 12, 13, 14, 15, 20}, {11, 16, 17, 18, 19, 24}, {2, 7, 8, 9, 10, 12}, {13, 16, 17, 18, 19, 23}, {9, 12, 13, 14, 15, 19}, {4, 6, 7, 8, 9, 12}, {12, 16, 17, 18, 19, 22}, {2, 7, 11, 12, 13, 18}, {14, 15, 17, 18, 19, 22}, {3, 7, 8, 13, 18, 19}, {9, 11, 12, 13, 14, 16}, {9, 10, 12, 13, 14, 17}, {8, 12, 13, 14, 15, 18}, {12, 17, 18, 19, 20, 25}, {4, 7, 8, 9, 10, 14}, {8, 13, 18, 19, 22, 23}, {7, 8, 13, 14, 15, 20}, {12, 16, 17, 18, 19, 23}, {4, 7, 8, 9, 10, 15}, {1, 2, 7, 12, 17, 18}, {3, 7, 8, 13, 14, 18}, {4, 8, 9, 14, 15, 20}, {8, 12, 13, 14, 16, 17}, {4, 7, 8, 9, 10, 14}, {6, 7, 12, 13, 14, 19}, {12, 17, 18, 19, 20, 23}, {14, 17, 18, 19, 20, 23}, {8, 12, 13, 18, 19, 24}, {8, 12, 13, 18, 23, 24}, {8, 12, 13, 14, 15, 19}, {4, 5, 8, 9, 12, 13}, {6, 11, 12, 13, 17, 22}, {7, 11, 12, 13, 14, 19}, {8, 13, 17, 18, 19, 24}, {13, 16, 17, 18, 19, 23}, {2, 6, 7, 8, 12, 17}, {7, 12, 16, 17, 18, 22}, {14, 17, 18, 19, 20, 23}, {4, 8, 9, 10, 14, 19}, {6, 11, 12, 13, 14, 18}, {5, 8, 9, 10, 14, 19}, {9, 14, 18, 19, 20, 24}, {9, 13, 14, 15, 17, 18}, {6, 11, 12, 17, 18, 22}, {7, 11, 12, 17, 22, 23}, {7, 11, 12, 13, 14, 18}, {10, 13, 14, 15, 19, 24}, {12, 13, 18, 19, 20, 23}, {13, 16, 17, 18, 23, 24}, {8, 13, 17, 18, 19, 22}, {12, 16, 17, 18, 19, 23}, {11, 16, 17, 18, 19, 22}, {13, 16, 17, 18, 19, 23}, {14, 15, 17, 18, 19, 23}, {14, 17, 18, 19, 20, 24}, {6, 7, 12, 17, 22, 23}, {8, 11, 12, 13, 18, 19}, {8, 9, 11, 12, 13, 18}, {8, 13, 18, 19, 22, 23}, {18, 19, 20, 21, 22, 23}, {7, 12, 13, 14, 16, 17}, {7, 12, 16, 17, 18, 22}, {3, 4, 6, 7, 8, 13}, {2, 7, 8, 9, 10, 14}, {3, 8, 12, 13, 18, 19}, {9, 11, 12, 13, 14, 17}, {9, 13, 14, 17, 18, 23}, {7, 11, 12, 13, 18, 19}, {9, 11, 12, 13, 14, 16}, {10, 12, 13, 14, 15, 20}, {2, 7, 11, 12, 13, 16}, {14, 17, 18, 19, 20, 22}, {8, 12, 13, 14, 15, 18}, {15, 17, 18, 19, 20, 24}, {13, 17, 18, 19, 24, 25}, {12, 17, 18, 19, 20, 22}, {4, 5, 7, 8, 9, 14}, {9, 14, 18, 19, 24, 25}, {2, 7, 8, 11, 12, 17}, {12, 17, 18, 19, 20, 23}, {3, 8, 9, 13, 17, 18}, {6, 7, 12, 13, 14, 19}, {9, 14, 15, 18, 19, 24}, {10, 14, 15, 18, 19, 23}, {7, 12, 16, 17, 18, 22}, {12, 16, 17, 18, 19, 22}, {15, 17, 18, 19, 20, 22}, {3, 8, 13, 14, 17, 18}, {4, 8, 9, 12, 13, 18}, {7, 11, 12, 13, 18, 19}, {8, 11, 12, 13, 14, 17}, {7, 12, 16, 17, 18, 22}, {14, 17, 18, 19, 21, 22}, {1, 2, 7, 8, 13, 14}, {12, 16, 17, 18, 19, 23}, {2, 6, 7, 8, 9, 12}, {7, 11, 12, 13, 14, 19}, {5, 7, 8, 9, 10, 12}, {5, 7, 8, 9, 10, 15}, {4, 7, 8, 9, 14, 15}, {9, 12, 13, 14, 19, 20}, {3, 4, 9, 14, 15, 19}, {8, 13, 14, 17, 18, 23}, {2, 7, 8, 11, 12, 17}, {2, 7, 8, 13, 18, 19}, {8, 12, 13, 14, 18, 23}, {5, 10, 14, 15, 19, 24}, {4, 7, 8, 9, 10, 15}, {14, 15, 17, 18, 19, 23}, {8, 9, 14, 15, 19, 24}, {3, 7, 8, 9, 10, 13}, {9, 14, 15, 18, 19, 24}, {5, 7, 8, 9, 10, 15}, {7, 12, 17, 21, 22, 23}, {3, 8, 9, 14, 15, 20}, {12, 13, 18, 19, 20, 24}, {8, 13, 17, 18, 19, 23}, {7, 12, 13, 14, 16, 17}, {9, 13, 14, 19, 24, 25}, {8, 12, 13, 14, 16, 17}, {4, 9, 13, 14, 15, 18}, {1, 2, 7, 8, 12, 17}, {8, 9, 11, 12, 13, 16}, {9, 14, 18, 19, 20, 23}, {12, 17, 18, 19, 20, 24}, {3, 8, 12, 13, 17, 22}, {9, 14, 19, 20, 23, 24}, {2, 7, 8, 13, 14, 18}, {9, 14, 19, 23, 24, 25}, {8, 9, 11, 12, 13, 18}, {2, 7, 11, 12, 13, 17}, {1, 6, 11, 12, 17, 22}, {1, 6, 7, 12, 13, 17}, {7, 8, 12, 17, 21, 22}, {14, 16, 17, 18, 19, 24}, {12, 16, 17, 18, 19, 22}, {7, 12, 13, 16, 17, 22}, {8, 13, 14, 17, 18, 23}, {9, 13, 14, 19, 20, 24}, {2, 7, 8, 12, 16, 17}, {3, 4, 6, 7, 8, 13}, {4, 5, 7, 8, 9, 14}, {2, 6, 7, 8, 9, 13}, {7, 12, 16, 17, 22, 23}, {2, 6, 7, 8, 12, 17}, {7, 11, 12, 17, 18, 22}, {9, 10, 14, 18, 19, 23}, {8, 9, 13, 17, 18, 23}, {14, 16, 17, 18, 19, 23}, {7, 8, 12, 17, 21, 22}, {14, 16, 17, 18, 19, 23}, {10, 12, 13, 14, 15, 18}, {3, 6, 7, 8, 9, 12}, {6, 11, 12, 13, 17, 22}, {8, 11, 12, 13, 17, 22}, {8, 12, 13, 14, 15, 19}, {8, 9, 11, 12, 13, 16}, {7, 12, 16, 17, 18, 21}, {8, 11, 12, 13, 17, 22}, {8, 13, 14, 19, 24, 25}, {3, 8, 9, 10, 12, 13}, {6, 11, 12, 13, 17, 22}, {7, 12, 13, 14, 15, 17}, {14, 15, 17, 18, 19, 23}, {4, 9, 10, 13, 14, 18}, {8, 9, 10, 11, 12, 13}, {2, 7, 8, 13, 18, 19}, {3, 8, 12, 13, 18, 19}, {9, 13, 14, 19, 20, 24}, {2, 7, 12, 13, 18, 23}, {6, 11, 12, 17, 18, 23}, {4, 9, 14, 18, 19, 20}, {9, 12, 13, 14, 15, 17}, {6, 11, 12, 13, 14, 16}, {9, 13, 14, 15, 19, 24}, {14, 17, 18, 19, 20, 25}, {14, 17, 18, 19, 21, 22}, {7, 12, 13, 18, 19, 23}, {3, 4, 9, 14, 15, 20}, {2, 6, 7, 8, 9, 13}, {3, 4, 6, 7, 8, 13}, {8, 13, 14, 15, 19, 24}, {13, 16, 17, 18, 19, 22}, {8, 9, 11, 12, 13, 17}, {2, 6, 7, 12, 13, 17}, {1, 2, 7, 8, 9, 12}, {3, 8, 9, 12, 13, 17}, {2, 3, 8, 9, 10, 14}, {13, 16, 17, 18, 19, 23}, {8, 11, 12, 13, 14, 17}, {2, 7, 8, 11, 12, 16}, {6, 7, 12, 13, 14, 17}, {13, 18, 19, 20, 22, 23}, {10, 12, 13, 14, 15, 20}, {1, 6, 11, 12, 17, 22}, {2, 7, 8, 11, 12, 17}, {8, 9, 14, 19, 20, 25}, {2, 7, 11, 12, 17, 18}, {2, 6, 7, 12, 17, 18}, {8, 13, 18, 19, 22, 23}, {4, 8, 9, 10, 14, 19}, {9, 14, 18, 19, 20, 23}, {4, 6, 7, 8, 9, 12}, {14, 17, 18, 19, 20, 24}, {4, 8, 9, 10, 14, 19}, {2, 6, 7, 8, 9, 12}, {13, 17, 18, 19, 20, 25}, {7, 11, 12, 17, 22, 23}, {3, 6, 7, 8, 9, 13}, {7, 11, 12, 13, 14, 17}, {7, 11, 12, 13, 14, 19}, {4, 9, 13, 14, 18, 23}, {7, 12, 13, 16, 17, 22}, {4, 8, 9, 12, 13, 17}, {11, 12, 17, 18, 19, 23}, {12, 16, 17, 18, 19, 23}, {3, 8, 12, 13, 14, 19}, {1, 2, 3, 7, 12, 17}, {8, 9, 11, 12, 13, 18}, {7, 12, 13, 16, 17, 21}, {7, 11, 12, 13, 17, 22}, {2, 7, 11, 12, 13, 18}, {12, 16, 17, 18, 19, 23}, {7, 11, 12, 17, 18, 22}, {3, 8, 9, 10, 12, 13}, {7, 8, 9, 13, 18, 23}, {2, 7, 8, 9, 13, 18}, {11, 12, 13, 18, 19, 20}, {7, 12, 13, 14, 15, 20}, {4, 7, 8, 9, 10, 12}, {2, 7, 11, 12, 13, 17}, {9, 11, 12, 13, 14, 18}, {9, 11, 12, 13, 14, 17}, {9, 11, 12, 13, 14, 19}, {3, 8, 12, 13, 18, 19}, {4, 8, 9, 13, 17, 18}, {12, 13, 18, 19, 24, 25}, {7, 11, 12, 13, 17, 22}, {13, 18, 19, 20, 22, 23}, {6, 11, 12, 13, 17, 22}, {4, 5, 7, 8, 9, 13}, {8, 9, 10, 11, 12, 13}, {8, 9, 11, 12, 13, 17}, {8, 9, 10, 14, 19, 24}, {9, 14, 19, 23, 24, 25}, {13, 17, 18, 19, 20, 23}, {9, 14, 18, 19, 20, 24}, {11, 12, 17, 18, 19, 23}, {13, 16, 17, 18, 19, 23}, {7, 12, 16, 17, 18, 21}, {7, 8, 9, 13, 18, 23}, {5, 8, 9, 10, 12, 13}, {5, 7, 8, 9, 10, 12}, {2, 6, 7, 12, 13, 17}, {9, 12, 13, 14, 15, 19}, {14, 16, 17, 18, 19, 23}, {7, 12, 17, 18, 21, 22}, {16, 17, 18, 23, 24, 25}, {8, 12, 13, 14, 15, 17}, {8, 13, 17, 18, 19, 23}, {2, 3, 8, 9, 10, 14}, {4, 9, 14, 15, 18, 19}, {15, 17, 18, 19, 20, 25}, {4, 9, 10, 14, 18, 19}, {13, 16, 17, 18, 19, 23}, {7, 12, 13, 16, 17, 21}, {9, 11, 12, 13, 14, 19}, {3, 7, 8, 9, 10, 14}, {4, 8, 9, 14, 15, 19}, {2, 7, 12, 13, 16, 17}, {6, 7, 8, 13, 14, 15}, {7, 8, 12, 16, 17, 22}, {8, 9, 14, 15, 19, 24}, {3, 8, 13, 14, 17, 18}, {7, 12, 13, 14, 15, 19}, {15, 17, 18, 19, 20, 24}, {3, 6, 7, 8, 9, 11}, {3, 7, 8, 11, 12, 16}, {8, 12, 13, 16, 17, 22}, {6, 7, 8, 12, 17, 22}, {11, 16, 17, 18, 19, 21}, {8, 13, 14, 15, 19, 24}, {2, 6, 7, 12, 17, 18}, {7, 8, 13, 18, 23, 24}, {3, 6, 7, 8, 9, 13}, {4, 7, 8, 9, 13, 18}, {13, 17, 18, 19, 20, 24}, {3, 7, 8, 9, 10, 12}, {1, 6, 7, 12, 13, 17}, {3, 8, 9, 14, 15, 19}, {13, 16, 17, 18, 19, 22}, {12, 16, 17, 18, 19, 22}, {3, 7, 8, 9, 10, 14}, {14, 17, 18, 19, 20, 22}, {7, 11, 12, 17, 18, 22}, {1, 6, 7, 12, 13, 17}, {1, 6, 7, 8, 12, 17}, {2, 6, 7, 12, 17, 18}, {3, 7, 8, 9, 10, 15}, {7, 12, 13, 16, 17, 21}, {4, 6, 7, 8, 9, 13}, {1, 6, 7, 8, 9, 11}, {2, 7, 11, 12, 13, 17}, {13, 17, 18, 19, 20, 24}, {9, 13, 14, 15, 19, 24}, {8, 9, 13, 17, 18, 23}, {4, 9, 10, 13, 14, 19}, {1, 2, 3, 7, 12, 17}, {9, 10, 14, 18, 19, 24}, {7, 12, 13, 14, 15, 18}, {4, 5, 9, 14, 18, 19}, {3, 4, 7, 8, 11, 12}, {8, 13, 18, 22, 23, 24}, {1, 6, 7, 12, 17, 18}, {2, 7, 8, 9, 10, 13}, {3, 6, 7, 8, 9, 14}, {8, 9, 14, 19, 20, 24}, {4, 6, 7, 8, 9, 11}, {9, 12, 13, 14, 18, 23}, {1, 2, 7, 8, 9, 13}, {8, 9, 13, 18, 22, 23}, {2, 3, 8, 13, 14, 18}, {1, 2, 7, 12, 13, 18}, {14, 16, 17, 18, 19, 22}, {1, 2, 7, 12, 13, 17}, {6, 7, 12, 13, 14, 18}, {3, 6, 7, 8, 12, 17}, {3, 7, 8, 11, 12, 17}, {1, 6, 7, 12, 13, 18}, {8, 12, 13, 17, 21, 22}, {4, 5, 8, 9, 14, 19}, {1, 6, 7, 8, 9, 12}, {7, 8, 11, 12, 17, 22}, {13, 16, 17, 18, 19, 21}, {6, 7, 12, 17, 18, 23}, {8, 11, 12, 13, 14, 19}, {2, 7, 11, 12, 16, 21}, {5, 9, 10, 14, 18, 19}, {4, 9, 14, 15, 20, 25}, {7, 8, 12, 16, 17, 21}, {9, 10, 12, 13, 14, 18}, {9, 10, 12, 13, 14, 19}, {2, 3, 7, 11, 12, 17}, {9, 13, 14, 17, 18, 22}, {7, 11, 12, 17, 18, 23}, {13, 14, 16, 17, 18, 23}, {14, 17, 18, 19, 24, 25}, {8, 13, 14, 15, 17, 18}, {7, 11, 12, 13, 14, 16}, {3, 8, 13, 14, 19, 24}, {2, 3, 8, 9, 13, 18}, {3, 4, 7, 8, 13, 18}, {4, 5, 9, 13, 14, 18}, {4, 9, 13, 14, 15, 20}, {2, 3, 8, 13, 14, 19}, {2, 7, 12, 16, 17, 18}, {10, 14, 15, 19, 23, 24}, {5, 7, 8, 9, 10, 14}, {3, 4, 6, 7, 8, 11}, {2, 7, 8, 13, 14, 19}, {3, 6, 7, 8, 13, 14}, {7, 12, 13, 14, 18, 23}, {7, 12, 13, 18, 23, 24}, {2, 3, 8, 13, 18, 19}, {14, 16, 17, 18, 19, 21}, {11, 12, 17, 18, 19, 22}, {8, 9, 12, 13, 18, 23}, {3, 4, 5, 6, 7, 8}, {4, 5, 9, 13, 14, 19}, {8, 9, 12, 13, 16, 17}, {8, 11, 12, 13, 14, 16}, {2, 7, 8, 9, 10, 15}, {4, 7, 8, 9, 11, 12}, {5, 9, 10, 13, 14, 18}, {13, 16, 17, 18, 19, 24}, {14, 18, 19, 20, 22, 23}, {8, 12, 13, 14, 19, 20}, {3, 8, 13, 17, 18, 19}, {3, 4, 9, 14, 19, 20}, {14, 15, 17, 18, 19, 24}, {7, 8, 13, 18, 19, 23}, {2, 3, 8, 9, 14, 15}, {6, 7, 12, 17, 18, 22}, {3, 4, 8, 12, 13, 17}, {13, 14, 16, 17, 18, 21}, {2, 3, 7, 12, 16, 17}, {2, 6, 7, 8, 9, 14}, {13, 14, 16, 17, 18, 22}, {2, 3, 7, 11, 12, 16}, {2, 7, 8, 9, 11, 12}, {3, 4, 5, 9, 14, 19}, {8, 9, 13, 17, 18, 22}, {10, 12, 13, 14, 15, 17}, {7, 8, 13, 14, 15, 18}, {7, 12, 13, 14, 19, 20}, {2, 6, 7, 8, 9, 11}, {7, 8, 13, 18, 19, 24}, {3, 7, 8, 9, 14, 15}, {11, 16, 17, 18, 23, 24}, {7, 8, 13, 14, 15, 19}, {2, 3, 4, 8, 13, 18}, {8, 13, 14, 17, 18, 22}, {7, 12, 13, 18, 19, 24}, {9, 13, 14, 18, 22, 23}, {9, 10, 13, 14, 17, 18}, {9, 10, 14, 19, 23, 24}, {12, 13, 18, 19, 20, 25}, {6, 11, 12, 13, 14, 19}, {8, 13, 14, 18, 22, 23}, {3, 7, 8, 12, 16, 17}, {1, 2, 7, 8, 9, 14}, {8, 12, 13, 16, 17, 21}, {6, 7, 12, 13, 18, 19}, {9, 12, 13, 14, 16, 17}, {15, 18, 19, 20, 22, 23}, {5, 7, 8, 9, 10, 13}, {6, 11, 12, 13, 18, 19}, {8, 13, 14, 19, 20, 25}, {8, 9, 14, 19, 24, 25}, {12, 17, 18, 19, 24, 25}, {1, 2, 3, 8, 9, 10}, {11, 12, 17, 18, 19, 24}, {3, 8, 9, 14, 19, 20}, {11, 12, 17, 18, 23, 24}]

def fold_cube(num_list):
	return set(num_list) in db


U = 1
F = 2
R = 3

def fold_cube(num_list):
	print(num_list)
	nums = set(num_list)
	faces = set()
	print(nums)
	try:
		face = Face(nums.pop(), nums, faces)
	except:
		return False
	faces.add(face)
	fs = set((f.z for f in faces))
	return fs == {1, 2, 3, -1, -2, -3}

class Face:

	def __init__(self, val, nums, faces, x=F, y=R, z=U):
		self.nums = nums
		self.val = val
		self.z = z
		self.x = x
		self.y = y
		for num in self.borders():
			self.nums.remove(num)
			(z, x, y) = self.fold(num)
			faces.add(Face(num, self.nums, faces, x=x, y=y, z=z))

	def borders(self):
		borders = []
		ds = (self.val - 1) // 5
		ms = (self.val - 1) % 5
		for num in self.nums:
			dn = (num - 1) // 5
			mn = (num - 1) % 5
			if ds - dn == 0 and abs(ms - mn) == 1 or (ms - mn == 0 and abs(ds - dn) == 1):
				borders.append(num)
		return borders

	def fold(self, num):
		relation = self.val - num
		if relation == 5:
			z = self.x
			x = -self.z
			y = self.y
		elif relation == -5:
			z = -self.x
			x = self.z
			y = self.y
		elif relation == 1:
			z = -self.y
			x = self.x
			y = self.z
		elif relation == -1:
			z = self.y
			x = self.x
			y = -self.z
		return (z, x, y)

	def __hash__(self):
		return self.val

	def __eq__(self, other):
		return self.val == other.val


def fold_cube(number_list):
	print(number_list)
	connections = 0
	number_list.sort()
	for i in range(len(number_list)):
		if number_list[i] - 1 in number_list and number_list[i] not in [6, 11, 16, 21]:
			connections += 1
	for j in range(len(number_list)):
		if number_list[j] - 5 in number_list:
			connections += 1
	if connections != 5:
		return False
	horizontal = [[1, 2, 3, 4, 5], [6, 7, 8, 9, 10], [11, 12, 13, 14, 15], [16, 17, 18, 19, 20], [21, 22, 23, 24, 25]]
	vertical = [[1, 6, 11, 16, 21], [2, 7, 12, 17, 22], [3, 8, 13, 18, 23], [4, 9, 14, 19, 24], [5, 10, 15, 20, 25]]
	lenghts_horizontal = []
	height = 0
	for h in horizontal:
		lh = 1
		any_in_row = False
		for i in range(len(number_list)):
			if number_list[i] in h:
				any_in_row = True
			if number_list[i] - 1 in number_list and number_list[i] in h and (number_list[i] not in [6, 11, 16, 21]):
				lh += 1
		if lh > 1:
			lenghts_horizontal.append(lh)
		elif any_in_row:
			lenghts_horizontal.append(1)
		if any_in_row:
			height += 1
	if lenghts_horizontal[0] == 4 or lenghts_horizontal[-1] == 4:
		return False
	is_3_horizontal_fist = False
	if lenghts_horizontal[0] == 3 or lenghts_horizontal[-1] == 3:
		is_3_horizontal_fist = True
	lenghts_horizontal.sort(reverse=True)
	if lenghts_horizontal[0] > 4:
		return False
	width = 0
	lenghts_vertical = []
	for v in vertical:
		lv = 1
		any_in_row = False
		for i in range(len(number_list)):
			if number_list[i] in v:
				any_in_row = True
			if number_list[i] - 5 in number_list and number_list[i] in v:
				lv += 1
		if lv > 1:
			lenghts_vertical.append(lv)
		elif any_in_row:
			lenghts_vertical.append(1)
		if any_in_row:
			width += 1
	if height == 3 and width == 3:
		return False
	if lenghts_vertical[0] == 4 or lenghts_vertical[-1] == 4:
		return False
	is_3_vertical_first = False
	if lenghts_vertical[0] == 3 or lenghts_vertical[-1] == 3:
		is_3_vertical_first = True
	lenghts_vertical.sort(reverse=True)
	if is_3_vertical_first and height == 4:
		return False
	if is_3_horizontal_fist and width == 4:
		return False
	if lenghts_vertical[0] > 4:
		return False
	if lenghts_vertical[0] == 3 and lenghts_vertical[1] == 3 and (width == 2):
		return True
	if lenghts_horizontal[0] == 3 and lenghts_horizontal[1] == 3 and (height == 2):
		return True
	upper = False
	upper_list = [1, 2, 3, 4, 5]
	lower = False
	lower_list = [21, 22, 23, 24, 25]
	left = False
	left_list = [1, 6, 11, 16, 21]
	right = False
	right_list = [5, 10, 15, 20, 25]
	for n in number_list:
		if n in upper_list:
			upper = True
		if n in lower_list:
			lower = True
		if n in left_list:
			left = True
		if n in right_list:
			right = True
	if upper and lower:
		return False
	if right and left:
		return False
	if lenghts_vertical == [2, 2, 1, 1] and lenghts_horizontal == [3, 2, 1] and (width == 3) and (height == 4):
		return False
	if (lenghts_vertical[0] == 3 and width == 4 and (height != 2) or (lenghts_horizontal[0] == 3 and height == 4 and (width != 2))) and lenghts_vertical != [3, 1, 1, 1]:
		return True
	if lenghts_vertical[0] == 4 and width == 3 and (height != 2) or (lenghts_horizontal[0] == 4 and height == 3 and (width != 2)):
		return True
	if lenghts_vertical[0] == 3 and height == 4 and (width != 2) or (lenghts_horizontal[0] == 3 and width == 4 and (height != 2)):
		return True
	if lenghts_vertical[0] == 4 and height == 3 and (width != 2) or (lenghts_horizontal[0] == 4 and width == 3 and (height != 2)):
		return True
	if lenghts_vertical == [2, 2, 2] and lenghts_horizontal == [2, 2, 1, 1]:
		return True
	if lenghts_horizontal == [2, 2, 2] and lenghts_vertical == [2, 2, 1, 1]:
		return True
	return False