# 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)

```python

USER_FILENAME = "<user_code>"
import sys, json, builtins
import math, heapq, bisect, itertools, functools, collections, statistics, fractions, decimal, operator, array, random, re, string, os, pathlib
try:
    import resource as _res
except Exception:
    _res = None
from collections import Counter, defaultdict, deque, OrderedDict
from functools import lru_cache, reduce
from itertools import accumulate, permutations, combinations, product
from math import gcd, ceil, floor, sqrt
from decimal import Decimal
try:
    from typing import *
except Exception:
    pass
try:
    sys.setrecursionlimit(1_000_000)
except Exception:
    pass
try:
    import site as _site
    try:
        _sys_sites = _site.getsitepackages() if hasattr(_site, "getsitepackages") else []
        for _p in _sys_sites:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
    try:
        _usp = _site.getusersitepackages()
        if isinstance(_usp, str):
            _usp = [_usp]
        for _p in _usp:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
except Exception:
    pass

def _disabled_input(*args, **kwargs):
    raise RuntimeError("input() is disabled in callable mode")
builtins.input = _disabled_input

ns = {"__name__": "__judge__"}  # prevent if __name__ == "__main__" blocks
for _k, _v in list(globals().items()):
    if not _k.startswith("_"):
        ns[_k] = _v
_CODE = 'U = 1\nF = 2\nR = 3\n\n\ndef fold_cube(num_list):\n    print(num_list)\n    nums = set(num_list)\n    faces = set()\n    print(nums)\n    try:\n        face = Face(nums.pop(), nums, faces)\n    except:\n        return False\n    faces.add(face)\n#     print(face.val)\n#     print(face.borders())\n#     for face in faces:\n#         print(face.val, face.z)\n#     return sum(f.z for f in faces) == 0\n    fs = set(f.z for f in faces)\n    return fs == {1, 2, 3, -1, -2, -3}\n\n\nclass Face:\n    def __init__(self, val, nums, faces, x=F, y=R, z=U):\n#         print(val, nums)\n        self.nums = nums\n        self.val = val\n        self.z = z\n        self.x = x\n        self.y = y\n\n        for num in self.borders():\n            self.nums.remove(num)\n            z, x, y = self.fold(num)\n#             print(z, x, y)\n            faces.add(Face(num, self.nums, faces, x=x, y=y, z=z))\n\n    def borders(self):\n        borders = []\n        ds = (self.val - 1) // 5\n        ms = (self.val - 1) % 5\n        for num in self.nums:\n            dn = (num - 1) // 5\n            mn = (num - 1) % 5\n#             print(ds, ms, dn, mn)\n            if ds - dn == 0 and abs(ms - mn) == 1 or ms - mn == 0 and abs(ds - dn) == 1:\n                borders.append(num)\n#             if abs(self.val - num) in [1, 5]:\n#                 borders.append(num)\n#         print(borders)\n        return borders\n\n    def fold(self, num):\n        relation = self.val - num\n        if relation == 5:\n            z = self.x\n            x = -self.z\n            y = self.y\n        elif relation == -5:\n            z = -self.x\n            x = self.z\n            y = self.y\n        elif relation == 1:\n            z = -self.y\n            x = self.x\n            y = self.z\n        elif relation == -1:\n            z = self.y\n            x = self.x\n            y = -self.z\n        return z, x, y\n\n    def __hash__(self):\n        return self.val\n\n    def __eq__(self, other):\n        return self.val == other.val\n'
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

_target = None
_fn_name = 'fold_cube'
if _fn_name:
    if "Solution" in ns:
        try:
            _obj = ns["Solution"]()
            if hasattr(_obj, _fn_name):
                _target = getattr(_obj, _fn_name)
        except Exception:
            _target = None
    if _target is None and _fn_name in ns and callable(ns[_fn_name]):
        _target = ns[_fn_name]
# Fallback: if there's exactly one user-defined function, use it
if _target is None:
    _cands = [v for k, v in ns.items() if callable(v) and getattr(v, "__module__", "") in ("__judge__", "__main__")]
    _user_funcs = [c for c in _cands if getattr(c, "__name__", "").startswith("_") is False]
    if len(_user_funcs) == 1:
        _target = _user_funcs[0]
if _target is None:
    raise RuntimeError(f"Could not resolve callable '{_fn_name}'")

_raw = sys.stdin.read().strip()
if _raw:
    _parsed = json.loads(_raw)
    if isinstance(_parsed, dict):
        _kwargs = _parsed.get("kwargs", {})
        _args = _parsed.get("args", [])
    else:
        _args, _kwargs = _parsed, {}
else:
    _args, _kwargs = [], {}
if not isinstance(_args, (list, tuple)):
    _args = [_args]
_res = _target(*_args, **_kwargs)
try:
    sys.stdout.write(repr(_res) + "\n")
except Exception:
    pass

```

Measured memory_bytes: 12226