# 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 = '# Today I shall be mostly using geometry, group theory, and massive overkill to solve a problem... \n\n# These are the archetypes of the nets from a cube\nnets = [\n    [\n        [1, 0, 0, 0],\n        [1, 1, 1, 1],\n        [1, 0, 0, 0],\n    ],\n    [\n        [1, 0, 0, 0],\n        [1, 1, 1, 1],\n        [0, 1, 0, 0],\n    ],\n    [\n        [1, 0, 0, 0],\n        [1, 1, 1, 1],\n        [0, 0, 1, 0],\n    ],\n    [\n        [1, 0, 0, 0],\n        [1, 1, 1, 1],\n        [0, 0, 0, 1],\n    ],\n    [\n        [0, 1, 0, 0],\n        [1, 1, 1, 1],\n        [0, 0, 1, 0],\n    ],\n    [\n        [0, 1, 0, 0],\n        [1, 1, 1, 1],\n        [0, 1, 0, 0],\n    ],\n    [\n        [1, 1, 0, 0],\n        [0, 1, 1, 1],\n        [0, 1, 0, 0],\n    ],\n    [\n        [1, 1, 0, 0],\n        [0, 1, 1, 1],\n        [0, 0, 1, 0],\n    ],\n    [\n        [1, 0, 0, 0],\n        [1, 1, 1, 0],\n        [0, 0, 1, 1],\n    ],\n    [\n        [1, 1, 0, 0],\n        [0, 1, 1, 0],\n        [0, 0, 1, 1],\n    ],\n    [\n        [1, 1, 1, 0, 0],\n        [0, 0, 1, 1, 1],\n    ],\n]\n\n# Various transformations we can apply to the nets...\ndef h_mirror(m):\n    # Mirror a matrix horizontally\n    return m[::-1]\n\ndef v_mirror(m):\n    # Mirror a matrix vertically\n    return [r[::-1] for r in m]\n    \ndef transpose(m):\n    # Transpose a matrix\n    return list(map(list, zip(*m)))\n\ndef rotate_cw(m):\n    # Rotate a matrix clock-wise\n    coords = [(x, y) for y, row in enumerate(m) for x, v in enumerate(row) if v == 1]\n    new_coords = normalise([(y, -x) for x, y in coords])\n    result = [[0 for _ in range(len(m))] for _ in range(len(m[0]))]\n    for x, y in new_coords:\n        result[y][x] = 1\n    return result\n\ndef rotate_ccw(m):\n    # Rotate a matrix counter lock-wise\n    coords = [(x, y) for y, row in enumerate(m) for x, v in enumerate(row) if v == 1]\n    new_coords = normalise([(-y, x) for x, y in coords])\n    result = [[0 for _ in range(len(m))] for _ in range(len(m[0]))]\n    for x, y in new_coords:\n        result[y][x] = 1\n    return result\n\ndef identity(coords):\n    # No transformation\n    return coords\n\ndef normalise(coords):\n    # Translate all coordinates to that all x>=0 and y>=0\n    min_x = min(x for x, y in coords)\n    min_y = min(y for x, y in coords)\n    return tuple([(x - min_x, y - min_y) for x, y in coords])\n\ndef compose(*fns):\n    # Compose transformations\n    def apply(*args, **kwargs):\n        result = fns[0](*args, **kwargs)\n        for fn in fns[1:]: \n            result = fn(result)\n        return result\n    return apply\n\n# Transformations needed to generate all the symmetries of the nets\ncompositions = [\n    identity,\n    h_mirror,\n    v_mirror,\n    transpose,\n    rotate_cw,\n    rotate_ccw,\n    compose(rotate_cw, rotate_cw),\n    compose(rotate_cw, rotate_cw, transpose),\n]\n\n# Build all possible transformations of each net\nsolutions = []\nfor net in nets:\n    for composition in compositions:\n        net_prime = composition(net)\n        if net_prime not in solutions:\n            solutions.append(net_prime)\n\ndef to_relative(m):\n    # Find the coordinates of the 1s in the supplied matrix\n    return tuple(sorted([(x, y) for y, row in enumerate(m) for x, v in enumerate(row) if v == 1]))\n\n# All possible solutions using a relative coordinate system\nrelative_coordinate_solutions = set(to_relative(s) for s in solutions)\n\ndef fold_cube(number_list):\n    # Convert from numbered cells to relative coordinates\n    relative_coordinates = normalise(sorted([((n - 1) // 5, (n - 1) % 5) for n in number_list]))\n    # See if we have a known solution\n    return relative_coordinates in relative_coordinate_solutions'
_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: 6622