Perhaps it would be convenient to solve first ```Probabilities for Sums in Rolling Cubic Dice``` see at: http://www.codewars.com/kata/probabilities-for-sums-in-rolling-cubic-dice

Suppose that we roll dice that we never used, a tetrahedral die that has only 4 sides(values 1 to 4), or perhaps better the dodecahedral one with 12 sides and 12 values (from 1 to 12), or look! an icosahedral die with 20 sides and 20 values (from 1 to 20).

Let's roll dice of same type and try to see the distribution of the probability for all the possible sum values.

Suppose that we roll 3 tetrahedral dice:
```
Sum   Prob    Amount of hits           Combinations
3   0.015625     1             {1,1,1}
4   0.046875     3             {1,2,1},{2,1,1},{1,1,2}
5   0.09375      6             {3,1,1},{1,1,3},{1,2,2},{1,3,1},{2,2,1},{2,1,2}
6   0.15625     10             {2,2,2},{1,3,2},{3,2,1},{1,4,1},{2,1,3},{1,1,4},{4,1,1},{3,1,2},{1,2,3},{2,3,1}
7   0.1875      12             {2,3,2},{1,2,4},{2,1,4},{1,4,2},{3,2,2}, {4,2,1},{3,3,1},{3,1,3},{2,2,3},{4,1,2},{2,4,1},{1,3,3}
8   0.1875      12             {2,3,3},{3,3,2},{2,2,4},{4,3,1},{2,4,2},{4,1,3},{4,2,2},{3,4,1},{1,3,4},{3,2,3},{3,1,4},{1,4,3}
9   0.15625     10             {3,3,3},{4,4,1},{2,4,3},{4,2,3},{4,3,2},{1,4,4},{4,1,4},{3,4,2},{2,3,4},{3,2,4}
10  0.09375      6             {3,4,3},{4,4,2},{3,3,4},{4,2,4},(2,4,4),{4,3,3}
11  0.046875     3             {4,3,4},{3,4,4},{4,4,3}
12  0.015625     1             {4,4,4}
tot: 1.0000
```
Note that the total sum of all the probabilities for each end has to be ```1.0000```

The register of hits per each sum value will be as follows:
```
[[3, 1], [4, 3], [5, 6], [6, 10], [7, 12], [8, 12], [9, 10], [10, 6], [11, 3], [12, 1]]
```
Create the function ```reg_sum_hits()``` that receive two arguments:

- number of dice, ```n```
- number of sides of the die, ```s```

The function will output a list of pairs with the sum value and the corresponding hits for it.

For the example given above:
```python
reg_sum_hits(3, 4) == [[3, 1], [4, 3], [5, 6], [6, 10], [7, 12], [8, 12], [9, 10], [10, 6], [11, 3], [12, 1]]
```
You will find more examples in the Example Test Cases.

Assumption: If the die has n sides, the values of it are from ```1``` to ```n```

```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 = 'from itertools import product\n\ndef reg_sum_hits(n, s):\n    pairs = [[i, 0] for i in range(n, n*s+1)]\n\n    for i in product(range(1, s+1), repeat=n):\n        sum = 0\n        for j in i:\n            sum += j\n        index = sum - n\n        pairs[index][1] += 1\n\n    return pairs'
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

_target = None
_fn_name = 'reg_sum_hits'
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: 6888