# Task
 Consider a `bishop`, a `knight` and a `rook` on an `n × m` chessboard. They are said to form a `triangle` if each piece attacks exactly one other piece and is attacked by exactly one piece. 
 
 Calculate the number of ways to choose positions of the pieces to form a triangle.

 Note that the bishop attacks pieces sharing the common diagonal with it; the rook attacks in horizontal and vertical directions; and, finally, the knight attacks squares which are two squares horizontally and one square vertically, or two squares vertically and one square horizontally away from its position.

 ![](https://codefightsuserpics.s3.amazonaws.com/tasks/chessTriangle/img/moves.png?_tm=1473934712872)

# Example

 For `n = 2 and m = 3`, the output should be `8`.

 ![](https://codefightsuserpics.s3.amazonaws.com/tasks/chessTriangle/img/combinations.png?_tm=1473934713038)

# Input/Output


- `[input]` integer `n`

    Constraints: `1 ≤ n ≤ 40.`


 - `[input]` integer `m`

    Constraints: `1 ≤ m ≤ 40, 3 ≤ n x m`.


 - `[output]` an integer

```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 = 'clamp  = lambda moves, w, h: {(x, y) for x, y in moves if 0 <= x < w and 0 <= y < h}\nknight = lambda x, y: {(x+u, y+v) for u in range(-2, 3) for v in range(-2, 3) if u*u + v*v == 5}\nrook   = lambda x, y, w, h: {(x, v) for v in range(h)} | {(u, y) for u in range(w)} - {(x, y)}\nbishop = lambda x, y, w, h: {(x+d*k, y+k) for d in (-1,1) for k in range(-max(x, y), max(w-x, h-y))} - {(x, y)}\n\ndef search_triangle(x, y, w, h):\n    return sum(\n        len(rook(x, y, w, h) & clamp(bishop(u, v, w, h), w, h)) +\n        len(rook(u, v, w, h) & clamp(bishop(x, y, w, h), w, h))\n        for u, v in clamp(knight(x, y), w, h)\n    )\n\ndef chess_triangle(h, w):\n    wm, hm = min(6, w), min(6, h)\n    return (\n        sum(search_triangle(min(x, wm-x-1), min(y, hm-y-1), w, h) for x in range(wm) for y in range(hm))\n        + max(0, w-6) * sum(search_triangle(3, min(y, hm-y-1), w, h) for y in range(hm))\n        + max(0, h-6) * sum(search_triangle(min(x, wm-x-1), 3, w, h) for x in range(wm))\n        + max(0, h-6) * max(0, w-6) * search_triangle(3, 3, w, h)\n    )'
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

_target = None
_fn_name = 'chess_triangle'
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: 12508