Alice and Bob take turns playing a game, with Alice starting first.
Initially, there are n stones in a pile.  On each player's turn, that player makes a move consisting of removing any non-zero square number of stones in the pile.
Also, if a player cannot make a move, he/she loses the game.
Given a positive integer n. Return True if and only if Alice wins the game otherwise return False, assuming both players play optimally.
 
Example 1:
Input: n = 1
Output: true
Explanation: Alice can remove 1 stone winning the game because Bob doesn't have any moves.
Example 2:
Input: n = 2
Output: false
Explanation: Alice can only remove 1 stone, after that Bob removes the last one winning the game (2 -> 1 -> 0).
Example 3:
Input: n = 4
Output: true
Explanation: n is already a perfect square, Alice can win with one move, removing 4 stones (4 -> 0).

Example 4:
Input: n = 7
Output: false
Explanation: Alice can't win the game if Bob plays optimally.
If Alice starts removing 4 stones, Bob will remove 1 stone then Alice should remove only 1 stone and finally Bob removes the last one (7 -> 3 -> 2 -> 1 -> 0). 
If Alice starts removing 1 stone, Bob will remove 4 stones then Alice only can remove 1 stone and finally Bob removes the last one (7 -> 6 -> 2 -> 1 -> 0).
Example 5:
Input: n = 17
Output: false
Explanation: Alice can't win the game if Bob plays optimally.

 
Constraints:

1 <= n <= 10^5

```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 = 'class Solution:\n    \n    def winnerSquareGame2(self, n):\n        dp = [False] * (n + 1)\n        for i in range(1, n + 1):\n            dp[i] = not all(dp[i - k * k] for k in range(1, int(i**0.5) + 1))\n        return dp[-1]\n    \n    def winnerSquareGame(self, n: int) -> bool:\n        # 尝试同样的解法？\n        # solve(s), s=当前的剩余的数字下，score diff? 需要的最小步数？\n        # 如果s=0，则返回False\n        # 如果s=任意一个平方数，则return True, 因为alice可以都拿走\n        # 否则，alice可以从中切走任意大小的平方数x，然后把剩下的s-x扔给bob\n        \n        # 只要其中有一个切割，bob无法在s-x下获胜，那就是alice获胜\n        # 如果bob在所有切割下都获胜，那alice lose\n        \n        # 只从alice的角度出发，是否足够？\n        cache = dict()\n        \n        def solve(s):\n            if s in cache: return cache[s]\n            if s == 0: \n                cache[s] = False\n                return False\n            \n            if pow(int(sqrt(s)), 2) == s: \n                cache[s] = True\n                return True # s is a square number and current player can take it directly, so win\n            \n            iswin = False\n            #for x in range(s-1, 0, -1): # from 1 to s-1, since s is not a square number\n            #    if pow(int(sqrt(x)), 2) == x:\n            #        if not solve(s-x):\n            #            iswin = True\n            #            break\n            for k in range(1, int(sqrt(s))+1):\n                if not solve(s - k*k):\n                    iswin = True\n                    break\n                \n            cache[s] = iswin\n            return iswin\n        return solve(n) # 方法是对的，但是超时了，n=31250的时候\n'
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

_target = None
_fn_name = 'winnerSquareGame'
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: 6124