探寻平方幻方：一种高效的基于约束的构建策略

import math
from collections import Counter

def find_solutions_initial(N):
    solutions = set()
    sqrt_N = int(math.isqrt(N))
    for a in range(sqrt_N + 1):
        a_squared = a**2
        for b in range(sqrt_N + 1):
            b_squared = b**2
            for c in range(sqrt_N + 1):
                c_squared = c**2
                remaining = N - a_squared - b_squared - c_squared

                if remaining < 0:
                    break

                d = int(math.isqrt(remaining))
                d_squared = d**2

                # 检查和是否等于N，且四个数字互不相同
                if a_squared + b_squared + c_squared + d_squared == N and len({a, b, c, d}) == 4:
                    solutions.add(tuple(sorted([a, b, c, d])))
    return solutions

# 示例：N = 8515
# solutions_initial = find_solutions_initial(8515)
# print(f"初始方法找到的解决方案数量: {len(solutions_initial)}")

import math

def find_solutions_optimized(N):
    solutions = [] # 使用列表而非集合，追加操作更快
    sqrt_N = int(math.isqrt(N))
    # 循环范围调整为 a < b < c < d
    for a in range(0, sqrt_N):
        a_squared = a**2
        # b 从 a+1 开始，确保 b > a
        for b in range(a + 1, sqrt_N):
            b_squared = b