✨ 新增 0039、0040 题解及配套文档

solutions/0040.ChampernownesConstant/euler_40.py

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+"""
+An irrational decimal fraction is created by concatenating the positive integers:
+    0.123456789101112131415161718192021...
+
+It can be seen that the 12th digit of the fractional part is 1.
+
+If d_n represents the nth digit of the fractional part, find the value of the following expression.
+
+    d_1 × d_10 × d_100 × d_1000 × d_10000 × d_100000 × d_1000000
+"""
+
+import time
+
+def timer(func):
+    def wrapper(*args, **kwargs):
+        start_time = time.time()
+        result = func(*args, **kwargs)
+        end_time = time.time()
+        print(f"{func.__name__} time: {end_time - start_time:.6f} seconds")
+        return result
+    return wrapper
+
+class ChampernownesConstant:
+    def __init__(self):
+        self._cache = "123456789"
+        self._level = 1
+        self._nth = 1
+
+    def __iter__(self):
+        if self._nth > len(self._cache):
+            self._level += 1
+            self._cache += "".join(str(i) for i in range(10**(self._level-1), 10**self._level))
+        yield self._cache[self._nth-1]
+        self._nth += 1
+
+    def __next__(self):
+        return next(self.__iter__())
+
+    def get_nth_digit(self, nth:int) -> int :
+        while True:
+            if nth > len(self._cache):
+                self._level += 1
+                self._cache += "".join(str(i) for i in range(10**(self._level-1), 10**self._level))
+            else:
+                return int(self._cache[nth-1])
+
+
+@timer
+def main_longstring() -> None:
+    gen = ChampernownesConstant()
+    ziel = [1,10,100,1000,10000,100000,1000000]
+    res = 1
+    for i in ziel:
+        res *= gen.get_nth_digit(i)
+    print(res)
+
+
+def length_numbers(k:int) -> int :
+    """计算前k位数的总位数"""
+    if k == 0:
+        return 0
+    return (10**k * (9*k - 1) + 1) // 9
+
+
+def get_nth_digit(nth:int) -> int :
+    if nth < 0 :
+        raise ValueError("nth must be non-negative")
+    level = 1
+    while length_numbers(level) < nth :
+        level += 1
+    # 计算在k位数中的偏移量
+    offset = nth - length_numbers(level-1) # 在k位数中的第几个数字
+    num_index = (offset - 1) // level      # 第几个k位数（从0开始）
+    num = 10**(level-1) + num_index        # 具体的数字
+    digit_index = (offset - 1) % level     # 在数字中的位置
+
+    return int(str(num)[digit_index])
+
+
+@timer
+def main_quick() -> None :
+    ziel = [1,10,100,1000,10000,100000,1000000]
+    res = 1
+    for i in ziel:
+        res *= get_nth_digit(i)
+    print(res)
+
+
+if __name__ == "__main__":
+    main_longstring()
+    main_quick()