Project Euler 45

题目

Triangular, pentagonal, and hexagonal

Triangle, pentagonal, and hexagonal numbers are generated by the following formulae:

Triangle	\(T_n=\dfrac{n(n+1)}{2}\)	\(1, 3, 6, 10, 15,\dots\)
Pentagonal	\(P_n=\dfrac{n(3n−1)}{2}\)	\(1, 5, 12, 22, 35, \dots\)
Hexagonal	\(H_n=n(2n−1)\)	\(1, 6, 15, 28, 45, \dots\)

It can be verified that \(T_{285} = P_{165} = H_{143} = 40755\).

Find the next triangle number that is also pentagonal and hexagonal.

解决方案

每次先对当前(因变量，函数)进行排序。如果排完序后，发现三个函数值相同，那么说明找到了一个同时是三角形数、五边形数和六角形数的数，如果需要找下一个，就把所有自变量加一。

否则，就将函数值最小的那个函数自变量加一，以此保持三个函数值之间的大小平衡。

代码

N = 3


def T(n):
    return (n * (n + 1)) >> 1


def P(n):
    return (n * (3 * n - 1)) >> 1


def H(n):
    return n * (2 * n - 1)


ls = [[1, T], [1, P], [1, H]]

while True:
    ls.sort(key=lambda x: x[1](x[0]))
    if ls[0][1](ls[0][0]) == ls[-1][1](ls[-1][0]):
        N -= 1
        if N == 0:
            ans = ls[0][1](ls[0][0])
            break
        ls[0][0] += 1
        ls[1][0] += 1
        ls[2][0] += 1
    else:
        ls[0][0] += 1
print(ans)