Project Euler 55

题目

Lychrel numbers

If we take $47$, reverse and add, $47+74=121$, which is palindromic.

Not all numbers produce palindromes so quickly. For example,

$\begin{array}{rl}& 349+943=1292\\ & 1292+2921=4213\\ & 4213+3124=7337\end{array}$

That is, $349$ took three iterations to arrive at a palindrome.

Although no one has proved it yet, it is thought that some numbers, like $196$, never produce a palindrome. A number that never forms a palindrome through the reverse and add process is called a Lychrel number. Due to the theoretical nature of these numbers, and for the purpose of this problem, we shall assume that a number is Lychrel until proven otherwise. In addition you are given that for every number below ten-thousand, it will either (i) become a palindrome in less than fifty iterations, or, (ii) no one, with all the computing power that exists, has managed so far to map it to a palindrome. In fact, 10677 is the first number to be shown to require over fifty iterations before producing a palindrome: $4668731596684224866951378664$ ($53$ iterations, $28$-digits).

Surprisingly, there are palindromic numbers that are themselves Lychrel numbers; the first example is $4994$.

How many Lychrel numbers are there below ten-thousand?

NOTE: Wording was modified slightly on $24$ April $2007$ to emphasise the theoretical nature of Lychrel numbers.

解决方案

本题将迭代上限定为 $50$，因此如果 $50$ 次迭代完成后仍不是回文数，那么这个数是所要求的。

因此通过 Python 直接进行暴力枚举迭代。

代码

N = 10000
M = 50
ans = 0
for i in range(1, N):
    w = i
    for i in range(M):
        s = str(w)
        w = int(s) + int(s[::-1])
        s = str(w)
        if s == s[::-1]:
            break
    s = str(w)
    if s != s[::-1]:
        ans += 1
print(ans)