Project Euler Solution 55: Lychrel numbers
In Project Euler Problem 55: Lychrel numbers we find another curious number property builds on palindromes.
If we take 47, reverse and add, 47 + 74 = 121, which is palindromic.
Not all numbers produce palindromes so quickly. For example,
349 + 943 = 1292,
1292 + 2921 = 4213
4213 + 3124 = 7337That 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.
We can write a test from the examples:
def test_is_lychrel() -> None: assert is_lychrel(47) assert is_lychrel(349) assert not is_lychrel(196)
In order to implement this, we need a function which does this reversing and adding. We can have another test:
def test_reverse_and_add() -> None: assert reverse_and_add(47) == 121
Implementing this function is easy with the string manipulations of Python:
def reverse_and_add(number: int) -> int: return number + int(str(number)[::-1])
Then we can write a predicate:
def is_lychrel(number: int) -> int: for iteration in range(50): number = reverse_and_add(number) if is_palindrome(number): return False else: return True
I have used function is_palindrome
from Solution 36: Double-base palindromes.
With that predicate it is easy to just go through all the numbers and see how many we find.
def solution() -> None: lychrel_numbers = list(filter(is_lychrel, range(1, 10_000))) return len(lychrel_numbers)
That runs in 31 ms, so it seems fast enough. No deep insight needed in order to solve this one.