Palindrome Number

Problem

Given an integer x, return True if x reads the same forwards and backwards, and False otherwise. A negative number is never a palindrome because the leading minus sign has no trailing counterpart. Aim to solve it without converting the number to a string.

Examples

• x = 121 → True — reversing the digits gives 121.
• x = -121 → False — reversed it reads 121-, which differs from -121.
• x = 10 → False — reversed it reads 01, i.e. 1, which differs from 10.

Constraints

• -2^31 <= x <= 2^31 - 1

Think about it first

TL;DR

Reverse only half the digits arithmetically and compare to the other half — O(log₁₀ x) time, O(1) space.

Approach 1 — Brute force (string conversion)

The direct reading: turn the number into a string and check it against its reverse.

class Solution:
    def isPalindrome(self, x: int) -> bool:
        s = str(x)
        return s == s[::-1]

Complexity: O(d) time where d is the number of digits, O(d) space for the string. Correct and simple — but it uses extra space, and the problem specifically invites an integer-only method.

Approach 2 — Reverse the whole number arithmetically

The insight: rebuild the number with its digits reversed using only arithmetic, then compare. Peel digits off the right with % 10 and push them onto a growing rev with rev = rev * 10 + digit.

class Solution:
    def isPalindrome(self, x: int) -> bool:
        if x < 0:
            return False
        original = x
        rev = 0
        while x > 0:
            x, digit = divmod(x, 10)
            rev = rev * 10 + digit
        return rev == original

Complexity: O(d) time, O(1) space (in languages with fixed-width ints the full reversal can overflow; Python’s big integers make this safe, but reversing only half avoids the issue entirely).

Approach 3 — Reverse only the second half (in-place, no overflow)

The insight: you never need the whole reversal. Build rev from the trailing digits while shrinking x from the front; once x <= rev, you have consumed half the digits. Then a palindrome satisfies x == rev (even length) or x == rev // 10 (odd length — the middle digit sits alone in rev and is discarded).

class Solution:
    def isPalindrome(self, x: int) -> bool:
        if x < 0 or (x % 10 == 0 and x != 0):
            return False
        rev = 0
        while x > rev:
            rev = rev * 10 + x % 10
            x //= 10
        return x == rev or x == rev // 10

Walkthrough with x = 12321:

Now x = 12 is not greater than rev = 123, so the loop stops. The middle digit 3 lives in rev; drop it with rev // 10 = 12, which equals x → True.

For x = 1221: the loop runs to x = 12, rev = 12; x == rev → True.

Complexity: O(d) time (half the digits), O(1) space, and — importantly — rev only ever holds half the digits, so it cannot overflow a fixed-width integer.

Common pitfalls

• Forgetting that negatives are never palindromes.
• Missing the trailing-zero case: numbers like 10, 100 end in 0 but can’t start with 0, so they’re not palindromes (except 0 itself).
• In the half-reversal, mishandling odd-length numbers — you must compare x against rev // 10, not just rev.

Pattern takeaway

Digit-manipulation problems rarely need string conversion: % 10 peels the last digit and // 10 drops it, while rev * 10 + digit grows a reversed value. When you only need to compare halves, reverse just half the digits — it halves the work and sidesteps integer overflow.