788. Rotated Digits

788. Rotated Digits

Difficulty: Medium

Topics: Senior, Math, Dynamic Programming, Weekly Contest 73

An integer x is a good if after rotating each digit individually by 180 degrees, we get a valid number that is different from x. Each digit must be rotated - we cannot choose to leave it alone.

A number is valid if each digit remains a digit after rotation. For example:

• 0, 1, and 8 rotate to themselves,
• 2 and 5 rotate to each other (in this case they are rotated in a different direction, in other words, 2 or 5 gets mirrored),
• 6 and 9 rotate to each other, and
• the rest of the numbers do not rotate to any other number and become invalid.

Given an integer n, return the number of good integers in the range [1, n].

Example 1:

• Input: n = 10
• Output: 4
• Explanation:
  • There are four good numbers in the range [1, 10] : 2, 5, 6, 9.
  • Note that 1 and 10 are not good numbers, since they remain unchanged after rotating.

Example 2:

• Input: n = 1
• Output: 0

Example 3:

• Input: n = 2
• Output: 1

Example 4:

• Input: n = 100
• Output: 40

Example 5:

• Input: n = 30
• Output: 13

Example 5:

• Input: n = 10000
• Output: 2320

Constraints:

• 1 <= n <= 10⁴

Solution:

The solution counts integers between 1 and n that are good — meaning when each digit is rotated 180°, the resulting number is different from the original and contains no invalid digits (3, 4, 7).

It iterates over each number, checks digit-by-digit using the rotation rules, and increments a counter for good numbers.

Approach:

• Loop through all integers from 1 to n.
• For each number, check if it is “good” using digit-by-digit validation.
• A number is not good if:
  • Any digit is 3, 4, or 7 (invalid).
  • After rotation, the number remains unchanged (i.e., contains only 0, 1, or 8).
• A number is good if:
  • No invalid digits.
  • At least one digit from {2,5,6,9} exists (since those change on rotation).
• Return the total count.

Let's implement this solution in PHP: 788. Rotated Digits

<?php
/**
 * @param Integer $n
 * @return Integer
 */
function rotatedDigits(int $n): int
{
    ...
    ...
    ...
    /**
     * go to ./solution.php
     */
}

/**
 * Check if a number is "good"
 *
 * @param Integer $num
 * @return bool
 */
function isGood(int $num): bool
{
    ...
    ...
    ...
    /**
     * go to ./solution.php
     */
}

// Test cases
echo rotatedDigits(10) . "\n";          // Output: 4
echo rotatedDigits(1) . "\n";           // Output: 0
echo rotatedDigits(2) . "\n";           // Output: 1
echo rotatedDigits(100) . "\n";         // Output: 40
echo rotatedDigits(30) . "\n";          // Output: 13
echo rotatedDigits(10000) . "\n";       // Output: 2320
?>

Explanation:

• Digit rotation mapping:
  • 0,1,8 → rotate to themselves.
  • 2,5 → rotate to each other.
  • 6,9 → rotate to each other.
  • 3,4,7 → become invalid.
• Why hasRotating flag?
  • If all digits are in {0,1,8}, the rotated number equals the original → not good.
  • If any digit is in {2,5,6,9}, the rotated number is different → good.
• Early exit: If an invalid digit 3,4,7 is found, the number is immediately rejected.

Complexity

• Time Complexity: O(n * d), where d is the number of digits in n (at most 5 for n ≤ 10⁴). So roughly O(n log₁₀ n).
• Space Complexity: O(1) — only a few variables used per iteration.
• Trade-off: Simple implementation but linear in n, which is fine given the constraint n ≤ 10⁴.

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