Problem Explanation:

Given an array of integers, you can perform the following operations any number of times: - Subtract 1 from any element. - Subtract 2 from any element. - Subtract 5 from any element.

The goal is to determine the minimum number of operations required to make all the elements in the array equal.

Solution Approach:

1. Find the Minimum Element: Identify the smallest element in the array since this will guide us on how much we need to reduce the other elements.
2. Consider Different Targets: To minimize operations, consider making all elements equal to min_element, min_element-1, min_element-2, min_element-3, and min_element-4. This is because these slight variations might help avoid some larger reductions.
3. Calculate Operations: For each of these target values, calculate the total number of operations required to reduce all elements in the array to that target.
4. Choose the Best Option: The answer will be the minimum number of operations required among all these target values.

PHP Implementation:

Here is the implementation in PHP:

<?php

function minOperations($arr, $target) {
    $operations = 0;
    foreach ($arr as $value) {
        $diff = $value - $target;
        $operations += intdiv($diff, 5);
        $diff %= 5;
        $operations += intdiv($diff, 2);
        $diff %= 2;
        $operations += $diff;
    }
    return $operations;
}

function equal($arr) {
    $min_value = min($arr);
    $min_operations = PHP_INT_MAX;

    // Check reducing to min_value, min_value-1, ..., min_value-4
    for ($i = 0; $i < 5; $i++) {
        $target = $min_value - $i;
        $operations = minOperations($arr, $target);
        if ($operations < $min_operations) {
            $min_operations = $operations;
        }
    }

    return $min_operations;
}

// Test cases
$arr1 = [2, 2, 3, 7];
$arr2 = [10, 7, 12];
$arr3 = [1, 1, 1, 1];

echo equal($arr1) . "\n"; // Output: 2
echo equal($arr2) . "\n"; // Output: 3
echo equal($arr3) . "\n"; // Output: 0
?>

Explanation of the Code:

1. minOperations Function: This function calculates the number of operations needed to reduce all elements in the array to a specified target.
   • It uses integer division and modulo operations to determine how many steps of 5, 2, and 1 are needed.
2. equal Function: This function iterates over possible target values (from min_value to min_value-4) and uses the minOperations function to determine the minimum number of operations required.
3. Test Cases: The test cases check for various scenarios to ensure the solution works correctly.

This implementation ensures that all possible target reductions are considered, and the minimum number of operations required to make all elements in the array equal is found.