Explaining How To Find PHP Equilibrium Indices - php

Question

So, this question is more about needing an explanation of why certain code works than it is about needing the code problem answered. I recently did a Codility sample test where you had to write a function in any desired language that could accept an array of varying integers and return each index in the array that was an Equilibrium Index. I wrote my answer in PHP and got a pretty horrible score so I went looking at other people's solutions and found some that worked great and got perfect scores, but they did something very specific that I couldn't figure out the mathematical logic for making that leap.
First, here is my function I wrote to answer the question:
function solution($A) {
for($i=0; $i<=count($A); $i++){
$leftTotal = 0;
$rightTotal = 0;
for($j=0; $j<count($A); $j++){
if($j < $i && isset($A[$j])){
$leftTotal += $A[$j];
} else if($j > $i && isset($A[$j])) {
$rightTotal += $A[$j];
}
}
if($leftTotal === $rightTotal){
return $i;
}
}
return -1;
}
This answer received a 95% Correctness score but only an 18% Performance score. I knew nested for loops was going to kill me on Performance, but I couldn't think of any way to get rid of it until I went and looked at some other solutions.
This is an example that I found on this Github page:
function equi($A) {
$lower_elements_sum = 0;
$higher_elements_sum = array_sum($A);
for ($i = 0, $cnt = count($A); $i < $cnt; $i++) {
if (isset($A[$i - 1])) {
$lower_elements_sum += $A[$i - 1];
}
$higher_elements_sum -= $A[$i];
if ($lower_elements_sum == $higher_elements_sum) {
return $i;
}
}
return -1;
}
So, I understand everything that this code is doing, and structurally, it's not really THAT different from what I did. What is driving me crazy is how they knew that in order to get an Equilibrium Index they could do this whole function working from the outside in and starting their $higher_elements_sum and the array_sum value of the array passed into the function.
How did they know that would work that way? I literally had to write out their function's results on pen and paper to double check because I just couldn't make the leap from having to find the left and right values each time vs what they did and just kept building off of the same value for left and right totals.
Can anyone help me understand and fill in the gaps between my answer and theirs because I'm at a complete loss for how I could have gotten from A to B?

Answer 1

similar but different, the solution I found scored 100 and 100
function solution($A) {
// write your code in PHP7.0
$left_sum = 0;
$right_sum = array_sum($A);
foreach($A as $k => $v) {
if($k > 0) {
$left_sum += $A[($k -1)];
}
$right_sum -= $v;
if($left_sum === $right_sum) {
return $k;
}
}
return -1;
}