Welcome! Are you ready to embark on a captivating journey into the world of array manipulations? Today, we're going to explore a fascinating scenario involving a wonderful small town, its houses, and a fun balloon game. Without further ado, let's dive right in!

Picture a quaint, small town where every house is numbered sequentially from 1 to n. One day, a festive town event is held, and balloons are tied to each house. The festivities do not end there. At the conclusion of the event, a fun game is played: at each step of the game, each house sends half of its balloons to the neighboring house simultaneously (the neighbor to the right side, and for the last house, the neighbor is the first house). When counting half of the balloons, we consider the existing balloons before receiving any balloons from a neighbor. The game goes on until, at some step, there are no changes in the number of balloons compared to the previous step.

The task is to create a PHP function, function solution($balloons), where $balloons is an array representing the number of balloons at each house. The function should simulate this game and return the number of steps in the game.

For example, if $balloons = array(4, 1, 2), the output should be solution($balloons) = 3. After the first step, the list becomes array(3, 3, 1). This is because the first house sends 2 balloons and gets 1, the second house sends nothing but gets 2, and the third house sends 1 but receives nothing. Note that when the number of balloons $x is odd, the house sends ($x - 1) / 2 balloons. After the second step, the list becomes array(2, 3, 2) and never changes after that. So after the third step, the process finishes.

Firstly, it's essential to note that we're dealing with a cyclical event. In other words, when iterating over our $balloons array, we need to perceive the array as circular, meaning the last element should be considered the neighbor of the first element. This concept of cyclicity is crucial when we consider the last house passing balloons to the first.

Confident in our understanding of the problem, we move on to programming our solution. First, we need to set up a loop to iterate through the rounds of balloon sharing. This loop should continue as long as the array changes.

php
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1<?php
2
3function solution($balloons) {
4    $steps = 0;
5    while (true) {
6        $steps += 1;
7        $newBalloons = $balloons;  // Store updated balloon counts
8        // TODO: Share the balloons
9        if ($newBalloons === $balloons) {
10            break;
11        }
12        $balloons = $newBalloons;  // Update balloons with new counts.
13    }
14    return $steps;
15}
16
17$balloons = array(4, 1, 2);
18echo solution($balloons);  // Output should be 3
19?>

Our next step delves into the core game mechanics: sharing the balloons. Throughout each cycle, each house must share half of its balloons with the next house.

We must also ensure that the last house shares balloons with the first house at the end of each cycle — for this, we'll use the modulo operation.

Here's the updated solution, complete with the mechanics of balloon sharing:

php
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1<?php
2
3function solution($balloons) {
4    $n = count($balloons);
5    $steps = 0;
6    while (true) {
7        $steps += 1;
8        $newBalloons = $balloons;  // Store updated balloon counts
9        for ($i = 0; $i < $n; $i++) {
10            $share = intdiv($balloons[$i], 2);  // Balloons to share
11            $newBalloons[$i] -= $share;  // Decrease balloons of current house
12            $newBalloons[($i + 1) % $n] += $share;  // Increase balloons of next house
13        }
14        if ($newBalloons === $balloons) {
15            break;
16        }
17        $balloons = $newBalloons;  // Update balloons with new counts.
18    }
19    return $steps;
20}
21
22$balloons = array(4, 1, 2);
23echo solution($balloons);  // Output should be 3
24?>

Congratulations on mastering this crucial programming scenario! You've successfully navigated a task involving the simulation of real-world events using array manipulation.

What's next? Now is the time to put into practice everything we've learned today. Try designing different versions of this balloon-sharing game. As always, happy coding!